600 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
The Plane.—% 232-235. 
232. If ( Xfxvpcr ) be any plane L, ( xyzivv ) any point P on it, we have at once, since 
But by (256) 
lienee the equation 
/P, 1 . 2 , 3 , 4 , 5 \ 
Hi, 1 , 2 , 3 , 4 , 5 j U ’ 
01 lr 0-vir 0-Jr d\Ir ddr 
+ V Wp + v~- = 0. 
oX J dp dv dp da 
I. lit _J_7. ''l l 7. '-'I i /. S t_A . 
+*»^ +* s 0„ + A) P +A «- -° - 
oo- 
will represent a plane when 
ax + by + cz + chv-\- ev — 0, 
ak-y -b bJc. 2 ~\- c/i'a-)- clk±-\-ek- 0 — 0. 
(265) 
And if this condition be satisfied, we have to determine its coordinates, 
0-Vp 01p 0->y 01p 0Tp 
0\_o/z_ dv _ dp _ da _ _2_ 
abode b, c, d, e)’ 
(266) 
where 'l'(a, h, c, d, c ) is defined by equation (262). 
233. The power of the sphere (trjpj—) and the plane 
is given by 
therefore 
ax -b bycz + die + ev = 0 
o _ t Pt I , jty, 4_l Ci/r 
ax + % +4 &. +%+*% ’ 
77 = 
£ + b?7 + c£"+ dm + t’nr 
&, Cf d, e) 
(267) 
534. The angle of intersection of the planes 
will be given by 
ax by —fi cz -\-dw —b ev —0, 
a x-\- b y~\~ c z-\- d w-\- e / r= 0, 
COS (f) = — 
JdV jdV ,dy-JV , ,0^ 
" 5 r +S 3 s +c V + h\ 7 +c a 7 
\/^(«> O ib e).4 / '(a / , 6', P, of, <f)‘ 
(268) 
