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DR. INGLEBY is curious to know what Prof. Helmholtz would say on this vexed question. If Dr. Ingleby will turn to page 630 of the “Physiologische Optik,” he will find that Prof. Helmholtz has anticipated his wishes. As others of your readers may be interested in seeing how the matter is treated by one who is facile princeps in this department, I subjoin a translation of the passage. If the curious experiment mentioned by Dr. Ingleby had referred only to the vertical diameter of the disc, it would have seemed to be another illustration of our inveterate tendency to ascribe an exaggerated value to vertical lines or angles, at or near the horizon. It is said that if ten men be required to fix off-hand on a star half way between the zenith and horizon, nine, at least, will choose one very much too low. If an exact square be cut out in paper and pinned against the wall opposite to the eye, the sides will appear longer than the top or bottom. If an equilateral triangle be placed in the same position, the angles at the base will appear larger than the angle at the vertex. If a line be drawn parallel to the bottom of a sheet of paper, and a second line, making with it an angle of 20° or 30°, any one attempting, without moving the paper, to draw a third line through the point of intersection, so as to make an angle with the second line equal to that which the second makes with the first, will make the second angle too large. (This experiment is guaranteed by Helmholtz.) After reading Helmholtz's theory, metaphysicians may be willing to allow that all these illusions are to be derived, after his example, from the clouds. As metaphysicians have, before now, contributed a good deal to the clouds, it is perhaps only fair that the clouds should contribute something to the metaphysicians.

IN his second letter (NATURE, vol. x. p. 263) Governor Rawson makes the following remarks:—“Mr. Meldrum, in his letter (vol. viii. p. 547), writes, that I have ‘taken 1846 and 1871 as middle maxima years [in my first paper I also took 1848], whereas 1849 and 1872 are probably more correct.’ Mr. Meldrum is in error as to my having taken 1846 as a middle maximum, as a reference to my former letter will show.... I demur to the changes to 1849 and 1872: to the first because, without any sufficient reason, a dry year (48.10 in.) is discarded, and a wet year (67.88 in.) is added; and to the second, not because it affects my calculations, but because no reason is given.”

THE laws of metachromism, enunciated by Mr. W. Ackroyd (in his recent paper read before the Chemical Society, NATURE, vol. xiii. p. 298), have an apparent parallel in the order of colours shown by various series of combinations; there being but few exceptions to the following rule, in its application to binary compounds. Increase of the electro-negative element produces a colour change towards the red end of the spectrum, and vice versâ. Thus the sub-oxides are generally blue, and the per-oxides yellow; the sab-sulphides white or yellow; and the per-sulphides red.

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IN reference to Mr. Backhouse's letter last week (p. 145) we have nowhere stated that “similarity in colour in electric discharges is sufficient to indicate similarity of constitution, eyen when their spectra are quite unlike” If Mr. Backhouse will again read our paper on the Aurorae, he will see that we say that in the same gas the colour of the discharge varies with the degree of exhaustion. In the Phil. Trans., 1878, part 1, we have shown also that in the same gas at the same pressure the colour of the discharge varies with the amount of current; at pp. 180–181, for example, the strata in hydrogen at a pressure of 1.2 mm., 1579 M, were of a cobalt blue with a current of 0.019940 W, and pink with a current of 0.008504. W. In the index at p. 239 are given several references to the same phenomena; and at p. 240 references are given to a number of observations on the “spectra of strata and of glow on terminals.”

DR. DUDGEON'S first letter under this heading contained the suggestion of a friend that his enigmatical thermometric readings were to be accounted for by the high temperature “caused by the condensation of the moisture of the breath by the silk handkerchief.” The discussion that followed has not only brought as back to this solution, but has also furnished us with an authoritative expression of opinion that the clinical thermometer is not sensitive to pressure. F. J. M. P. first hinted the contrary proposition only to have it thrust aside by Dr. Dudgeon with blunt denial, neglected by Dr. Roberts, and finally discarded by himself for no other apparent reason than that aqueous vapour in condensing liberates heat. Yet I venture to as ert that readings as high as any obtainable by Dr. Dudgeon's method, less the pressure, can be obtained by a very similar mode of experimenting, without the developed heat: 1. If the bulb of a thermometer, protected by paper or other nonconductor, be squeezed in an intermittent manner between finger and thumb, it will be found that the mercury can readily be made to dance up and down through about a degree on the scale with a celerity not attributable to changes of temperature. 2. If eighteen inches of cotton thread be tightly wound about the bulb, on immersing the thermometer in water it will exaggerate the temperature sometimes by as many as 12° F. 3. If a tube filled with cacao butter be substituted for the thermometer the batter beneath the thread will be longer in melting than that in other portions of the tube, a result which I think proves that the high readings of experiment No. 2 are not temperature, but (in the light of No. 1) pressure readings.

