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In consequent continuation of the investigations into the increase functions dW/dt = kWm(En--Wn), dW/dt = kWm(E--W)n and dW/dt = ktp (E--W)n the type dW/dt = kWm(tE--t)q with q greater than 0 is added. As in the last paper, 2 cases of integration have to be distinguished, namely for m = 1 and 0 less than m less than 1 respectively. In both cases adultness W = E is reached after a finite time tE. A quite different aspect results for the turning point of the growth function with the ordinate covering values between E/e and E for m = 1 and from 0 up to a maximum and down to E/e when 0 less than m less than 1. As in former papers graphs give an illustration of the variability of the growth curves and the increase functions. The already used example is taken up and shows a fargoing neighbourhood of the growth functions resulting from dW/dt = kWm (tE--t)q and dW/dt = ktp (E--W)n, thus hinting that quite different increase equations can lead to similar results.

The paper gives in a series of 3 parts a survey, quantitative statements and conclusions on the healing process of denatured orthotopic osteoresections with the initial bone obtained by, polarization optical measurements (this first part), morphological evaluation method (part II), (in these cases the collagen polymorphism served as indicator for the regeneration course) and, semiquantitative method of observation after conventionally histological treatment as well (part III). The investigations were carried out on the mandible of pigs depending on the treatment of the resected bone parts (chemical or hyperthermal denaturation in comparison with the native state). Samples of osteoreimplantions were taken in intervals of one week up to the 70th postoperative day. Histological sections were used from the different regions (contact region, gap region, border and internal regions of the resections) and investigated by the methods mentioned above. The results of the first part (polarization microscopical measurements) revealed the regular course of differentiation in the bone regeneration process: The resection is fully integrated into the initial bone after a mean duration of 112 +/- 25 days. The fastest regeneration takes place in the contact region, the slowest is found within the resection (in the case of chemical denaturation investigated).

After a summary of the author's work on the allometry principle with regard to the original and generalized Bertalanffy and Gompertz functions of organic growth attention is drawn to the analogous topic concerning the Janoschek growth function. At first the Janoschek function is resettled a little as to fit also for starting growth with values unequal to zero. After giving the main characteristics illustrated by graphs the allometric principle is applied to the growth function enforcing a repetition of all derivations due to the changed structure against the growth function proper. Secondly, the increase ansatz is changed with integration now leading to curves of finite growth time. Examples for allometry are added taking up values of recent investigations thus allowing for comparison. Finally an approximation method for the calculation of the inflexion points rounds up the paper.

After the investigations into 4 increase functions of organic growth a fifth ansatz should be added as dW/dt = ktp(tE--t)q with only the time t involved on the right side of the equation. In the course of researches about approximations the author has dealed with this type of functions, which can only be integrated within the limits t = 0 and t = tE leading to the Beta-function with W = E as a restricted result. Therefore the basic equation was thoroughly modified as to fit the type dW/dt = cf(t) df(t)/dt. Surprisingly the integration results in a growth function formally equal to the solution for the increase function dW/dt = ktp(E--W)n. Although the determination of the coefficients runs along other lines, the traditional example given in the preceding papers shows values so close to each other as to suppose genuine identity, which is finally proved. The fact, that seemingly quite different increase functions can lead to identical results must be given serious account: judging about increase functions should be taken as a rather delicate matter.

In a preceding paper increase functions have been treated following the Bertalanffy conception of organic growth given in the transformed shape as dW/dt = k Wm (En--Wn). As the growth functions gained by integration reach the final value E only for t leads to infinity, the increase function is changed to dW/dt = k Wm (E--W)n. Now integrations yield growth functions with asymptotic behaviour W leads to E as well as such reaching W = E in finite time, and are so better corresponding with natural development. The mathematical treatment shows a broad variety of resulting integrals given as graphs, some of them having chances as possible growth functions. Future investigations into increase functions should be focussed to the aim of choosing an ansatz allowing allround integrations leading to a variation of the parameters (m and n) in the growth function proper. This would mean avoiding a series of structurally different integrals without continuous transition between them and could be an aggravating mathematical simplification.