NOVEMBEE 27, 1914] 



SCIENCE 



769 



independent generators of (?. In the present in- 

 vestigation the (/>-subgroups of various important 

 groups G are studied, — in particular, of groups of 

 order the power of a prime, of direct product 

 groups, of abelian groups, of the Sylow sub- 

 groups of the symmetric group on n letters. Two 

 of the principal results are: (1) The ^-subgroup 

 of a Sylow subgroup G of the symmetric group on 

 n letters is the commutator subgroup of this Sy- 

 low subgroup G. (2) The various sets of inde- 

 pendent generators of a group G of order the 

 power of a prime contain the same number of gen- 

 erators, so that this number is an invariant of the 

 group G. 



Eliakim H. Moore: On the Integration by Suc- 

 cessive Approximations of the Ordinary Differ- 

 ential Equation of the First Order in General 



The classic problem of integration of a simul- 

 taneous system of n ordinary differential equa- 

 tions of the first order may be expressed as fol- 

 lows: To determine the m-partite number or point 

 f=(aji, •■•, Xn) in m-space Sn as a function of 

 the real variable t in such a way as to satisfy the 

 differential equation 



with the initial condition: 



|(t„) =a^ (Oi, ■••,o„). 

 Here for every t of a certain interval T' of the 

 real number system T and point J of a certain 

 region ;S' of Sn- J^{t, denotes a point of Sn] to 

 lies within T' ; a lies within S' ; and K satisfies the 

 continuity and Lipschitz conditions. A point ? may 

 be thought of as a function of i; i{i)=xi 

 (i = l, ..., n). If in this problem we replace 

 systematically the special variable i with the spe- 

 cial range i = 1, • ■ ■ , « by a general variable p with 

 the general range P we obtain the corresponding 

 problem in general analysis. Then by imposing 

 suitable conditions on S', K and a we validate for 

 the general equation with initial condition the proc- 

 ess of integration by successive approximations. 

 The general treatment covers the classical case and, 

 for example, certain types of infinite simultaneous 

 systems of differential and of integro-differential 

 equations. 

 P. B. MoDLTON : An Extension of the Process of 



Successive Approximations for the Solution of 



Differential Equations. 



"With the exception of the Cauchy polygon proc- 

 ess, which is not of practical value, the existing 



methods of solving differential equations have, in 

 general, only a limited domain of applicability. 

 The processes defined in this paper apply to a 

 very general class of differential equations, they 

 are convenient in practise, and they furnish the 

 solution with any prescribed accuracy in an arbi- 

 trary part of the domain of its existence. The 

 range on the independent variable for which the 

 solution exists is not known in advance, but the 

 process enables one to determine when he is safely 

 within that range. 

 H. S. White: The Synthesis of Triad Systems At 



in t Elements, in Particular for t^=31. 



This note reviews earlier and recent studies in 

 triad systems, and signalizes one advance step. 

 The field is largely unexplored, and progress re- 

 quires study of specimens and induction. Twenty- 

 three specimens of Ajj's showing odd and even 

 structure, many new, are available in Miss Cum- 

 mings's dissertation (Bryn Mawr, 1914) ; and one 

 of different structure, headless. Upon these one 

 new theorem is verified, then demonstrated, re- 

 lating t with 2t + 1- This makes possible the pre- 

 cise enumeration of A31 's of odd and even structure 

 which can be compounded from a headless A^ and 

 any other. Headless systems (yet unpublished) 

 prove the resulting A^i's to number above 14! 



E. J. "Wilczynski: Conjugate Systems of Space 

 Ciirves with Equal Laplace-Darboux Invariants. 

 It is well known that four linearly independent 



solutions of a linear differential equation of the 



form 



d'e 36 ^ .Be 



5ni:. + «^ + 6 5;; + <:« = 



dudv 



du 



dv 



determine a surface upon which the parametric 

 curves form a conjugate system. A great deal of 

 work has been done upon the special case when the 

 Laplace-Darboux invai'iants of the equation are 

 equal to each other. The geometrical significance 

 of this condition, which up to the present time 

 seems to have escaped notice, is the object of Mr. 

 Wilczynski 's communication. 



II. Astronomy 

 E. E. Babnard: Explanation of Certain Phe- 

 nomena of the Tail of Comet Morehouse (III., 



Within certain limitations, there is nothing so 

 wonderfully effective for the study of cometary 

 phenomena as the stereoscope. The author has 

 applied this method for the study and explana- 

 tion of the remarkable phenomena of the tail of 