AS the chief object of Mr. Preston's paper under the above title in NATURE, vol. xxiii. p. 461, seems to be to support Le Sage's “shelter theory” for gravity, you will perhaps let me point out one objection to that theory in any form which has hitherto been deemed conclusive, and with which Mr. Preston does not deal. It is that under it gravity would not vary, as it is known to do, equally with mass, but would vary not equally.

IN my paper on “Chemical Affinity and Solution,” published in NATURE, vol. xxxiii. p. 615, I gave some general proofs (taken from Thomsen's researches on thermo-chemistry) of the truth of my theory of solution. I shall now show that there are certain well-marked and definite laws of solution which are in complete accord with that theory, and seem to me to place it beyond doubt. In all chlorides, bromides, iodides, sulphates, and nitrates, for which data are available, the heats of solution in water vary directly—

THE following simple lecture-experiment may interest teachers of botany. It is described by Georg Klebs in his paper “Ueber d. Organisation d. Gallerte bei einigen Algen u. Flagellaten,” published in the most recent part of Unters. a. d. bot. Inst. z. Tubingen. A description of the experiment I give in Klebs's words, translated:—“It is easy to demonstrate, by addition of a watery solution of phenolphtalein, that Algae make the water in which they live alkaline when they are fixing carbon in light. In proportion as the fixation of carbon proceeds, the water gradually assumes a deep red tinge, which gradually disappears when light is excluded.” The explanation given is:—“The Alga takes up not only the CO2 absorbed in the water, but also in part that which is in combination in acid carbonates, in consequence of which alkaline combinations arise; in darkness, owing to respiration, the reverse process takes place.”

MR. MADAN's description of a device for showing that metals and solids expand when exposed to heat is very interesting, especially as such an arrangement, but with important modifications, is capable of giving very excellent scientific results, results which are only surpassed by M. Fizeau's method. One necessary alteration is the substitution of a spring-pressure for the weight on the strip of metal. This and other points will be made quite clear by a perusal of a short description contained in my paper, “A Strain-Indicator for Use at Sea,” read before the Institution of Naval Architects. The numerous tables and diagrams there given would, I am afraid, hardly interest your readers, but the repeated experiments in Table I. would be a subject of interest, as they show how well the experiments agree amongst themselves. The errors, though small, are due, in my opinion, not only to the difficulty of reading the dial (each unit being equal to about half an inch, and the second decimal therefore about 1/200 inch), but also to the difficulty of reading the exact position of the weight on the steel-yard of the testing-machine. Far more accurate results are obtained when, instead of a jockey weight being run out, small weights are added one by one.

IN NATURE of March 10 (p. 437), Mr. Maxwell Hall gives an interesting table of the vertical distribution of temperature and pressure in Jamaica, and, apparently in happy ignorance of the dangers of the process known as extrapolation, goes on to apply the results of observations extending to a maximum height of only 7400 feet to the determination of the probable temperature of meteorites in extra-terrestrial space. As he expresses a desire to know whether any similar results have been found in India, and as I have on several occasions during the past ten years discussed the vertical distribution of temperature and pressure in this country, I gladly take this opportunity of referring him to my papers on the meteorology of the North-West Himalaya, and on the temperature of North-Western India, published in the “Indian Meteorological Memoirs,” vols. i. and ii. From the latter I extract the following table on the mean decrement of temperature up to a height of 12,000 feet, computed from the observations of twenty-five stations combined in various ways. For each month an interpolation formula of the form T = T0 + ah + bh2 + ch3, was computed, and by its means the decrements from sea-level to 1000 feet, 1000 to 2000 feet, &c., were calculated. Finally, the average decrement for the twelve months was computed, and is here given in an abridged form. The curves for the several months differ very widely from one another, those for the summer giving the most rapid decrement at sea-level, and the decrement increasing with altitude in winter: —

EVERYONE interested in the theory of statistics is aware how strongly Quetelet was under the conviction that there is only one law of error (or curve of facility, to use the corresponding expression for the graphical representation of the law) prevalent for the departure from the mean of a number of magnitudes or measurements of any natural phenomenon. I have done what I can to protest against this doctrine as a theoretic assumption; and recently Mr. F. Galton and Mr. F. Y. Edgeworth have shown in some very interesting and valuable papers in the Philosophical Magazine and elsewhere how untenable it is, and how great is the importance of studying the properties of other laws of error than the symmetrical binomial, and its limiting form the exponential.

