240 PROCEEDINGS OF THE AMERICAN ACADEMY. 



I shall number the equations the same, and shall use the same nota- j 

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

 demonstration. 



Lie first shows that if the n equations • i 



(1) ^/ =/i {^1^ • • • ^n^ «i . . . a,) (i — 1, 2, . . . n) ] 



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

 a's satisfy certain differential equations of the form ! 



9x' »■ ' 



(9) ■— = 1j xPj^ (ai . . . a,.) ^ji (x/ . . . x^) j 



{i=\,2,...n;k=\,2,...r),  • \ 



in which the determinant of the i(/j/. ^ ; — that, consequently, these i 

 equations may be written in the form 



^ 9x' ' 



(10) 4(x/ . . . x,^) = Ij. aj^ («!... «^) -^ ; 



1 do^ I 



(« = 1,2, . . .71; J = 1,2, . . . r),  



where the determinant of th'e a^t i ; and that, further, no linear rela- '■ 

 tion of the form ; 



e,^,,{x') + ... + e,^,,{x') = 0, . 



with constant coefficients e, persists, simultaneously, for ^' = 1, 2, . . . w. ' 



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

 equations 



Xi —Xi-\- 02 =/i (x, a), I 



Xo = e'^'iX.2 + Oi r=/2 (^j «)> I 



define 00" of transformations T^^ which constitute a group. For, by the 1 

 elimination of xl, xJ from (1) and 



(2) 



we derive 



(3) 

 where 



(4) 



^' = x^ + b, =fi(x',h), 



X 



Xo!' — e''2x/ + 61 =/o ix', h). 



x{' = .ri + Co =/i {x, c), 

 x^' = e''2 iCa + <^i = f-i (x, c) , 

 Ci = «! e''2 -}- bi = (fii (a, b), 



C2 = 02 + ^2 = ^2 (Sh ^)* 



