586 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



-^) ^k =r ^k Xj 



(j,k = s+l,s + 2,... r) 



We then have Cj^i = for / = 1, 2, . . . s, aud k, 1= 1^2, . . . r. 

 And, as a consequence of the differential equations satisfied by the func- 

 tions Qj = (fj (a, t h), it will be found that 



5 (fj (a, b) _ ^ 



9 cfj (a, ^) __ Q 



9 b, 



r k=zl,2, . . . s). 



9 (h 

 {j = s+l,s + 2, 



Moreover, we shall have 



% {a, b) = ttj + b^ + ipj (a,4.i, . . . a„ b, + i . . . b,) 

 U = 1,2, ... s). 



From the differential equations satisfied by the functions Oj = q)- (a, t h) 

 it also follows that, if Cj^i = forj/', ^- = 1, 2, . . . r, we then may put 



(Pi (a, h) =7ti + bi. * 



§5. 



If r = 2, group G either contains no extraordinary infinitesimal trans- 

 formation or two linearly independent extraordinary infinitesimal trans- 

 formations. In the first case, the infinitesimal transformation 2 Oy A^ is 

 commutative with no other infinitesimal transformation of G. In the 

 second case, every two transformations of G are commutative. 



If r = 3, and the structural constants are such that 



^122> ^\"2i ^-232 

 •^123' ^l-«' ^2.'!.3 



* This theorem, for the case in which (r is a sub-group of the projective group, 

 was given by Mr. Rettger in tlie American Journal of Mathematics, XXII. p. 73. 



As an example of this theorem let X, = oc-, -z — , A'o = a:., -r — , Xo = x.y -^ — . 



Then 

 And if 



Cj-ti = 0, C;i2 = 0', ^•= 1, 2, 3). 



TbTn= Tr, c■^ — a-^ + h-^ + ^lcn y^ —\, c., — a^ + h2 + 2h'i7 y/ —\, where h and 

 k' are integers which may both be taken equal to zero. 



t E. g., .Y, 



dx. 



, X: 



()x. 



X, = x{^ 



dx{ 



