TRANSACTIONS OF THE SECTIONS. 



19 



the diffei'ential resolvent reijuirecl. The developed form i3 



2a{ac-h')x' - { a'i^b" - ac)—2b'ab->rc'a-\x - a'bc+2b'ca - cab = 0, 



a result ■which had been otherwise obtained previously by both Sir James Cockle 

 and Mr. Harley. 



Proceeding to the cubic 



(a, h, c, d) [x, 1)^=0, 



Mr. Spottiswoode, with some assistance in the redactions from the author, finds 

 that the resolvent may be concisely written in the form 



+ 



i. E'E 



1 F' F 



\xG'G 



-2E 



. . a' 3&' 3c' d' 

 . a'Sb' 3c' d' . 

 . . a 36 3c f^ 

 . a 36 3c d . 

 1 . . rt 26 c 

 ,r . a '2b c . 



=0, 



in which the values of A, E, F, G are as follow : 

 A 



a 26 c . 



6 2c d . 

 . b2c d 

 . a 26 c 



the discriminant of the cubic. 



^^ a'Sb'Sc'd 

 a 36 3c rf 

 a 26 c . 

 . «26 c 



8F. 

 a 



3G 



rt"rf2— 6f/6c</4-lrtc''4-46V/-36V, 



a'36'3c'fZ' 

 a .36 3c d 

 a 2b c . 

 b2c d . 



= a' (—acd+ib^'d—'Abr) 

 -i3b'a(bd —c') 

 -\-3c' a(ad — 6c) 

 -2da{ac -62). 



n'{—ad'+7bcd—Qc') 

 -36' ( rtcf/+26\/— 36c^) 

 +3c'( «6(/+2«c- -36*0) 

 + d( a-d—7(tbr-\-Qb->). 



= 2a'd(bd —c^) 

 — :ib'd(ad —be) 

 ■^Qc'd(ac —b-) 

 + d (abd—lac^ + P,h'c). 



Attempts have been made to exhibit the cubic resolvent as a single deter- 

 minant, but hitherto without success, the only result obtained (a determinant of 

 the IGth degree) having proved illusory. The author has developed the resolvent 

 in the case of a = l, and he finds that it contains 203 terms. He has also nearly 

 completed the calculation of the cubic resolvent when the coefficients are all un- 

 modified. He hopes shortly to publish these results. 



Eight years ago, at the Meeting in Birmingham, Mr. Spottiswoode communi- 

 cated to the author a method of solving algebraic equations by integration which 

 may be conveniently noticed liere. 



Let the general equation of the «th degi'ee be represented by 



(«,6, ..)0i-, 1)" = 0; 



(1) 



then, differentiating on the supposition that the coefficients are all functions of a 

 single variable, we have 



n(fu b, . . ) (.r, D" \r + (^„, ^6, . .) f.r, 1)" = 0. 



o* 



