352 SCIENCE PROGRESS 



group to the determination of these properties when G is of 

 type (i, I, I, . . .)• 



P. Faton contributes two memoirs on functional equations 

 to the Bull. Math. Soc. France, 47, 1919 ; 48, 1920. 



G. A. B\\ss {Trans. Amer. Math. Soc, 21, 1920) gives results of 

 a general character with respect to Differential Equations con- 

 taining arbitrary functions. The investigations were sug- 

 gested by the problems which arise in ballistics in the computa- 

 tion of trajectories disturbed by the wind and other factors. 

 The results given cover these problems, but are of a more 

 general character than the ballistic problem requires. 



Applications of these results are given in a second paper by 

 the same author {ibid., 21, 1920). 



P. Boutroux {Ann. Math., 22, 1920) suggests a method by 

 which in the case of a number of differential equations of the 

 first order the general solution in its whole field of its existence 

 is represented to any required degree of approximation, the 

 representation displaying the fundamental properties of the 

 equation, showing how the internal conditions are involved. 

 The method is also of interest as introducing certain multiform 

 functions (associated with the solutions of the differential 

 equations) which have remarkable automorphic properties. 



J. L. Walsh, in a paper " On the Solution of Linear Equations 

 in infinitely many Variables by Successive Approximations " in 

 the Amer. Journ. Math., 22, 1920, gives a number of new condi- 

 tions under which a system of equations of the type 



ciriXi + aysXz + a^^Xi + • . . = C",. (r = i, 2, . . .) 



can be solved by successive approximations. This method had 

 previously been used chiefly for Hilbert Space [i.e., the space 



of points {xk) for which S {XkY converges] with corresponding 



restrictions on a„ and c,. 



R. D. Carmichael, in a paper entitled " On Sequences of In- 

 tegers defined by Recurrence Relations " {Quart. Journ., 48, 1920) 

 develops certain general properties of the sequence of integers 

 Uo, u\, U2 . . . defined uniquely in terms of the given initial 

 numbers UgUi . . . u^-x by the recurrence relation 



tix + k -\- cciUx + ft -1 -f a2Ug -I- ft _3 -f- . . . -\- cif^Ug = a 



in which «,«!...«* are given integers. The theory of such 

 sequences affords an extensive generalisation of the theorems 

 of Fermat and Wilson. 



