MATHEMATICS ^ 



self-adjoint linear differential equations of the fourth order the 

 work of Weyl and Hilb on those of the second order. 



G. Giraud {C.R., 173, 1921, 543-6) treats non-hnear differen- 

 tial equations of the second order in n variables, of the elliptic 

 type, by a modification of Picard's method of successive 

 approximations, and on the lines of Hadamard and Gevrey. 



R. Gosse (C.R., 173, 1921, 903-5) reduces the non-linear 

 partial differential equation r + f(x, y, z, p, q, t) = O, with two 

 invariants of the second order, to one of three integrable types ; 

 Riquier {ibid., 754-5) extends to most general systems of 

 partial differential equations results recently obtained by him 

 for completely integrable systems of the first order ; and G. 

 Cerf {ibid., 1053-6) advocates the use of Pfaffian systems for 

 the transformation of partial differential equations of the third 

 order in two variables with second order characteristics. 



The integral equations which arise in the problem of the 

 motion of a fluid past a body have only one solution of physical 

 significance ; H. Villat {C.R., 173, 192 1 , 816-18) examines the 

 other solutions, infinite in number, by means of the Theta- 

 Klein functions. 



G. Usoni {Rend. Palermo, 45, 1921, 271-83) finds solutions 

 in finite terms of integral equations with nucleus x — y. 



E. L. Ince {Proc. Roy. Soc. Edin., 42, 1922, 43-53) investi- 

 gates the connection between linear differential equations and 

 integral equations. The case in which the nucleus of the 

 integral equation arises as a Green's function (with discontinuous 

 derivates) is well known ; the author here by another method 

 obtains nuclei with continuous derivates. 



The application of the theory of limited quadratic forms 

 in an infinite number of variables to integral equations is 

 familiar ; there is a similar theory in functional calculus, 

 corresponding to the quadratic forms being linear, real, sym- 

 metric, limited functional operations, P. Nalli {Rend. Palermo, 

 46, 1922, 49-90) studies such operations. 



P. Humbert {Proc. Edin. Math. Soc, 39, 192 1, 21-4) 

 investigates polynomials which are generalisations of Pincherle's, 

 being the coefficients of powers of t in the expansion of 

 (1-3/::^+ /*)-"; and B. B. Baker {ibid., 58-62) expresses them 

 as sums of three hypergeometric functions of the second order. 



J. Kampe de Feriet {C.R., 173, 1921, 902-3) continues his 

 work on generalised hypergeometric functions of two variables ; 

 and E. G. C. Poole {Proc. Lond. Math. Soc, 20, 1921, 374-88) 

 treats of certain classes of Mathieu functions. 



E. Kogbethantz {Rend. Palermo, 46, 1922, 146-64), ex- 

 panding work which has appeared in the Comptes Rendus, 

 examines the summation of " ultraspherical " series by the 

 method of de la Vallee-Poussin. 



