70 



THE UNIVERSITY SCIENCE BULLETIN. 



The system of equations for the invariants involving also the 

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

 but three more variables. The three solutions may be taken 

 to be 



(44)^ 4 J, 7/'i, - 15J'iT^ir„ 

 4 Jx e'ls - 18J'iW,8. 



The invariants involving the next higher derivatives of J\, 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 tttyth > t)y replacing in (43) Ji by J — 17', J'l 



by ^(J-]P), J"iby ^^(J-i7-^)-47 {J-\P),Kx by 



d 



d 



K - 1 (7')--', TC'i by ^ ] 7C - i (7')^ ( - 2/^ ( J - 1 7'^) and U 



by L - 1 (7")2+ 47 \K-l- {I'r f - 27 ^ ( J - ] P) + 



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



re,=j-\r\ 



dux = 9(0',)- - 80,6% + 3210,', 



e,, = {0',y-4.f>,\K-\ii'r\, 



H,, = 5t),u O'i - 2 0\oO,, 



0,, = 0,, [L - 1 (7")--' + 47 { K - 1 (7')' \-2I0",+ 



APoq + \K-\{rf\ \o",-uo,-2K+\{Py-\' 

 + o,(K'-^pr-2P->',y- 



y, {K'-\ir-2U)',) j 0", -^I0,-2K+ \ {Pf\ 



= Ou.\L-Hi'y\ + \K-}{Py\{j"-^ir-2Kr 



+ 0,{K' - \I ry- - 0',{K' - i,U") {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)-^ 



