100 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



By supposition y = IjCjXj-, and since the r infinitesimal transformations 



1 



X^ . . . X^ satisfy the Lieschen criterion, 



r r r 



{a, y) = (^jajXj, Vt-Yfc) = 2 [(01^2 — «2Cl)ei2j + («lC3 — «3Cl)Ci3j + .. .]Xj. 



1 1 1 



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

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

 Ci . . . c^. Since Xi . . . X^ are independent, the coefficients of cor- 

 responding X's in the two members of (26) are equal. Therefore 



bi = GuCi + G12C2 + . . + Gi,.c,i 



(27) 



b,. — G,i f 1 + G^^c^ + . . . + G,, c^ , 

 in which the (r's are integral functions of «! ... a,.* 



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



A = 



Gwi 6ri2 • • • Gxr 

 21' 22 ' ' • ^2/' 



(?,.i, G, 



r1 



G. 



The symbolic equation 

 may be written 



^_„^a + 6/7 _ g8/^ 



gag5/^^ga + 6^V_ 



If A =j: we may take b^ . . . h^ arbitrarily, and by means of equa- 

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

 case 



^W J , ^12 ; . 1 -^Ir 7 



^rl 7 , ^(-2 7 I I ^/r 7 



* Schur and Engel. See Transformationsgruppen, III. 754 et serj. and 788 et seq 

 See also These Proceedings, XXXV. 584, 585. 



