544 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
lienee 
so that 
or since 
Hence 
vo / 7 JSu uSx\ 
-o . ,v/8m 
S ~ 2 =(a-6)( T -— , 
/ \/ -,.v?/a—b c—a\ 
= (c—a)(a—b)— l l- — ->• 
s /2 3 / 3 I ’ 
(a—&);?/ 3 +(c&“-c)z' 2 +(a— d)w' 2 =0, 
g=(a-b)(a-c)^(d-a}w'z. 
K 
(O 
(a — b)(a—c)(a—d)x' s ( b—a)(b—c)(b — d)y ' 3 (c — «)(c — b)(c — d)2 /3 ( d—a)(d—b)(d—c)ii / i ’ 
So that the equation of the circle of curvature at the point (x'y'z'w') on the curve 
is 
0 ££c s + by % -\-c$-\-dvP= 0, 
(a—b)(a—c)(a—d)x ,3 x-\-(b—a)(b—c)(b—d)y' i y 
-f-(c— ci)(c — b){c—d)z'h-\-(d— a){d— b)(d— c)iv'hv=0 ;. 
. (133) 
and the points of inflexion of the curve lie on the tricircular sextic, 
(a — b)(ci — c)(a—d ) s , ( b—a)(b — c)(b — d) s (c — a)(c — b)(c—d) . A 
~~ - «£' “i y “T" ~ 
a 
r 9 
? l o 
_j_ (d-a)(d-b)(d-c) ^ _ (( 
(134) 
124. If R be the radius of curvature at the point (xy'z'iv) of the curve 
ax 2 by lJ r cz~ -\-dw~ = 0, 
we shall have 
R3 
f 3 + r + ? 3 + « 2 
