TABEll. 



ON SINGULAR TRANSFORMATIONS. 



681 



§2. 



The trausformation obtained by the successive application to the mani- 

 fold Xi, Xo, . . . x„, in the order named, of the transformation Ta~ ^ = T_ „, 

 inverse to T^, and the transformation Ta^say where Sa^, S 02, . . . 8 a^ are 

 infinitesimal (consequently, Ta-\-sa is infinitely near to T^), is one of the 

 QO '""^ of infinitesimal transformations of G. If we denote the parameters 

 of this infinitesimal transformation by Stbj (J ^ 1, 2, . . . r), 8t being 

 an infinitesimal constant, we have 



(H) 

 or 



(11a) 



That is to say, 



^Stb Ta — Ta-i^ 



Sw 



. x,„ Stbi, . . . 8 tb^) 



= fi (/i (^) — «),•••/« (^, — «), «! + 8ai, 

 (1 = 1,2, . . . n) 



a,. + 8 a,) 



From this system of equations, which hold for all values of the x's, we 

 derive, for the determination of bi, b^, etc., r equations independent of 

 the x's and linear in Soi, 803, etc., namely, 



(12) 



Stbj — Aji 8ai + Aj2 802 + • • • + ^jr 8a^ 



(j = 1, 2, . . . r), 



where the -4's are functions of «!, a^, . . . a^. These equations written 

 in Cayley's " abbreviated notation " are 



(12 a) 8t {bu bo, . . . b,) = (^u J12 . . . ^i,()Sa.i, 802, .. . 8 a,)* 



-^21 -^^22 • • • -^2r 



^7-1 -^7-2 * ' ' ^  



* In this paper I employ the notation of Cayley's "Memoir on the Theory of 

 Matrices," Fhilosopliical Transactions, 1858, with the exception that the identical 

 transformation will be denoted by 7, whereas Cayley denotes this transformation 

 by the symbol 1. In the notation and nomenclature invented by Cayley a linear 

 substitution and a bilinear form is each represented by the square array of its 

 coefficients, the matrix of the bilinear form or of the linear substitution. In 

 accordance with Cayley's theory, if A denotes the matrix of the linear substitution 



n 



x'i = 2^ aiv Xi ((' = 1, 2, . . . n), 

 1 



and B the matrix of the linear substitution 



