TRANSACTIONS OF THE SECTIONS. 



21 



or iutegrating and determining the constjiut by the condition that, when c = 0, .r = 0, 

 we finally obtain the usual solution, 



Next, for the cubic («, b, c, d) (x, 1)'=0, 2FFj becomes 



« 36 3c d . . 



a' 3b' Sc' d . . 



. a 3b 3c d . 



. a 3b' 3c' (f . 



a 3b 3c d 



a' 3b' 



d' 



= 0; 



and when written under this form it is seen that it is a cubic function of the 

 determinants 



\\ a b c d 



11 a' b' c' d' 



or writing ab' — a'b=(ab), &c., SFFj becomes 



81(«6) (6c) {cd) + 18{ab) (ad) (cd)—27(tib) {bd)- 



+ 9(ac) (ad) (bd)— 27 (cd) (acy-—(ady'=0. 



Also v(«6) = «-, v(ffc)=2«6, v(«(^=3«r, 



V(,bc)=2b-—ac, w(bd) = 3bc—ad, \7(cd) = 3<J'—-lbd. 



By means of these formulae vFFj maybe easily calculated: and thence, with the 

 help of (8), the entire value of the resultant for the cubic will be found. If, how- 

 ever, as in the case of the quadratic, we make (rt6)=0, and then reduce by means 

 of the identical equation 



b{cd) + c(db)+d{bc)=0, 

 we find that 



'lFF^='^^{-a(bdf+m{bdf{bc)-27{bd)(bcf+21(bcy) 



and 



V2.1^Y^=Q'^^{iac-W){bdf-3(ad-bc)(bd)(be) + 9(bd-cf(bcy], 



so that V2FF, is a une facteur 2^1'es, the Hessian of SFFj. In fact the whole 

 equation (5) takes the form 



V-|6H(V)6Hd(V) = 0, 

 in which 



\=a(bdy-db(bdy{bc)+27{bd){bcy+27(bcy. 



If, further, we make a'=0 and 6' = 0, the above expression retains the same form, 

 only in it d' takes the place of (bd), and c' of (be). Finally, if we also make c'=0, 

 we have 





3(6=-«c)-r-3-j9^ 

 whence, substituting — fZ'=:3(«.r + 26.i-f c). 



-dd 



3dx 



V-n »^{Mb--ac)-(ax+by-y 



D being now regarded as a function of d, the only remaining variable ; so that .r 

 may be determined by integration, as in the case of the quadratic. 



Those who are interested in this subject may compare the foregoing method 

 with that exemplified by the author in his paper entitled " On the Theory of 

 the Transcendental Solution of Algebraic Equations," Quarterly Journal of Mathe- 

 matics, vol. V. pp. 337-360. 