BY reference to the last number of the Electrical Review (vol. xxi. p. 262), wherein is printed a short abstract of our paper on “A Null Method of Electro-calorimetry” read before the British Association on September 1, Mr. Huntly will find that the method of measuring specific heats suggested by him is in principle similar to that described by Mr. Gee and myself. The method has been employed for determining specific heats during the last two years, but we have delayed publication till the best working details of the method have been elaborated.

WITH reference to the remarks of “R. B. H.” (NATURE, June 28, p. 197) on my interpretation of the differential equation to all conics, I wish to point out that the objections he seems to take do not appear to be well founded. The difficulty he finds is that the geometrical interpretation given amounts to the fact that “a conic is a conic.” But it is easy to see that there is no peculiarity in this; it arises simply from the well-known fact that all the geometrical properties of any given figure are inter-dependent: one of them being given, the others may be deduced as legitimate consequences from it. “R. B. H.” takes the proposition which constitutes my interpretation, and then, coupling it with the other theorem that the osculating conic of any conic is the given conic, comes to the conclusion that a conic is a conic, and, apparently, he takes it to be very strange; bat, as a matter of fact, given any two properties of a conic (or of any other curve), we can only come to the conclusion that the conic is a conic (or that the given curve is what it professes to be). Take, for example, the geometric interpretation of the differential equation of all right lines, which is q = o; it simply means that the curvature vanishes at every point of every right line, which is equivalent to the fact that a straight line is not curved, or that a straight line is a straight line. There is certainly nothing strange in this: it is the legitimate effect of the process employed. Would “R. B. H.,” on this ground, reject the geometrical interpretation of the differential equation of all straight lines? Surely the process is nothing but a piece of quite unobjectionable verification. Similarly, the differential equation of all circles, (1 + p2)r 3pq2 = o, means that the angle of aberrancy vanishes at every point of every circle. Combining this with the self-evident proposition that the normal and the axis of aberrancy coincide in the case of a circle, we may come to the conclusion that a circle is a circle; but I submit that this is really a verification, and surely no ground for rejecting the interpretation. Indeed, the question whether such processes are to be regarded as verifications or not seems to me to be much the same question whether every syllogism is a petitio principii or not. But as I have elsewhere, in the papers referred to in my last letter (p. 173, ante), fully discussed what a geometrical interpretation properly ought to be, I need not enlarge further on this point.

THE errors made in measuring the height of the head are certainly much larger in the instance given by “F. M. T.” than they should be; still they do not seem to me large enough to throw doubt on the truth of the general conclusions to which he refers. (1) As regards the difference between the means of the “products” in the high honour and the poll men. Those means are 224 and 237, and they depend on 258 and 361 observations respectively, which numbers are much the same as 162 and 192. Therefore the probable error in the determination of each of these means will be equal to the probable error of a single “product” divided by 16 in the one case and by 19 in the other. I have as yet no data to determine the probable error of a single “product,” due to faults of measurement alone, other than those given by “F. M. T.,” which suggest, though in the rudest way, that it is about 4 units. Accepting this for the moment as a basis, the probable error of the means of the two sets of “products” would be 4 divided by 16 and by 19, equal, say, to ¼ and 1/5 respectively. Now, the chance of an error exceeding 4 or 5 times the probable error is not worth regarding; therefore safety, so far as regards the effects of inaccuracy of measurement, is practically to be found in each mean value beyond a range of about 1 unit. In the differences between the mean measures, safety will be found beyond the range √2, say 1½ units. But the two means 224 and 237 differ by as much as 7 units. It should, however, be remarked that the seven observations fall into two well-marked groups, each of which is very consistent within itself, but which differ from one another by 10 units. This raises strong suspicion of some peculiarity in the shape of the head, which caused doubt as to the exact line of maximum height, and that one line was followed in three of the measurements and another line in the remaining four. (2) As regards the differences between the high honour and the poll men at different ages, the observations at each stage are, of course, much less numerous than in the sum of all of them, still they range in all cases but two between 25 (or 52) and 102 (or say 102). Each person must judge for himself, from the diagram that accompanied my little paper, how far the run of those differences confirms my conclusions. I think they do well enough to give “an approxirnately true” idea of what we should find if we had the opportunity of discussing a much larger number of observations, and this was all that I claimed.

