152 Proceedings of the Royal Irish Academy. 



solving these equations we find that we may take for /, m, n, 2s the four 



quantities 



alcd'' [«' {W + r) - 3&'c- + 2alc (5 + c - a)] + 2da%''c^ [Ic - ca - ah) + a^b'c\ 



abed' \]f {a" + c") - Sah'' + 2abc {a + c-h)'] + 2dcv'V& {etc -he- ah) + a''h''c\ 



abed- [c' (a' + ¥) - Za-h' + 2ahc (a + b- e)] + 2da^bh- (ah - ca - he) + a%H\ 



- [ (d {he + ca + ah) - alef - 4d^ (a + b + c) (d (ah + hc + ca) - ahe) ahc + lMh(?bh^] . 

 The expanded expression is then ■ 



l^ {hcx^ + ?>caxy' + 3cd)xz'') + mi (cay^ + dhcyx^ + Zahyz^) 



+ iiy {cd)z^ + Zhezx^ + 2>cazy'') + Gahcsxyz, 

 where h = dl, mi = din, ni = dn, 



or, removing the factor cdic, and replacing xyz by tangential coordinates 

 A, ju, V, we have 

 (hc\^ + 3caXju' + 3«&ApO 

 X [d^ (a?h'' + «'c' - W& + 2abe (& + c - a)) + 2cl\thc (he -ca- ah) + a^b'^cH] + &c. 

 - 3 { [d{ab + he+ca) - ahcf - M^ {a+h+c)(d(ah+be+ca) - ahe) ahe -f 16fZ^a^6V} A//y 

 ^-^(8SQ-QTF), whence^. 



The covariant is found in the usual manner. 



(2) As another example we take the important form 



[x + y + z)^ + Q (m - 1) xyz. 

 Here 

 Fi = (m - l)^ Gi = (m - If, Hi = (m - 1)1 



F, = (m - 1)\ G, = (m - 1)\ H, = (m - 1)1 



F, = (m - ly, G^ = (m - l)^ H, = (m - If, K = (m - Vf (m + 2). 



F\, G\, H'l, &c., are all zero. K' = 0. 



T = %(GiG^ + F^F^ + HiH^) - 8K^ = - 8(m - ly (m"" + Im + 1). 



;S' = - (m - ly (m + 3). 



H = (m - iy{- x^ - y'^ - z^ + x^y + s^z + y'^x + y-s + z^x + z^y + 2 (m + 2) xyz}. 



P = (m- 1)M2A> + 2A^^ + 2^^A + 2,x\ + 2v-A + 2i;V - 4(m + 2) A^i/}. 



Finding the cubic of which the polar conies of the given cubic are 

 annihilating conies, as in the last example, we obtain the identity 



64 (m - 1)M2 (m + 3) (A^ + ^u^ + v') - 32 (A» - 6A/ivl - 8SQ - QTP, 



which determines the contravariant Q. Thus 



Q^-4:(m- ly {8 (A^ + /u^ + v^) - 6 (m + 1)(AV + A-i. + ^-A 4 ^uV + v'X + v^) 



+ 12m (m + 3) Ajuv}. 



We find 12P (m^ - 1) - $ = 32 (m - 1)^ (A^ + fx^ + v^ - 3A|uv). 



"Whence the common tangents to the Cayleyan and Q contravariants of a 

 system of cubics which have three fixed lines as asymptotes pass through 

 three fixed points, two of which lie on the line at infinity, and the third of 

 which is the centre of gravity of the triangle. 



