or LINEAR DIFFERENTIAL EQUATIONS. 239 



For tlie quintic, it is 

 D(D-i)(D-2)(D-3)2/ 



-{''iX-'-^X"-¥X''-T)-''-°- 



or, what is the same thing, 



whence the general form (A) or (B) assigned to the re- 

 solvent of the algebraic equation (I). 



7. We take now the form (II), and give to n the suc- 

 cessive values 2, 3, 4, 5. There result the equations 



- y'- — 2y + <2? =0, 

 2^^ — 32/^ + 2.2? =0, 

 2/4- -42^3 + 3^ = 0, 



2/^— 52/'*■ + 4^=o• 

 We have already (Art. 2) calculated the resolvent of the 

 first of these equations ; it only remains to calculate the 

 resolvents of the last three. 



8. For the cubic, we have 



3*.r^(2-^)^^,=3(i-^)2^^ 



— (3* — 2 . 5x + 2x^)y 

 — (i — 2^)<r ; 



whence the differential resolvent 



3'«^(2-.^)^. + 3'(i-<^)^ + 2/=i. 



9. For the biquadratic, we have 



