100 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



By supposition y = ^ r, Xj ; and since the r infinitesiuiul transformations 

 i 



A\ . . . A' r satisfy the Lieschen criterion, 



r r r 



(a,y) = (Sj-ajXj, ^ifik^t) — -jl^i^ ~ « 2 ^i)ci2j + («ic 3 — a 3 c 1 )c 13j + . . .] X r 



Whence it follows that (a, (a, y)), etc, are linear in X x . . . X r . It is 

 to he observed that each term in the right member of (26) is linear in 

 c x . . . c r . Since X t . . . X r are independent, the coefficients of cor« 

 responding X } 8 in the two members of (26) are equal. Therefore 



b x — G n c 1 + G l2 c 2 + . . -f G lr c r1 



b, = G. n c x + G. 2 ,c 2 + . . . + G ir c r , 

 (27) 



b r = G ri c l + 6> 2 c 2 -f ... +G rr c r , 



in which the G's are integral functions of a x ... a r * 



Let the determinant of the G^'s be denoted by A, that is, let 



If A 4= we may take b x . . . />,. arbitrarily, and by means of equa- 

 tions (27) determine the e's to satisfy this symbolic equation ; in which 

 case 



A A A 





A.,., 



■ I ] 

 A 



f 

 A 





inn- and Rngel. tnsformationsgruppen, III. 7~» 1 <t .« </. ami 7s w ,is,,/ 



I Proceedings WW. 584, 



