TABER. — ON SINGULAR TRANSFORMATIONS. 



583 



(13) (8 «!, 8 a„, ... 5 Or) = 



e" - I 



{K h,, . . . b,) 



. .(J^i, 6.,, . . .b,.), 



or 



(13a) 





+ ajr ^r) 



(7 = 1.2, 



»■). 



where a^v is the first minor of A„ relative to A^fj.. The quantities 

 8ai, 8«2j S^ri as determined by these equations, are infinitesimal if 

 A„ ^ 0, since then the constituents a^„ A„~ ^ of the matrix <^„ (e '' — /) ~ ^ 

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

 may take ^i, bo, . . . b,., arbitrarily, and, if Soi, hao, ... S«y, are 

 determined by equations (13), we have 



Tub Ta = Ta + Sa, 



where 8«i, Sa2> • • • S a^. 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 certainly iu 

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

 itesimal increments 8 a-^, 8 a<^^ ... 8 a,., of the pai'ameters a to satisfy 

 the symbolic equation 



■htb 



T — T 



-t a — -'■II 



a -\- Sw 



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

 values Ci, c^, . . . c,., of the parameters such that T^h Ta = T ; but group 

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

 finite system Ci, c^, . . . c,., of the parameters can be found to satisfy this 



5 -,, 9 . 9 

 9x^' 



2 TT V— 1, A„ = ; and if b^ ^0,b.2 = 0, T^ T^;, is essen- 

 0. 



symbolic equation. E. 



^., let )' = 2 and X^ = ^^--, ^2 = ^ 1^2 ^ — • 



Then, if (to 



tial singular for all values of t 



From what precedes we have therefore the following theorem : 

 If T ^5 an arbitrary transformation of G for which A„ 4^ 0, the trans- 

 formation Ti^T,,, tlie iKtrameters bi, bg, . . . b^, being arbitrary, can be 

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



