22 PROCEEDINGS OF THE AMERICAN ACADEMY. 



one of the b's. Then — — b — b va — is an invariant of the second 



dx J dx 



degree. But so also is b 2 . Therefore 



d fdb pq db\ n fdb PQ db\db 



Ty\^ h - b ^x) h -\l^ h - h ™Tx)Ty 



is an invariant of the third degree ; and so on. These invariants are 



evidently merely the numerators of the various derivatives of -p. 



With regard to them we have the following proposition : 



Proposition 8. Let b be any chosen one of the b's. Then every in- 

 variant can be expressed rationally, and save for the possible presence 

 of a power of b as a denominator, integrally, in terms of b and the 



numerators of -f^ , —^- , . . . and the numerators of the derivatives 

 b b 



of ~ , -— , ... all these numerators being themselves rational, 



integral invariants. The notation chosen for the enunciation refers to 

 the case of partial differential expressions in two independent variables : 

 the proposition is valid in every case. 



Let the invariant 1(a) be expressed in terms of the b's : 1(a) = J(b). 



Put u = r'V} an d let L(u) go over into A(t]). Since the adjoint of 

 A (77) is r times the adjoint of L(u) we shall get 1(a) by substituting 

 in J (b), for b pq , b plg >, . . . and their derivatives, -~ , -^-, . . . and 

 their derivatives. But 1(a) = y-~ 1(a). This gives us 



T ( h h h- • — d J?2± . d A db ™ 



V ' PQ ' P ' q '' ' ' '' dx' dx ' • ' •' dy' dy ' ' ' 



- h» j ( '1 hi h^ . o — fhi\ . n — (hs\ \ 



J V b ' b ' " -' U ' dx V b J' '• •' "' dy V b )'" •)' 



As to a determination of all invariants of the second degree, see below, 

 page 26. 



§ 6. Particular Covariants. 



The simple set of covariants which we now go on to deduce will be, 

 apart from such interest as they may possess in themselves, of use to 

 us later in another connection. For ordinary differential expressions 

 the n + 1 expressions 



