434 
ME. A. CAYLEY’S MEMOLE OX CLOVES OE THE THIEH OEHEE. 
and if &c., a!, &c. have the values before given to them, then the coefficients of the 
equation are 
«^'+«'j_2M'=r{-r + 4/Z'(^+/')(f+''?') + (16//'-2^^-2r)^;;2:, 
^/'4.Ay_^y_5'^==;;{(Z^+r)(f+^^+r)+(2/+2^'+8/^r)?;5r} + (l + 4Z/(Z+0)m 
and we thence obtain 
(t(/+l/c—2ff,..gh:+(/h—af—a:f,..Xl, n, If 
->r(P+P+mt){i.‘+?i‘+C)iril 
+ (6«+6?+24Z^r)P,T 
+ (4+16??(HO)(2V+Pi;’+rt*) 
= 0 
as the condition which expresses that a line cuts harmonically its hneo- 
polar envelopes with respect to the cubic and mth respect to a syzygetic cubic. 
27. To find the locus of a point such that its second or line polar with respect to the 
cubic may be a tangent of the Pippian. Let the coordinates of the point be (x, y. r); 
then if ^x-\-ny-\-tz=0 be the equation of the polar, we have 
'^:n- t=o(f‘-\-‘2ilyz\y'^-\-2lzx\z^-\-2lxy, 
and the line in question being a tangent to the Pippian, 
-«(P+p+2:’)+(-i+4P)f>ir=o, 
But the preceding values give 
= {xf'-\-y'^ + z^f -\-Ql {x^ -\-y^ z^)xyz + 36 Z fi- ( — 2 + + z^.i^-\-j^y^) 
= A.l\x^-\-y'^-\-z^)xyz-\-{l-\-^l^)x^y‘^z‘^-\- 21 
and we have therefore 
or introducing U, HU in place of x'^-{-y^-\-z^, xyz, the equation becomes 
-S.U^+(HU)^=0, 
Avhich is the equation of the locus in question. 
28. The locus of a point such that its second or line polar ufith respect to the cubic is 
a tangent of the Quippian, is found in like manner by substituting the last-mentioned 
values of t in the equation 
QU=(l-10Z^)(f+;?'^+r)-6/^(5 + 4/^)^;jr. 
