ON AN m n 2 PARAMETER GROUP OF LINEAR 

 SUBSTITUTIONS IN m n VARIABLES. 



BY E. J. WILCZYNSKI, PH. D., 



Instructor in Mathematics, University of California. 



§ I. All substitutions of the form 



n 



Vi = 2 a ik y k 



k=l 

 n 



7 ?i (ll = 2 (a ik y k + a ik y k ) 



k=l 



> ^i' 2 ' = 2 (« ik " jy k + « ik ' _y k ' + « ik v k ") 



k=l 



i^ = 2 (« ik (m - 1, ^ k + «ik (ni - 2) ^' + • . • + «ik J k ,m - 1) ) 



k=l 



(/= I, 2, . . «) 



belong to a group G. There are m n variables and m n 2 

 parameters. If D denotes the determinant 



D 



tl: 



ik 



, (/, k= I, 2, . . n) 



then the determinant of (i) is D m . 



The /» ri 2 independent infinitesimal transformations of 

 (i) are 



d I y 4- d / y > + + d / 



/T (0) / _ *V v . «y v ' . 4- "■/ 1/ « m - 1 » 



ik / " J7. -^ k J7.' /k ^T TT.(m-l) 4 k 



m-1 



w/= 



d /, 



d^ 



A=o dji 



Q) 



U) 



Jk 



J_ d/ (m-2) 



T 5— (m-l) J k 



dyi 



'tt)*"^ 



A=i djjV 



ik y ~ d y r~ 2) yk H 5^, m Jk 1=: - 2 33^ y * 



U^-*f: 



d/ 

 djVi' 



(in-1) jVk 5 



L.59] 