IN a recent paper (Proc. Lond. Math. Soc., April 11, 1889), Prof. Karl Pearson has discussed at some length the possible forms of the additional terms which may be introduced into the general equations of elasticity by a consideration of the mutual action of the molecules and the ether, and has examined what physical phenomena may admit of explanation in this way. In particular, certain terms which thus appear admissible are made to yield a theory of the phenomenon known as “elastische Nachwirkung,” or “after-strain.” The attempt to explain such a comparatively slow process by the intervention of the ether certainly invites scrutiny, and in fact a very slight examination serves, I think, to show that the theory in question rests on a mistake. The author, after writing down the equations which (on his view) represent the steady application of stress to a portion of matter, proceeds to integrate them in the usual way by assuming a time-factor emt, and arrives at a quadratic in m2 whose roots are μ/μ' and (3λ + 2μ)/(3λ' + 2μ'), where λ, μ are the ordinary elastic constants of Lame, and λ', μ' are the coefficients of the additional terms referred to. He continues:—“Now m cannot be positive, so long at least as we are dealing with elastic-strain. For λ' and μ' are small as compared with λ and μ, the effects we are considering being only of the second order. Hence m2 is large, and if m were positive the strain would rapidly grow immensely large, which is contrary to experience. Thus, we must give m the negative values – √(μ/μ') and – √{(3λ + 2μ)/(3λ' + 2μ')}.” The positive values of m are certainly inconvenient, but they are on the same footing with the negative ones; all are solutions of the author's equations, and all are required for the purpose of satisfying arbitrary initial conditions. The proper inference is surely that the substance is unstable, so long as the constants μ/μ' and 3λ' + 2μ' are (as the author has tacitly assumed them to be) positive. If, to avoid this disaster, we change the signs of these constants, we get circular instead of exponential functions, and all analogy to elastic after-strain of course disappears. In its place we have vibrations (not molecular, but “molar”) whose period is intrinsic to the substance and independent of the dimensions of the portion considered. To what physical reality these may correspond I do not undertake to say.

IN connection with the interesting investigations of Dr. Chandler on the variation of latitude, permit me to suggest the desirability of calling attention to Maxwell's remarks on the same subject at the conclusion of his paper “On a Dynamical Top,” Ed. Phil. Trans., xxi., Part iv. p. 568 (read April 1857). Maxwell, taking the value of as 1.00309, as deduced from the amount of precession (Pontecoulant, ii. 268), finds the time of revolution of the pole of the earth about the true pole to be about 325.6 solar days. Dr. Chandler's period of 427 days would on the same principles correspond to the value = 1.00235.

DR. WATSON has shown in his letter to NATURE of March 31 (p. 512) how the general methods of Maxwell and Boltzmann may be applied to the particular problem discussed by Prof. Burnside. He has also pointed out an error in Burnside's reasoning—namely, the non-introduction of the factor u − U + c?, whereby Burnside's conclusions at variance with the Maxwell-Boltzmann law of partition of energy are vitiated.

IN his criticism on a paper of mine on the partition of energy in a set of non-homogeneous spheres (NATURE, March 31, p. 512), Mr. Watson says that the conclusions are vitiated owing to my having omitted to introduce the frequency factor of collisions before proceeding to take the averages. This is not exactly accurate, since a frequency factor is introduced, viz. the relative speed of the centres of inertia of the impinging spheres parallel to the line of impact.

IN last month's Philosophical Magazine there is a paper by Lord Rayleigh criticizing a demonstration by Maxwell of the equality of the products dp1…dpn, dq1…dqn, and dP1…dPn, dQ1…dQn, where the p's and P's are the momenta, and the q's and Q's the co-ordinates, of a system at the beginning and end of any interval of time.

IN a paper by Prof. C. G. Knott on “Recent Innovations in Vector Theory,” of which an abstract has been given in NATURE (vol. xlvii. pp. 590–593; see also a minor abstract on p. 287), the doctrine that the quaternion affords the only sufficient and proper basis for vector analysis is maintained by arguments based so largely on the faults and deficiencies which the author has found in my pamphlet, “Elements of Vector Analysis,” as to give to such faults an importance which they would not otherwise possess, and to make some reply from me necessary, if I would not discredit the cause of non-quaternionic vector analysis. Especially is this true in view of the warm commendation and endorsement of the paper, by Prof. Tait, which appeared in NATURE somewhat earlier (p. 225).

