228 MR. W. SPOTTISWOODE ON DIFFERENTIAL RESOLVENTS. 



whence 



= 0, . . . . (4) 



2X d 2b' 



d 



. a lb 



c 



I . a 



b 



X a b 



• 



the differential resolvent required. The developed form is 



'}• • (5) 



2a {ac— b^)x —{a' {2b^ — ac) — 2b' ab + c a*}a? 

 —a'bc + 2b'ca—c'ab = o. 



Proceeding to the cubic 



(ttj b, Cj dyjc, I V = o 



and differentiating^ we may form the system 



{ajb,c']^x,iy(^-'2^') + 

 («, b, cjxy i)*o 



■(«j bjC'jQVf i)*o 



{a,b,c^x, i)^i 

 •{a^bjcj^x, lyx 



(a,b,cyxj lyx^ 



. dx^ + 3^>'a?^ + 2c'x + c?' = o, "" 

 + . ax^ + 'ijbx^ + 3ca? -\-d—o, 

 + «^^ + 3^^' + 3ca?* -\-dx . =0^ 

 + . . ax^ + 2bx + c = o. 



>(6) 



+ «^''- + 2bx'^ + ca?* . . = O; 



^ 



whence 



-30?' . d 36' 3c' rf' 



. . a 3Z> 3c c? 



. a 36 3c 6? . 



I . . a 2b c 



X . a 2b c . 



x^ a 2b c , , 



Writing this in the form 



(7) 



(8) 



