240 PROCEEDINGS OF TI1K A.MEBICAN A.CADEMT. 



I shall anmber the equations the same, and shall use the Bame Dota- 

 tion for the special group considered as employed by Lie in bis general 

 demonstration. 



Lie first shows that if the n equations 



(0 J 'i —Si fan • • • J '> "i • • • O (* = 1, 2, . . . «) 



represent a group with r parameters, the a/'s as functions of the x's and 

 a's Batisfy certain differential equations of the form 



(9) ^- = %^ k (a l ...a r )$ Ji (x l ' ...x n f ) 



c n k i 



(t = 1, 2, . ..»; *=1, 2, ... r), 



in which the determinant of the tff Jt ^ 0; — that, consequently, these 

 equations may be written in the form 



r 3 . ' 



(10) iji ; (x/ . . . a:,') = 2* «,* (ai . . . a, I ■=- ' 



i c ", 



(t== 1, 2, . . .»;/ =1, 2, .. . r), 



where the determinant of the a jk i ; and that, further, no linear rela- 

 tion of the form 



e^ u (x') + ... + e r $ ri (x') = 0, 



with constant coefficients e } persists, simultaneously, for i'= 1, 2, . . . n. 

 "We shall consider a case for which both n and r are equal to two. The 

 equations 



Xx — an + (i-i =/i (a:, «), 



0) 



a: 2 = e"iar., + a Y ■=./» (x, a), 



define ex 2 of transformations T a which constitute a group. For, by the 

 elimination of x/, x./ from (1) and 



x x " = x( + b, EE./KV, b), 



(2) 



x 3 " = e h *xJ + b x =f,{x', b), 



we derive 



»i" = an + c a =/j (z, c), 



(3) 



./•/' = c'lj. + r t =^ (.r, c), 



where 



C, = a, e''* + ij = 0, (fl, b), 



(4) 



C a = «o -f b.. = <£„ (rr, i). 



