TABER. — ON SINGULAR TRANSFORMATIONS. 



583 



(13) (Sa 1} 8« 2 , . . .8a r ) 



<t> 



i a -I 



(h 1} b 2 , . . . b T ) 



8t(a 



11) tl 12) 



• -WuK • • -K)> 



or 



(18a) 



where a 



8* 



So, = •— - (an hi + a,- 2 6 2 + 



A„ 



JUV 



• + a ? , 4 r ) 

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



is the first minor of A a relative to A VfL 



The quantities 

 8 «i, 8 a 2 , 8 « r , as determined by these equations, are infinitesimal if 



A a 4= 0, since then the constituents a M „ A ( , _1 of the matrix <£„ (e a — I) ~ l 

 are finite. Therefore, if the parameters a are so chosen that A a 4= 0, we 

 may take b ly b 2 , . . . b r , arbitrarily, and, if 8 a l} 8a 2 , ... 8 a,., are 

 determined by equations (13), we have 



Tub T a = T a + Sat 



where 8 a^ S r? 2 , . . . 8 a r are infinitesimal. 



On the other hand, if the values assigned to the parameters a are 

 critical values of the parameters, that is, if A„ = 0, it will certaiuly in 

 general, for arbitrary values of the 5's, be impossible to determine infin- 

 itesimal increments 8 a v 8 a 2 , ... 8 a r , of the parameters a to satisfy 

 the symbolic equation 



Tub T a — T a _|_ 



h,l- 



In this case, it may, nevertheless, be possible to find a finite system of 

 values c u c 2 , . . . c,., of the parameters such that Tub T a = T c ; but group 

 G may be such that, for at least special systems of values of the b's, no 

 finite system c u c 2 , . . . c r , of the parameters can be found to satisfy this 



c) C/ C/ 



symbolic equation. E.g., let r = 2 and X l = = — ,X 2 = ^ h a? 2 ^ — • 



d x. 2 d x x dx 2 



Then, if a 2 = 2 tt V^T, X = ; and if ^ $ 0, b 2 = 0, 7 7 a 7^ is essen- 

 tial singular for all values of t 4= 0. 



From what precedes we have therefore the following theorem : 

 If T is an arbitrary transformation of G for which A„ =j= 0, the trans- 

 formation T tb T„, the parameters b x , b 2 , . . . b r , being arbitrary, can be 

 generated by an infinitesimal transformation of the group, provided t is 