HAVING now returned to England, I have had an opportunity of examining my paper on wires (Proc. Lond. Math. Soc. vol. xxiii.), and I find that the discrepancy between my results and those given by Mr. Love, on p. 169 of his book, is due to a slip in my own work. On comparing my equations (11 and 15), it will be seen that in the latter equation the term – ρ(σρ – σγ cos θ)–1dw′/dθ has been omitted. The value of w′ is correctly given by equation 31, and when the omitted term is inserted in equation 32, the resulting value of g will be found to lead to values of the couples identical with those given by Mr. Love.

A SERIES of papers, by M. Mercadier, on the dimensions of physical quantities, has recently been appearing in the Comptes Rendus. They are summarised and extended in the Journal de Physique (July, 1893, p. 289).

DR. DOBERCK, of whose paper an interesting abstract was given on June 25, informs us that shooting-stars decrease in average magnitude from evening to morning, their duration and length of path decreasing with the magnitude, while the velocity increases as the magnitude diminishes.

IT seems noteworthy not only that the theoretical investigation on this subject, given in Maxwell's “Theory of Heat” and Tait's “Heat,” is fallacious, as has been pointed out by Mr. Rose-Innes in his letter which appeared in your issue of May 26, but that there is no indication in either of these text-books that the result given is not in accordance with the experimental facts. The investigation in question, which is practically the same as that first given by Lord Kelvin before the experiments were performed, leads to the result that the cooling effect should be proportional to the difference of the logarithms of the pressures before and after passing the plug, instead of proportional to the difference of the pressures themselves, as was actually found to be the case; the discrepancy was, of course, noted'by the experimenters. The methods used in all the theoretical discussions in Lord Kelvin's papers on this subject prior to the final one in Part iv. of the papers “On the Thermal Effects of Fluids in Motion,” seem equally open to objection. The numerical results derived by the-last method, which seems perfectly sound, were given apparently for the first time in the article “Heat” in the “Encyclopaedia Britannica”; the investigation and results are reproduced in Preston's “Theory of Heat,” and substantially in Parker's “Thermodynamics.” The correction obtained to the number representing the temperature of melting ice on the absolute scale by this method was + ˙70, the previous methods giving in succession + ˙83, –˙3 and + .87; the uncorrected estimate finally used was also different from that taken at first.

A SHORT account appeared in NATURE, October 12, p. 584, of a paper read by me at the British Association, entitled the “Median Estimate,” which will appear in the forthcoming Journal of the Association. Its object was to solve a problem of the following kind:—40 per cent, of the members at a meeting vote that a proposed grant should be less than 100l., 80 per cent, vote that it should exceed 500l. What is the Median Estimate, supposing the normal law of frequency to hold good? That is to say, What is the sum that one-half of the members would think too little, and the other half too much, and which therefore presents the best compromise between many discordant opinions? I showed that the calculation was exceedingly simple if certain tabular values are used that will be spoken of later. But, on after reflection, it seems to me that further simplification is both desirable and feasible. The problem is representative of a large class of much importance to anthropologists in the field, few of whom appear to be quick at arithmetic or acquainted even with the elements of algebra. They often desire to ascertain the physical characteristics of races who are too timorous or suspicious to be measured individually, but who could easily be dealt with by my method. Suppose it to be a question of strength, as measured by lifting power, and that it has been ascertained that a per cent. of them fail to lift a certain bag A of known weight, and that b per cent. of them fail to lift another heavier bag B. From these two data, the median strength can be determined by the simple method spoken of above, and not only it but also the distribution of strengths among the people. Having indicated the utility and importance of the general problem, I will proceed to work out the particular case of the voters by the now further simplified method. In Fig. 2 let the base line G represent 100l. and let each successive horizontal line above it represent an increment of 100l. A dot A is placed on G, at the division 40°, and another dot B is placed on the ordinate at the division 80° at the level of the fourth line above G. Therefore A and B are plotted at their respective places. Join the two dots with a straight line. The place where this line cuts the ordinate at 50°, shows the Median value. The principle on which this exceedingly simple process rests must be explained by beginning with Fig. 1, where an ordinary curve of distribution is drawn about the axis H, with a quartile equal to 1. The centiles from the axis to the curve are given in the following small table (see my “Natural Inheritance,” Macmillan, 1889) which is reproduced here for convenience.

