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In an ecological study of the growth of two grasses, blue grama (Bouteloua gracilis) and side-oats grama (Bouteloua curtipendula), it was necessary to make a determination of the carbohydrate reserves. Both a qualitative identification and a quantitative estimation of sugars were needed to find growth and storage trends. Paper chromatography has been utilized in many instances for both qualitative and quantitative analysis of sugars (2, 9) with sufficient reliability and accuracy to be applicable to this study. Because a very large number of procedures are given in the chemical literature, with varying degrees of success reported in separation and identification, several were investigated and the methods which gave results that were reproducible and provided adequate separation were selected for the study and are described in the experimental discussion.

In their paper 'Some observations on the relations between H-ion concentration of the soil and plant successsion', Emmett & Ashby (1934) proposed a sampling method for separating the frequency distribution of a species over an area from that of soil acidity over the same area. It is unforttnate that the statistical test used in the analysis of their data was incorrect and consequently the hypothesis that the two species Pteridium aquilinum (L.) Kuhn and Vaccinium myrtillus L. occurred with the same frequency at every pH value within their respective pH ranges was accepted when in fact there is significant evidence to the contrary. We propose to discuss in detail the data which were given in Table 1 of that paper, and are reproduced in Table 1 below.

IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

By using arithmetic probability paper a normal distribution can be transformed into a straight line by plotting cumulative frequencies, expressed as percentages of the total frequency as the ordinate, against the mid-point of the class intervals as the abscissa. From such a line both the mean and standard deviation can be obtained. The mean is the point on the abscissa corresponding to the position where the line intersects the 5000 ordinate and the standard deviation is the distance along the abscissa lying between the mean and a point corresponding to either 15.870% or 84 13% on the ordinate. Conversely, if the mean and standard deviation are known, a distribution can be represented as a straight line. Recently, from an investigation into the movements of coarse fish (Stott 1961, 1967), data were obtained on the frequency with which recaptures were made and the distance they had moved from a release point which, when plotted in the above manner on arithmetic probability paper, produced sigmoid curves similar to that labelled AF in Fig. 1. The data for Fig. 1 are given in Table 1, and for the purpose of illustration they are hypothetical. Since it can be demonstrated that this type of curve results if two straight lines, such as XX' and YY' in Fig. 1, are compounded, it seemed reasonable to interpret the data from the investigation as being the resultant of two superimposed normal distributions of differing dispersions. This led to the postulate that the fish populations studied were divisible into two components on the basis of their observed mobility and indeed subsequent laboratory work confirmed this hypothesis (Stott 1967). In order to estimate the proportion and parameters of the mobile and static sub-populations observed in the field, a search was made for a technique to separate these components. Methods of separating polymodal frequency distributions have been described by Harding (1949), Cassie (1954) and Tanaka (1962), the two former using arithmetic probability paper. However, their solutions seem only to be of use if the amount of overlap between distributions is small and a similar limitation applies to the cases referred to by Cox (1966). It appeared, therefore, that with superimposed distributions the only approach possible would be one of trial and error based on the fact that the less dispersed normal distribution (XX' in Fig. 1) does not overlap the other one to a significant degree at the tails of the resultant sigmoid. By trial and error the shallower line (YY' in Fig. 1) could be obtained by progressively reducing the values of the frequencies between the points where it was obvious that the two distributions were 'superimposed (i.e. between B and E on line AF in Fig. 1) until the data from this common portion of AF plus that from the two tails, AB and EF would form a straight line when

near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.

The incorrectness of Birkhoff's result, with which the paper [2] is concerned, was noted already by F. R. Gantmacher in 1954 in his book Theory of matrices (Russian); cf. the recent English translation of the second part of this, [l, pp. 175-176]. Example C in [2] is, in fact, the analogue of Gantmacher's counter-example when z= a> is taken instead of 2 = 0 as the singular point. Gantmacher does not, however, discuss the cases in which the result is correct; cf. [2, D]. Unfortunately, the writer became aware of Gantmacher's work only when it was too late to have paper [2] withdrawn or even to have a note added to it.

