DE. J. CASEY ON CYCLIDES AND SPHEEO-QTTAETICS. 
591 
o'", b , V, b ", b"', See., and we have then the following substitutions to make - 
x=ax' -\-by' + cd -\-dvd , 
y—a!x ' + b'y' + dz' -{- d'ld , 
2 =a"x' -\-b"y l -\-c"z' -\-d"w' , 
w — a!"x' + b'"y’ + c"'z' + d"w' ; 
and hence, by the equation of art. 11, 
ay-\-vz + giv= ±.x'+y'±z'-\-w'. 
Consequently the transformed equation of J (corresponding to the choice of the double 
signs) is simply 
± (d ± y' ± z' + id) = S\ 
Hence 
( +#' ±y ± z' d= w'f = ( ax' + by' + cz' + dud) 2 + {dad + b'y' + c'z' + d'w') 2 
-\-{o"x' + b"y' + c"z' + d"w'f -J- (a'"x' + b'"y' -f- c'"z' + d'"w'f. 
This can be written in a more convenient manner by the following substitution, and by 
suppressing accents as being no longer necessary. 
Let us denote the result of substituting the coordinates of the pole of 
B in C by L and of A in D by P, 
C „ A „ M „ B „ D „ Q, 
A „ B „ N ,, C „ D „ B, 
and we shall have the equation of Jj in the following form : 
+=(1 - S> 2 + (1 - - S"> 2 + (1 - S">w 
+2(1— B)yz-\-2{l — M)2#+2(l — ~N)xy l (26) 
+ 2(1 — ¥)xw + 2(1 — Q)yw-\- 2(1 — R)zw =0. J 
15. The equation J, is the locus of all the double points of the quadric 
A(S*-A)+/a(S*-B)+i{S*--C)+£(S*-D). 
In fact this is equivalent to the equation 
(x+^/j+y+^)"S — (aA + + vC + ^D)-, 
the discriminant of which is easily found to be 
(a+j7j + j'+^>)“ = (yxj! + yib + vc + %dy + {xd + yib' + vd + °d!) 
+ (X«' r + + + (W” + + ; 
and A, (Jj, v, p being replaced by x, y, z, w, we have the equation of J,. — Q.E.D. 
It is instructive to compare the modes of investigation employed in this article and 
article 3 of the last section, 
