150 Proceedings of the Royal Irish Academy. 



Section II. — Use of the Coefficients Fi, G^, Hi, &c., in calculating the 

 Concomitants of a Numerical Cuhic. 



(1) The Hessian and Cay ley an are easily expressed by means of the 

 equations in section i. ; we have 



Hessian ^ H ^ - F^x' - Q^if - H,^ + x\j {C, -f G^) + xrz {H\ + H,) + fx {F^ + F,) 

 + fz {H\ + H,) + z'x (F, + F,) + zhj {G\ + G,) + {K + 2K) xyz ; 



Cayleyan - P - FA' + G\fx' + H'y + {2G, - G\) A> + {2H, - H\) Vv 



+ i2F,-F\) fx'X + {2Hi-H\) f/v + {2F,-F\) .rA + {2Gi-G\) v'^ + {K'-4.K) Xfxv. 



The invariant T of the sixth degree in the coefficients, which may be 



obtained by operating with the Cayleyan on the Hessian, takes the simple 



form 



T ^ QKr + 8 {G,G, + F,F, + H^H,) - 4 {G\G', + F^F', + H\H\) 



- 8 {F,F\ + G,G\ + H,H\) + {2K + K') {K' - 4.K), 



It does not seem possible to express any power of S, the invariant of 

 the fourth degree, less than the cube entirely in terms of Fi, Gi, &c. ; the 

 following expression is, however, a fairly convenient formula for calculating /S' : 



4.S - aF\ + hG\ + cH\ + a, {2G, - G\) + a, {2H, - H\) + h, {2F^ - F\) 



+ h [2H, - H\) + c, {2F, - F,) + c, (2G, - G\) + m [IC - 4:K), 



it arises from operating with the Cayleyan on the cubic. It is fairly easy in 

 numerical cases to obtain the contra variant D (as seen in the examples which 

 follow), which gives us the contravariant Q by means of the identity 



D = 8SQ- 6TF. 



(2) Concomitants of ax^ + hy^ + cz^ - d(x + y + zy, 



F, = 0, G, = 0, ^1 = 0, F\ = - led, G\ = - bed, H\ = - led, 



F^ = 0, G^ = 0, H^ = 0, F\ = - cad, G\ = - cad, H\ = - cad, 



Fs = 0, 6^3 = 0, H, = 0, F's = - aid, G', = - aid, H\ = - abd, 



K =0, IC = abe - d {ah + 2)0 + ca). 



Hence 



H = - bcdyz (y + z) - cadzx [z + x) - dcd)xy[x + £/) + [abc -d(bc + ccc + ab) ] xyz ; 



F = -bed (X^ - XV - A^') - cad [jj? - ju'X - //v) - aM {y^ - v"X - v~ii) 



+ \abc - d {ab + 5c + ca)\ Xfxv ; 

 S = - abed ; 



T reduces to K'~ - 4 [G\G\ + F^F^ + R\H\). 



