

PROCEEDINGS OP THE AMERICAN ACADEMY. 



X A A Y 

 (j,/: = s + l,i h2, . . . r) 



We then have r ikl = For ./ L, 2, . . . s, and /■. / -- 1,2, ...?•. 

 Ami. as a <■ osequence of the differential equations satisfied by the funo- 

 tions ", • i), (a, tl>), it will be found that 



3 vj («, J) _ Q 



- ''■ '"• /m -o 



5 a* 9 ft, 



(; = i + 1, i + 2, . . . r jfc = 1, 2, . . . ■). 



Moreover, we shall have 



■i , (a, b) = a, + b, + fo fe +l , . ..^i, +1 ... 6 r ) 

 = 1,2, . . . »). 



From the differential equations satisfied by the functions a = m, (a, t b) 

 it also follows that, if c jkl = for/, £ = 1, 2, . . . r, we then may put 



<fl («, ty = «; + b t . * 



§5. 



If r = 2, group Neither contains no extraordinary infinitesimal trans- 

 formation or two linearly independent extraordinary infinitesimal trans- 

 formations. In the first case, the infinitesimal transformation - ", A' is 

 commutative with no other infinitesimal transformation of G. In the 

 second case, ever) two transformations of G are commutative. 



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



r l'J. , t ^1321 ( '•-'"■ , .' 



* This theorem, for the case in which G is a Rub-group of the projective group, 



was given l>y Mr. Rettger in the American Journal of Mathematics, XXII p. 7:1. 



A.8 ui example of tins theorem let A', = r, -5 — , A", : , A'. = .r. 



Then c /n = 0, c >K = (././■ = 1,2,8). 



And if 



/ / '/' . i ■, Oj f &, 4- 2/- 7T .y/ -1. ro = «., + 6 a + 2/ '' 7r V~~*> wluTC * ;1 '" 1 



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



: I g-j A', - ' . A'. . r, ' . .V. -,' ■ . 



it 1 , ,1 1 , it 1 1 