THERE is much interesting theory in your paper on the bicycle fitted to the rider in crank and gear, by Mr. Crompton (p. 87). But what is the practice? I agree with Mr. Crompton's theories, if a slight modification be made. I think that the crank-length should be proportional not only to a man's thigh-length, but to the weight of a man's leg. The loss of power in a bicycle, as soon as it travels fast, arises from the loss of momentum at each up and down stroke of the leg according to the well-known equation:—

A YEAR ago Prof. Perry drew attention to a method by which a schoolboy knowing how to extract square roots could, with the help of squared paper, construct a table of logarithms (NATURE, February 23, 1899).

THE publication of Mr. Dufton's method will, I think, serve a useful purpose. It is a common exercise in schools to plot on squared paper, numbers and their logarithms to the base 2 (see Blame's “Methods of Calculating,” Spon), to give a general notion of how the logarithm varies as the number varies; but I have never known it to be made a method of calculation. Indeed, I do not think it right to give a boy the idea that he may find log. 10 by interpolation between log. 8 and log. 16. There is a specious appearance of accuracy due to the fact that log210 is so nearly 31/3; and Mr. Dufton heightens it by using squared paper on which the divisions are thirds of the unit, so that a boy will have it fixed in his mind that 31/3 is exactly log210.

A WELL-ATTENDED conference of science masters in public schools was held last month at the University of London, and among the many interesting papers read was one by Mr. W. D. Eggar, of Eton, on the coordination of mathematics and physics in our public schools.

THE historical note in your last number by Sig. Vacca regarding the graphical solution of a cubic, given by Mr. T. Hayashi, reminds me that I had intended, when Mr. G. B. Matthews published his suggestion for the graphical solution of a biquadratic by means of two parabolas (NATURE, Nov. 16, 1899), to point out that he too had been anticipated, as will be seen by referring to a paper by Mr. R. E. Allardice in the Proceedings of the Edinburgh Mathematical Society (April 7, 1890), where it is shown that, with the exception of the case where the roots of the biquadratic are equal in pairs, the real roots of the general biquadratic can be found graphically by means of two equal parabolas having their axes at right angles, the one fixed and the other movable; and also that every cubic can be reduced to the form y3 ± y + r = 0; and then solved graphically by means of the fixed curve y = x3 and the movable straight line x±y = r.

IN his first letter (p. 53), Prof. Perry very properly drew attention to the desirability of greater definiteness in the notation for partial differential coefficients in the case of functions of two variables, the essence of his remarks being that it is not enough merely to change d into ∂, but that the variable which is for the nonce held to be constant should also be indicated. Apropos of this, and for the sake of historical interest, I quoted from a paper since published (Proc. R.S. Edinb xxiv. pp. 151–194) a short paragraph regarding a passage1 in Jacobi's writings of the year 1841, and containing a footnote with an old suggestion on the matter of notation. In his second letter (p. 271), Prof. Perry undertakes to show that this latter notation is objectionable so far as thermodynamics is concerned, and not to be compared with that which he himself uses. I regret to have to say that I was quite satisfied with his notation, and had no intention whatever of bringing the two into comparison, mine having been designed for much more complicated cases than those which occur in ordinary text-books on thermodynamics. His words are:—

MY note in NATURE of October 8 may be extended. I found that one of my pupils, Mr. Glasgow, was assuming that the expansion and compression parts of a gas engine diagram followed laws of the type pvn constant; in cases where there was a probability that the clearance had not been measured accurately, so that v being the measured volume and c the constant error, he assumed p(v+e)n to be constaznt, and he was enabled to find c from the curve. There is no reason to believe that these curves ought to have such a law, although, curiously enough, following this assumption, the clearance obtained from the compression curve is usually not very different from that obtained from the expansion curve. Mr. Glasgow's method of finding c is much the same as what I shall now describe. An empirical formula of the type y=a+bxn would be exceedingly useful in many parts of pure and applied science if, when given a table of values of y and x, we could readily find a, b and n. I have often sought for a method of working, but without success. If a is zero, we have only to plot log y and log x as the coordinates of points on squared paper. If a is not zero, there is a clumsy method of using logarithmic paper which may be adopted, but it is not satisfactory. We now have a method easy of application. Thus values of x and y being given, draw the curve AB shown in the figure. Set off any convenient angle DOX. Select the point P. Draw PD, XF, FEQ, EH, &c, the lines XF, EH, &c, being at 45°. Project horizontally from the points PQR, &c, to M, U or N, W or V, &c, letting lines at 45° from M, U, &c, meet the horizontals at N, V, X, &c. If the above law holds, N, V, X, Z lie in a straight line. If they lie only approximately in a straight line, draw the line N′O′ lying most evenly among them. Then OO′ is the value of a, and n is log (1+tan N′O′Y)/log (1+tan DOX), and b is readily found.