In a preceding paper 'On rings of operators, III" J. von Neumann has developed a theory of normed operators for operators in an arbitrary factor 9R.2 In part the procedure is as follows: First [[A]], the norm of A, is defined for all A e tR of finite rank (Definition 1.3.1) and fundamental sequences of such operators and an equivalence relation for fundamental sequences are defined (Definitions 1.4.1, 1.4.2 and 1.4.3). Then the key theorem (Theorem III) is proved that every fundamental sequence is strongly convergent operatorially, that two fundamental sequences converge strongly to the same limit if and only if they are equivalent, and finally that the norms of a fundamental sequence converge numerically to a limit depending only on the operator limit of the sequence. The proof of Theorem III makes use of two important lemmas: LEMMA 1.4.1. If An, n = 1, 2, * * is a fundamental sequence with [[AJ]] + 0 as n -oo then An converges strongly to 0 as n -* ??. LEMMA 1.4.3. If An, n = 1, 2, * * is a fundamental sequence with An converging strongly to 0 as n -* , then [[An]] -0 as n + oo. The purpose of the present note is to point out how the proofs of these two lemmas can be simplified by using more fully the results of the paper 'On rings of operators, II' by Murray and von Neumann.3 We require the following lemma. LEMMA. If OR is a factor in a finite case and An E OR with An converging strongly to O as n -oo , then [[An]] -*0 as n + oo . PROOF. There is a finite set of elements gi, * * , Mg,, such that [[An]]2 = Tr9 (A*An) = ET-, (A*Angi, gi) = 2t7l (Angi, Angi) = o l| Ang 112 for all n.4 Since A gi -* 0 as n oo for all i, and m is finite, the lemma follows. PROOF OF LEMMA 1.4.3. Suppose the lemma false, if possible. Then for some e > 0 we could assume that [[An]] > e for all n, by choice of a suitable subsequence. For some fixed no , [[An Ar]] < e/2 for all n, r _ no. From the definition of a fundamental sequence, Ano is contained in some closed linear manifold 91 of finite dimensionality relative to 'DR. Let E be the projection on W1.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.

(1964). Remarks on a Recent Paper on Round-Off Errors in Regression Analysis. The American Statistician: Vol. 18, No. 4, pp. 26-40.

In the above-named paper (this JOURNAL, vol. 77 (1955), pp. 895-913), Theorem 6. 4 should have been stated only for rings of zero characteristic: the argument for the case of prime characteristic breaks down in the last formula line on p. 911. This involves jettisoning Theorem 6. 5 (the implied "proof " of which depended on applying Theorem 6. 4 to homomorphs which cannot be guaranteed to have zero characteristic, even when the given ring has). However, the writer has no evidence that Theorem 6. 4 as stated or Theorem 6. 5 is actually false. And in any case, since the valid part of the proof of Theorem 6. 4 establishes the weaker conclusion [Xm, ynp?] = 0 without characteristic hypothesis, the remarks following Theorem 6. 4 still hold good, provided that we modify the final parenthetical clause so as to read "which would at any rate imply that every weak K-ring R with kc, m, n satisfying (a) or (b) has R/J commutative."

