70 



THE UNIVERSITY SCIENCE BULLETIN. 



The system of equations for the invariants involving also the 

 next higher derivatives of Ji, K\, L\, contains no more equations 

 but three more variables. The three solutions may be take.n 

 to be 



(44) J 



4./1, ^'4.1 - 9J'l^4.1, 



4 Ji w'i5- lbJ'it)n, 



4 Jl 6^18 - 18J'l"^,8. 



The invariants involving the next higher derivatives of Ji, K\, 

 L\, may obviously be obtained by combining J\ and J'l with the 

 invariants (44). A continuation of this process evidently gives 

 all the independent relative invariants. 



The invariants (43) may be expressed in terms of I , J , K, L, 

 and their derivatives by means of (38). However, a compari- 

 son of (38) and (41) shows that this substitution can be made, 

 except for a factor 



by replacing in ( 43 ) Ji by J — \ P, J'l 



^yd^iJ-\i"-),J"^^y -d^iJ -\P)-^i iJ-iP),Ki by 



d 



d 



K - 1 {Py\ K'rhy^\K- ] (Py ( - 2/^ (J - 1 P) and Li 



by L - i {Py^AI \K-l (/')•-' f - 2/ ^ (J - i P) + 



4 7- (J — ] /'- ) . The results of these substitutions are as follows: 

 re, =J - IP, 

 ^4.1 = 9 (^'4)- - 8^4^4 4- 32/^4^ 



6,,=. ^e',y-4.H,\K-\{py\, 



e,s = ^10 [L - 1 (/")- + 4 / j X - 1 (py \-2ie% + 



4 7--'^4]+ JK- !(/')-( j ^"4-47^4 -2K + i(/')^'[2 



-\-ei{K'--\PP-2i(:>'4y- 



<-f'i{K'-lIP-2P-f\) I r4-4/^4-2K+ -Hiy\ 



= ei,iL-i{py\ + ]K-\{py \{r-\jp-2Ky 



-^6,{K' - \I py - e\{K' - ^I P) {J" - \I P -2K). 



The same reasoning as in the case of the seminvariants shows 

 that the expressions ( 45 ) are invariants of ( A ) and that all inde- 

 pendent invariants of (A) are obtained in this way. 



(45)^ 



