DR. .1. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
631 
be the projection of If on the plane of circular section, the locus of R x is a circle; there- 
fore OR bears a constant ratio to its projection. Now let the projections of K, P, P' 
by lines parallel to the greatest or least axis be K„ P 1? P' n and it is evident that we have 
the proportion 
KP . KP' : KjP, . KjP', : : OR 2 : O ,R 2 (89) 
Hence the rectangle KjPj . KjP^ is constant, and Kj is a centre of inversion of the pro- 
jection. The projection is therefore an anallagmatic curve, and being evidently of the 
fourth degree is a bicircular quartic. Hence the proposition is proved. 
96. On account of the importance of the proposition of the preceding article, we give 
another proof by forming the equation of the projection in Cartesian coordinates. 
Let U and V be given by the Cartesian equations 
(x — a ) 2 + ( y - /3) 2 + (z — y) 2 - r 1 = 0, 
CC 2 7/ 2 
a 2 !r 
1 = 0 ; 
and changing the planes of reference to xy, xz, and one of the planes of circular section, 
we get, if 6 denote the angle made by the plane of xy with the plane of circular section, 
\J=(x— af-\-(y— fi) 2 -\-(z — y) 3 +2(a‘— «)(z — y) cos 0, 
V = 
X+Z COS I 
a 
+ 
+ 
Z Sill I 
• 1 = 0 . 
If we eliminate # between these equations, we get the projection of the curve UY on the 
plane of circular section ; this elimination is most easily performed by the following sub- 
stitution, namely, 
a 2 1 
y~ z 1 sin 2 l 
:S, 
r 2 — (y— j3) 2 — (s — y) 2 sin 2 0=S', 
(cs + y cos 0) 2 =S ,/ , 
and we have at once the equation 
\/ 8 — \/ S' T x / ~ 
(S - S') 2 -f S"(S" - 2S ■ - 2S') = 0 
or, cleared of radicals. 
as the required projection ; and substituting the value of sin 2 Q, viz. 
■b°- 
. (90) 
in S — S', 
we get 
- b 2 
AV,, 
y +■ 
26 2 /3 
cr — 6 2 y nl 
2c 2 y 
a- — c~ 
c 2 y 2 "I 
2 ~ — 6 2 ‘ a 2 — c 2 
which, equated to zero, represents a circle. Hence the proposition is proved. 
Cor. S+S'= 
/fl 2 -i 2 \ r« 2 + 6 2 B t « 2 +c 2 , 
\ 6 2 )\a?-b°-y +a 2 -c* z ~ 
2b 2 (3 2 Cry (/3 2 — a 2 — y 2 ) b 2 c 2 y 2 1 
a*-b a ~y~ a *_ c 8 z + a 2 - b 2 + a 2 -c 2 j* 