THERE is, I fear, a difficulty in drafting Prof. Bryan's definition so as to be clear as well as accurate. This arises when the definition is first given with reference to the entropy of the working substance, because the non-available energy is not necessarily a portion of the energy of the substance. The terms available energy, free energy, bound energy, and non-available energy are continually used loosely in thermodynamics as if they referred to portions of the energy of the working substance. I know from experience the difficulty of defining the entropy of the working substance in terms of dissipation or degradation, without reference to the state of things outside the substance, and in a paper on the factors of heat I adopted the notion of reduction of “transfer credit,” so that increase of entropy went with lessening of capacity for transforming heat into work with change of volume. In my book on “Entropy” the whole treatment is essentially from the dissipation or degradation point of view, but entropy is first defined in connection with the irreversible increase of entropy in an isolated system. It is thus defined: “Increase of entropy is a quantity which, when multiplied by the lowest available temperature, gives the incurred waste.”

A complete and consistent summary of the paper is scarcely possible since the paper is in itself in a sense a summary. The following are, however, the most important points. 1) Regulation may be defined as consisting in the return or approach to physiological or functional equilibrium after such equilibrium has been disturbed. 2) The planarians serve as an excellent basis for the consideration of the problem of form-regulation since they exhibit various processes and degrees of form-regulation combined in various ways. Moreover, in these forms we have in many cases in the motor reactions of pieces and the relation between form-regulation and the nervous system, a clue to the character of the changes in functional conditions resulting from section and their localization. 3) The processes distinguished more or less artificially as redifferentiation and regeneration can be interpreted on a functional basis and it is possible to distinguish more or less exactly the conditions which determine the occurrence of one or the other. 4) Other processes of form-regulation are like these merely special cases and must be investigated as such. 5) Replacement of a lost part can occur only when the functional complex remaining retains the essential conditions of the whole so far as this part is concerned. Alteration of the functional complex beyond a certain limit leads to the establishment of a new equilibrium which may be accompanied by complete reorganization, heteromorphosis, or partial or total failure to replace the part removed.

THAT the formula of Gauss for finding the date of Easter fails in certain cases, of which the year 1954 is one, was pointed out by Gauss in his original paper in the Monatliche Correspondenz (vol. ii., p. 229), where he shows that there are the following two exceptions to the formula in the Gregorian calendar:—

I SHOULD fancy from the review by “G. H. B.” in NATURE of July 19 (p. 265) that the papers of Prof. Levi Civita relate largely to the mixed transformation of Lagrange's equations, the complete theory (Proc. Camb. Phil. Soc., vol. vi., p. 117; “Hydrodynamics,” vol. i., p. 171) of which was first given by myself so far back as 1887. But what I wish to point out is this, that this theory depends no more on any so-called theory of “ignored” coordinates (or kinosthenic coordinates as Prof. J. J. Thomson [Phil. Trans., 1885, part ii.] calls them) than it does on the existence of the hypothetical personage known as the Man in the Moon.

THE test described as new by Mr. Wood in NATURE of February 21 has been in use in my laboratory during the past year, where it forms part of the regular routine tests applied to flour. While I am fully in agreement with Mr. Wood's view that the volume of carbon dioxide evolved by a mixture of yeast and flour under standard conditions is a measure of the sugar content of the flour together with other fermentable matter produced during the fermentative change, it is important not to lose sight of the influence exercised by the character of the gluten on the volume of the loaf. A rotten gluten when distended by too much gas will break, and the gas will escape from the dough. From this point of view the character of the gluten is clearly of fundamental importance, but, after all, the problem is one in which no small number of variables must be dealt with.

MR. SPENCER PICKERING'S letter on the effect of soil sterilisation on the growth of apple trees (June 6, p. 126) is of interest in connection with experiments which have been made by Dr. Francis V. Darbishire and myself, and described in a paper read last year before the chemical section of the British Association. We find that most plants grow much better in heated than in unheated soils.

