DR. J. CASEY ON CYCLIDES AND SPHEEO-QUARTICS. 
705 
318. If we take the original conics S, S', and form the reciprocal of S with respect to 
S', we get a?x 2 + b 2 y 2 -|- c 2 z 2 ; if we denote this by S", and form the invariant angles of S, S", 
we find them to be 2d', 2d", 2d" 1 ; similarly, if S'" be the reciprocal of S' with respect to 
S", the invariant angles of S, S'" are 30 ' , 30", 30"', and so on. Again, if we denote by 
the conic which reciprocates S into S', the invariant angles of S, Si are \0', ^0", \0'", See.. 
Cor. If two conics S, S' be so related that a triangle circumscribed to S will be inscribed 
in S', and if we reciprocate S with respect to S', the reciprocal conic S" will be related to 
S by the condition that any triangle inscribed in S" ivill be self -conjugate with respect to S„ 
319. From the values given (art. 317), we get cos$' = 
b-\-c 
2 Vhc 
; and since —a=Jc', we 
have cos O' -D\/ k! — {b-\-c) : 2 \/ — abc , with similar values for cos 0" - f-\/ k", cos k"' . 
Hence we may write the equation of the conic <p, which is the envelope of a line cut 
harmonically by S, S' (see Salmon’s ‘ Conics,’ p. 334), in the following manner : — 
cos 5" 
( /j ~~ h 
/ cos Q "'\ 
\VW) 
>/=Q. 
(193) 
This equation is altogether metrical, having no reference to any particular system of 
axes, being in fact true for any system whatever of trilinear axes. 
Cor. 1. In like manner the equation of Salmon’s conic F, which is the locus of points 
whence tangents to S, S' form an harmonic pencil, may be written in the form 
y/ k’ cos d'x 2 -\-\ / k" cos d"y~ -f- \J k'" cos 0"'z 2 (194) 
Cor. 2. The discriminants of the covariant conics <p, F are the quotient and product 
of the expressions cos O', cos 0". cos d'" and k' . k" . k'" . 
Cor. 3. The reciprocal of S' with respect to F, that is, with respect to the conic (194), 
is 
cos 2 (5T ,2 + cos 2 0"y 2 -\- cos 2 O'"z 2 =0 (195) 
320. It is easy now to extend the results we have arrived at to the case of two 
quadrics. Let them be 
U = ax 2 -f- by 2 + cz 2 -f- dw 2 — 0 , 
V=x 2 -\-y 2 +z 2 -\-w 2 =0; 
and if the angles be determined thus, 
020’ V“ 1 
=k" 
: k'", 
g20(iv> Vn 1 — jg 
: k {,v \ 
020” V~ 
> =k "' : 
■.k' , 
g20(v) —IP 
: k^\ 
020'” 
~=k' : 
:k", 
^20(vi) V-i ~k '" : 
: k (iy \ 
the following Table gives the angles for the pairs of conics in which the faces of the 
self-conjugate tetrahedron intersect the quadrics: — 
