MATHEMATICS 35i 



J. F. Ritt {Annals of Math., 22, 192 1, 1 57) in a paper on the 

 conformal mapping of a region into a part of itself, shows that 

 a ring cannot be shrunk conformally into a ring lying in its 

 interior. 



E. Goursat {Annates de Toulouse, 10, 1921, 65, Bull, de 

 la Soc. Math, de France, 49, 1921, i) continues his researches on 

 the application of Backlund transformations to partial differen- 

 tial equations of the second order and to systems of Pfaffian 

 equations (see his Equations aux derivees partielles du second 

 ordre, t. 2, cap. 9). In this connection there is also a paper by 

 G. Cerf {C.R., 172, 1921, 518), who deals with equations of the 

 third order in two independent variables. 



R. H. Fowler and C. N. H. Lock {Proc. Lond. Math. Soc, 20, 

 1921, 127) led thereto by their work on spinning projectiles, 

 complete the theory of Schlesinger and Birkhoff on asymptotic 

 expansions of the solutions of linear differential equations, in- 

 cluding the particular integrals. 



Historically, transcendental problems have frequently 

 arisen as limiting cases of algebraical problems, as in the case of 

 the development of integral equations by Volterra and Fred- 

 holm and earlier by Sturm in his treatment of differential 

 equations of the second order with boundary conditions as the 

 limiting case of difference equations with boundary conditions, 

 which is an abbreviated form of a restricted system of algebraic 

 equations. R. D. Carmichael {Amer. Journ. Math., 43, 1921, 

 69), in a paper entitled " Boundary Values and Expansion 

 Problems," proposes to systematise this process and begins by 

 developing methods for algebraic systems corresponding to 

 " variation of parameters " and to Green's Function. 



T. Carleman {C.R., 172, 1921, 655) discusses a class of 

 integral equations with an asymmetrical kernel ; R. Wavre 

 {ibid., 172, 1921, 432), G. Juha {ibid., 1279), S. Pincherle {ibid., 

 1395), and G. Bertrand {ibid., 1458) deal with Fredholm's 

 equation. 



St. B6br {Math. Zs., 10, 1921, i) generalises von Koch's 

 theorem on the absolute convergence of an infinite determinant. 



K (5, t) 



_ a 



6 (s) 6 {t) dsdt is not negative, for every function 6 continuous 

 in the interval {a, b), K {s, t) is said to be of " positive type." 

 J. Mercer {Proc. Roy. Soc, A. 99, 1921, 19), examines the class 

 of functions arising by linear transformation of functions of 

 positive type. 



S. Szidon {Math. Zs., 10, 1921, 121) answers two questions 

 raised by W. H. Young on Fourier series ; he shows that it is 

 not sufficient for S q„ cosnx to be a Fourier series that the 

 sequence q„ should be monotonic and converge to zero as n 



