158 MR. J. J. WALKER ON THE DIAMETERS OP A PLANE CUBIC. 



20. The invariants of the equation (3), which forp = 3, is 



p 3 D 3 w + 3/> 2 D 2 u + 3/3 DM + u = 0, .- ..... (11) 

 and, for one of the conies, s, u { . . ., or 9 x w/3 . . ., 



p- D 3 /., + 2p D, + M! = 0, ....... (12) 



give the fundamental invariants of L and u, Uj . . ., in a very succinct form, involving 

 x'y'z' regarded as parameters connected by the equation lx' -\- my' -f nz' = ; thus, if 



A' = ax' + fty' + yz, 

 then (11) , i), being replaced in the operator D (4), 16, by /, m, n 



tt t DX - (Du^ 2 = A'Xj . . . ; , . i ... (13) 



8 D 2 tt 3 + 3 D 2 M 3 - 2 DM, D?f 3 = A'-., 3 . . . ; . . . . (14) 

 (' London Math. Soc. Proc.,' vol. 9, p. 232.) Also (1) 



u D 2 u - (Du) 2 = A' 3 *, . '. ' ...... (15) 



(w D 3 w - Dw D 2 M) 2 - 4{(Dw) 2 - ttD'tt} {(D%) 2 - DM D 3 w} = A'V . (16) 

 if 



v = b 2 cH 6 + . . . = 



is the standard form of the condition that L shall touch u. 



21. The above forms are very convenient for the comparison of related concomi- 

 tants essential to the objects of this Memoir. 



Thus, the condition that L should touch s (9) 



(wjaWjs - zt 2 23 / 4 ) l * + -I" ( Via/ 2 - ii2s) win +...( = 0), (17) 



multiplied by A' 2 is (if lx + my + z' = 0) equal to 



* D 2 s - (D.s) 2 . 

 Now, since D* (A'* 7 ) = 0, whatever integers k, kf may be, 



A'*{* D 2 * - (Ds) 2 } = A' 2 s6 D 2 (A' 2 *)* - 4 (D (A'%)} 2 ; 



* It is to be observed that if u is of order p, then D r is of order p r ; and that if = ^X> X being 

 of order q, -^ of order r, then 



y D0 = qty D x 4- r x DV', 



p.|) - 1 D 3 = q.,f - If D 2 X -I- 2qr D X D^ + r.r - 1 X 

 and so on : thus 