In a recent paper by Goodall (1961) criticisms are made of the method of analysis of pattern proposed by Greig-Smith (1952a). This method has been described fully elsewhere (Greig-Smith 1957, 1961b; Kershaw 1957) and need not be discussed in detail here. It depends on analysis of the variance of data, taken from grids or transects, into portions appropriate to blocks of increasing size. Peaks in the graph of mean square against block size are interpreted as corresponding to the scales of pattern present. Kershaw (1957) has used artificial 'communities' of known pattern to demonstrate the validity of the technique and Thompson (1955, 1958) has discussed the statistical aspects. Both Kershaw and Thompson have emphasized the importance of comparisons between analyses in assessing the significance of peaks (see also Greig-Smith 1961b). Goodall considers, we believe misguidedly, that the peaks can be explained entirely as chance fluctuations. The technique has already been used in a variety of situations and ecological conclusions have been drawn from the results (Agnew 1961; Anderson 1961a, b; Chadwick 1960; Cooper 1960; Greig-Smith 1952b, 1961a, c; Kershaw 1958, 1959, 1960; Kershaw & Tallis 1958; Phillips 1954). It therefore seems important to make clear our reasons for believing Goodall's criticisms to be invalid. Goodall has attempted to explain the form of the graphs in terms of a continued rise in variance with spacing between samples. (His use of 'spacing between samples' as opposed to 'block size' is unobjectionable but the latter term is not so misleading as he maintains. It is common practice in discussing analyses of variance to refer to 'variance within and between varieties', 'variance within and between treatments', 'variance within and between blocks', etc., although in fact it is the variance between mean values for constant sampling unit that are meant.) To test this he has applied a logarithmic transformation to both the variance and the 'spacing' and fitted a polynomial regression of log-variance on log-spacing. Of thirty-nine graphs prepared from data of his own or data made available to him he found twenty-three (59 %) to be fitted satisfactorily by a linear regression, six required a quadratic, six a cubic, two a quartic and two still had a significant amount of variation not accounted for after fitting a quartic regression. On testing twenty-two graphs from data of Kershaw (1957, 1958, 1959) and Kershaw & Tallis (1958) he found that only four (18 %) could be satisfactorily fitted by linear, four by quadratic, three by cubic and one by a quartic regression, with ten graphs still showing significant residual variation. Greig-Smith (1961a, b) has pointed out that the existence of an overall trend in representation of a species along the transects being sampled will necessarily result in high variance values at the larger block sizes which may mask scales of pattern present. The presence of such a trend may be tested by examining the regression of representation on position. (Its effect may be reduced sufficiently to expose the pattern present by deducting from the sums of squares at each block size a correction term for covariance

to show that certain estimates by Nehari are sharp. The range of the complex parameter a for which these functions are schlicht in the unit disk can be treated as an example of a one-dimensional subspace in the Teichmfiller space of an annulus, and as such can be used to illustrate some of the ideas developed by Bers in [1]. In these remarks we carry out the details of this point of view. If, to fit into the framework, the setting is translated from the unit disk to the lower half-plane L, the corresponding family of mappings arise as solutions to the schwarzian differential equation { W, z} = ( for ( in the subspace spanned by the element l/z2 in B2(L, r), r a cyclic fuchsian group generated by a loxodromic transformation. The intersection of this subspace with the Teichmiiller space T(r) can be described explicity, and we write out formulas for the various mappings and group homomorphisms associated to each element 9 in or on the boundary of T(r). These elements are then interpreted, both qualitatively and quantitatively, as deformations of an annulus, and we point out a natural correspondence between this deformation space and the Teichmiiller space of a torus or a punctured torus. Unfortunately the correspondence with the space of a punctured torus, in the form that we would like, is not explicit enough, and we have left it at equation (11). A solution of this equation, explicit enough to answer the question posed there, would be of interest in that it would lead to the first non-trivial example of a Teichmfiller space in the setting of the preceding paper. The example does illustrate that a cusp on the boundary can be reached by "twisting" without "pinching," showing along the way that the expression in (8), shown by Bers in [1, ? 6] to have a universal upper bound, does not have a positive lower bound. Although the example itself is trivial, the deformations involved have analogues for general Riemann surfaces, where we might expect the qualitative behavior to be the same. The "twisting" and "pinching" of the annulus, for example, can be carried out near a boundary curve of any open surface; while the corresponding deformation of the torus

