TABER. — ON SINGULAR TRANSFORMATIONS. 



581 



§2. 



The transformation obtained by the successive application to the mani- 

 fold Xi, x 2 , . . . x n , in the order named, of the transformation T~ 1 = T_ „, 

 inverse to T a , and the transformation T a + « a , where 8 a l5 8 a 2) • • -8 a r are 

 infinitesimal (consequently, 7^ + $ a is infinitely near to T a ), is one of the 

 oo '" _1 of infinitesimal transformations of G. If we denote the parameters 

 of this infinitesimal transformation by 8tbj (j = 1, 2, . . . r), 8t being 

 an infinitesimal constant, we have 



(11) 

 or 



(11 a) 



That is to say, 

 (11 b) f t (x u ■ 



■l&tb — J- a-j- 5a -L a — ■* a + Sa ■* — a, 



Tstb T a = T a _j_ Sa' 



. x n , 8t b u . . . 8 t b r ) 



~fi (/l fa — <*)> • ' • /» fa — «)» «1 + 8«!, 



(i = l,2, . . . n) 



a r + 8 a r ) 



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

 derive, for the determination of b u b 2 , etc., r equations independent of 

 the x's and linear in 8a u 8a 2 , etc., namely, 



(12) 



Stbj = Aji 8«i + A J2 8a 2 + . . . + A jr 8a r 

 U = 1,2 r), 



where the A's are functions of a x , a 2 , . . . a r . 

 in Cayley's " abbreviated notation " are 



(12 a) St (b u b 2 , . . . b r ) = (^l u J 12 . . . ^4 lr $Sai, k 2 , 



-^21 -^22 • • • -^2 



-^*rl -^-* r2 * 



These equations written 



. s«,.)* 



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

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

 transformation will be denoted by /, 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 



x'i — 2i* aiv xi (i = 1, 2, . 

 l 



and B the matrix of the linear substitution 



n), 



