128 UNIVERSITY OF COLORADO STUDIES 



Let 



be a differential form such that, in the symbolic notation of Boole, 

 (i) QLy=^Py . 



Evidently ^'^ .... ^'^ is a fundamental system of the equation Qy =o. 



The equation Py=o being irreducible, the coefficients q^ . . . . q„_^ 

 are not in the domain R. Indeed, by performing the operations 

 indicated in (i) and comparing coefficients there results: 



pi=qiv, p2=q2-qiv-(n-i)r)', 



(ft — I^ (ft — 2) 



p3 = Q3-Q2V-ifi-2)qiV' ^1 v", etc.. 



whence we obtain: 



qi=pj+v, q2=p2+piv+v^ + (^^-^W, 



, ,x , {n—\){n—2) ,, 

 ?3=/'3+M+/'i(^'-V)+^^+«^v'+^ -^_ n". etc. 



It is thus seen that the coefficients are rational in terms of />,.... ^"j 



'z, n'. r, ' 



By Appell's Theorem, an integral rational symmetric differential 

 function of the solutions y^ . . . . y„ oi Qy^o \s rational in the g's and 

 their derivatives. Therefore it is rational in />,, . . . . , pnyV, and their 

 derivatives. 



' Indeed, if 



Ty=^—^ + t, ^ + U -+ 'my=o 



dx"^ dx"'-' dx^-^ 



has m integrals in common with Py = o, then 



dyti-m ^ytt-m-j 



can be determined such that QTy= Py. The coeflScients t oi T are in domain R. The coefficients g can be 

 determined rationally in terms of the /)'s and /'s. This computation was made by Miss Ruby L. Carstens 

 who finds 



q, = Pi-li, gi=p2+Pit,-t\-mt,-t2, 



5j=^3_^,/,+/,,(/;_/,-m/!+/',) + 2/./a+TO/,/.-<3+<,/;(»«-i)-/^-(»t-i)/i-2ii^II^/,', etc. 



The corresponding theorem where, instead of a domain of rationality, one takes a monotopic domain 

 (i. e., a domain in which the coefficients are single valued) is given in Schlesingek's Handbuch,\ol. I, pp. 45, 46. 