PROFESSORS Friedman and Meiselman' recently have reported that a simple quantity theory model describes aggregate consumption more accurately than a simple autonomous expenditure model. They believe this result is evidence that the "quantity" theory is a better description of the American economy than the autonomous expenditure or "Keynesian" theory.2 If their interpretation were correct, the Friedman-Meiselman paper would be one of the most significant economic studies in many years. But it is not correct. Friedman and Meiselman have represented the autonomous expenditure theory in a very unorthodox form. Their statistical comparisons are extremely sensitive to how the autonomous expenditure theory is represented. Below, I employ a more conventional representation of the autonomous expenditure theory and demonstrate why Friedman and Meiselman's tests are misleading. Further, using this conventional model and some of their data, little empirical evidence is found which favors the quantity theory. Finally some other conceptual weaknesses of the Friedman-Meiselman tests are illustrated. Briefly, Friedman and Meiselman compare simple, partial, and multiple correlation coefficients obtained from the following equations, estimated from annual (1897-1958) and quarterly (1945-1958) data for the United States: C=al+8(A (1) C=a2 +82M (2) C = a3+/33A +13P (3) C = a4 +84M+y4P (4) C = a5 + 35A + 85M (5) C = a6 + 86A + 86M + Y6P (6)

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

If (aik) a ==a(t), where 0 ? t 1, then a C R is neither necessary nor sufficient for a C S (cf. the paper quoted in the title, which will be referred to as loc. cit.). On the other hand. if a C L means (as usual, with L =LI) that y e R holds for the norm, -y = I a J, of a also, then, according to a criterion which goes back to Bocher, a C L is sufficient for a C S (for references to the implication L > S, and for generalizations of L > S derived from L =>S itself, cf. loc. cit.). There is, however, something unsatisfactory in the standard criterion, L =>S. In fact, this criterion fails to contain the fact that, as readily seen from a quadrature, a C R is sufficient (and necessary) for a C S in the scalar case, n1 1. It is therefore of initerest that L > S can be improved to the following criterion: If ao (t) denotes the diagonal matrix the diagonal elements of which are those of a(t), then aO C R and a -aO L together are sufficient for a C S. It will be clear from the following proof that (and in which maIiner) this extended criterion can be combined with the criteria giveni loc. cit. First, it follows from L ) S by a known application of the method of the variation of constants (cf. vol. 68 (1946), p. 200, of this Journal), that if 3 =, /B(t) is any continuous matrix the real part of the trace of which has an indefinite integral posessing a lower bound (> -oo), then /3 C S and a -,B C L suffice for a C S. Since the trace condition is certainly satisfied if ,/ C R, it follows, by choosing / =ao, that the italicized assertion is proved

In a paper in this Journal, Watson (1961) has discussed the use of group screening methods for detecting effective factors. The methods are analogous to those suggested by Dorfman (1943) for the medical problem of detecting a rare defect among members of a large population by using pooled blood samples. One step in Watson's derivation of the probability that an effective factor is declared significant is invalid. The purpose of this note is to derive the correct expression for this probability and show how the correction affects the subsequent theory. None of Watson's qualitative statements about the efficiency of group screening methods nor his numerical example are seriously affected.

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

Theorem. If F is a df on [0,1], that is strictly increasing, and such that F(x) > x if O < x < ?, and F(x) < x if < x < 1, then F is not n-divisible for any n. There are similar, less simple results for df's with unbounded support. As a special case it follows that the uniform distribution is not n-divisible for any n though it is decomposable in many ways (see e.g., [2]). Kudina [1] uses a similar technique to prove the indecomposability of the arcsine df. It is well known that id df's with bounded support have zero variance, i.e., are degenerate. Similarly a df on [0,a] with variance exceeding a/(4n) is not ndivisible. This argument is extended to prove the following theorem (cf. [4], also for references).

The object of this note is to fill a gap in the definition of, and to make certain remarks concerning a group, hr, which was introduced in H. G. (p. 421).1 The definition should read as follows: Let r be a given group, whose elements we denote by small greek letters, and let H be an aggregate of individuals a, a-', b, b-1, *, such that a # a-', b 5 bV' etc.2 Let h(H) C F be a single-valued, but not necessarily (1 1), transformation of H in F, such that h(a-') = {h(a) }-'. We denote by hr the group generated by all pairs (a, i), to be denoted by at, with (a, 7 ' = (a-', ), subject to the relations (1.1) atbt,= brat, for each a, b C H, i, il C F, where r = th(a)t7 'n (if a = b, t = l, this reduces to at = ath(a)). We recall from H. G. the definitions of the automorphism Oi(hr) = hr. defined by