In the foregoing paperBaltzer's objections toHartog's account of the play of forces in mitotically dividing cells is combated. 1) The suggestion that the forces are physically unable to induce the discession of the daughter chromosomes is shown to be incorrect in theory and inconsistent with the behaviour of magnetic models as shown byHartog in his '07 papers whichBaltzer appears not to have seen. 2) Similarly the equilateral triaster, has been explained byHartog with a figured magnetic model in his second paper ('071). 3) The tetraster with four consecutive spindles and one diagonal spindle is said byBaltzer to be inconsistent withHartog's views. Yet this very case was explained as deducible from the inequilateral spindle and illustrated by a photograph from a magnetic model inHartog's paper ('05), cited byBaltzer. 4) The tetraster with two diagonal spindles as figured byBaltzer (Fig. 34) is inconsistent with every extant explanation of karyokinesis. The geometrical difficulties in its way are so great that we are bound to admit the probability of a mistaken interpretation of the specimen figured.

So far as I can ascertain, the method referred to is not known, at least in its complete form. It is a development of a method described by me in a previous paper (“Verb Functions, with Notes on the Solution of Equations by Operative Division”, Proceedings of the Royal Irish Academy, vol. xxv., Sec. A, No. 3, April, 1905), which was reviewed in NATURE of April 25, 1905. I give it here as briefly as possible.

MR. CUNNINGHAM'S inquiry (August 5) is aimed, apparently, at obtaining a kind of index mark for each candidate in an examination containing several papers. In getting a boy's percentage mark in any one paper there is no trouble; but the question is, By what law are percentages in different papers to be combined in order to get an index mark? Percentages may be combined in an infinite number of ways; which is the way Mr. Cunningham desires?

In recent years one of the most interesting fields of" geometrical investigations has been that of transformations of geometrical entities of a particular type into entities of the same type. As examples, we mention the well-known B~icklund transformations of pseudospherical surfaces, the BIASCHI transformations B k of surfaces applicable to quadrics ~), and the transformation of the surfaces of Voss ~). In the first two cases a surface S and a transform S, are the focal surfaces of a W-congruence. BIANCm a) has shown the analytic relation between W-congruences and the transformation of MOt;TARt) 4) and consequently we have general methods of procedure in the determination of any transformation for which the two surfaces are the focal surfaces of a W-congruence. Moreover, it should be said that there are other known transformations of this type than those mentioned above. In the third example referred to a surface S and a transform S are so related that the lines joining corresponding points on their surfaces form a congruence whose developables meet S and S~ in a conjugate system with equal tangential invariants; namely the geodesic conjugate system. It is the purpose of this paper to establish for any conjugate system with equal tangential invariants a transformation possessing the property just mentioned and determine other properties of this transformation. When a surface S is referred to a conjugate system with equal tangential invariants, one can find by quadratures an associate surface x referred to its asymptotic lines. For the transformation under discussion the associate surfaces X and X of S and S

IN a recent paper, entitled “Notes on Spherical Harmonics” (Proceedings of the Edinburgh Mathematical Society, vol. xxxii., 1914), Dr. John Dougall wrongly claims as new the expansion which he has given there for a homogeneous function of the coordinates of a point on a sphere. This expansion was first given in 1900 by Dr. G. Prasad, in the Messenger of Mathematics, vol. xxx., p. 13, and again, by a different method, in 1912 in the Mathematische Annalen, vol. lxxii., p. 436. The ethod of Dr. Prasad in the second paper is the same as that of Dr. Dougall.

IN two papers published in the Proceedings of the Amsterdam Academy (vol. xvii., 445, 1914; vol. xviii., 398, 1915) an experimental investigation concerning Fresnel's convection-coefficient for light of various colours was described. The main object of my repetition of Fizeau's experiment, in the improved form introduced by Michelson, was to decide between the expressions for the convection-coefficient given by Fresnel and by Lorentz. As a review of the papers mentioned has appeared in NATURE (vol. xcvi., 430, 1915), I may be permitted to give here a short account of further progress. It may suffice to recall that my results were largely in favour of the Lorentz expression with the dispersion term. For the wavelength 4500 U the difference between the two expressions under consideration amounted for water to quite 5 per cent. The probable error of the experimental result was estimated at somewhat less than 1 per cent.

The problem of obtaining a realistic guaranteeda posteriori bound on the accumulated error in a computed solution to the initial value problem in ordinary differential equations is difficult, because of the “wrapping” effect. This difficulty can sometimes, but not always, be avoided by making use of coordinate transformations. In this paper we propose that the wrapping effect be reduced by enclosing the accumulated error in a convex polygon of a certain form, and we describe one possible way of choosing the faces of such a polygon. The method is computationally expensive, but provides, in cases where other methods are unable to do so, a bound which does not grow exponentially too fast.