OF LINEAR DIFFERENTIAL EQUATIONS. 237 



plifying the result^ we obtain 



the resolvent required. 



4. In the case of the biquadratic, the same process gives 

 the equations 



4 (i-«^')^=-(2/' + «^y* + ^'2/-3). 



+ 3(1 + ,2?')^2/— 18^*}, 

 — {(43 + 65<r^)<2?2 

 + (6n-47<27^).2?Y 



4Hi-^')'^3= -{(43 + 65^^)^2/' 



,4 



.6\ 



4- ( I o + 7 7<2?^ + 2 1.2?^) 2/ 



-3(43 + 65^')^}. 

 which, combined so as to eliminate y"^ and y*, give 



-2.43^^ + 52/=Oi 



the differential resolvent for the biquadratic. 

 5. For the qiiintic we have in like manner 



5 (i-^)^=- (2/^ + ^2/' +^V + <^'2/-4), 



5*(i _^4)z _^=, _{io^V+ (i 4-9'^)2/' 



+ (3 + 7^)^2/* 



4- 2 (3 + 2^)a?*2/ — 40.2?'}, 



SUi-^)'^ =-{i5(9 + ii^'^)«^y 



4- 15(11 4-9,3?^) .27^ 



4-3(4 + 67^^+29^^)2^^ 



+ 3 ( 1 7 + 7 1 <2?''- + 1 2^^) i2?2/ 

 — 60 (9+ II a?*) a?^}. 



