190 Sir James Cockle on the 



5. The form of the quadricriticoid s does not vary with the 

 order of the equation. Consequently the formulae which I 

 gave in this Journal for February 1875* enable us to write 

 down at once 



L==/— a 2 + a, 



M.=f+g—Zae + (n + l)a—(n—l)e, 



N=g — e 2 + e; 



and the interchanges (a, e), (f,g), and (n, —n) reverse the 

 order L, M, 1ST. 



6. The last coefficient t, which I call the cubicriticoid (Phil. 

 Mag. for March 1870), is 



«3 o <h<h , 9 /«i \ 3 / oiY' 



TO" ^* + \5x/ ~UX/ ' 



where, in differentiating, it should be noticed that 



% __ e a — e 

 <#X x #X 



7. We obtain on development, 

 P=/ i -3a/+2a 3 -2a, 



Q l = 2h + k-3(ag + ef+af) + 6a 2 e 



+ (n + l)(n— 4)a— (n— 'l)(n— 2>, 

 R=/i + 2&— 3(a^ + ^/+^) + 6ae 2 



— (n+l)fn'+2)a + (n— l)(n+4)e, 



S = &-3^ + 2e 3 -2£, 



where the interchanges (a, e), (f,g), -(Jt } Is) 9 and (w, — w) 

 reverse the order P, Q, R, S. 



8. Hence, writing 



a— e — t), 



L-M + N=6, 

 P-Q + R-S=2B, 



3(P + S)-(Q + R) = 6C, 

 we find the following relations : — 

 l>=—y(v + n), 



B=(rj—n)7j(rj + n)= — (77 — n)b, 

 C=b + (N-L) v . 



* This paper of February 1875 has a sequel iu the Number for Mav 

 1880. 



