EXPLICIT FORM OF THE COMPLETE CUBIC DIFFERENTIAL RESOLVENT. 441 



We have, therefore, the differential equation 



aa'Ey" -aB.y' + [aWF + E{(a'd''-7ahcd + 6ac^)^a" 



+ (a'^cd + 2ab\l—3abc^)b"—{a'^bd + 2a'^c^-3ahhy 



— (aH-7a^c+6ab^)^d" + {5ad^-l8bcd + l8c'^)la'^ 

 + 3ac26'2-(2a2c-3afc2)c'2-a3(ic?'2) 



— (2acd—12i'^d + lSbc'^)^a'b'-{4iabd-6b^e)a'c' 

 _ (4a2d— S6abc + 36P)la'd' + (2a^d - 6abc)b'c' 



+ 2ahQb'd')]]ij + aB'G+'E{-(2abd^ + 2ac^d)^a" 



— (aH'^—alcd)b"+{2a^cd-2alH)c" 



+ (ci26cZ-4a2c2 + 3ab\')ld" - {Abd-^ - 6chl)}a'^ 

 + acdb'^—a%(ijd'^) + (ad^—2bcd)a'b' 



— (^l0acd-12bH)^a'c' —(ISabd—SOac^ + I8b^c)^a'd' 

 -abdb'c' + (ahl-abc)b'd'-i2a^c-Sab-)^c'd'} =0, 



the fnll development of which is an equation containing no fewer than 

 423 terms ; namely, 



y" + 



y' + 



