MR. R. LACHLAN ON SYSTEMS OE CIRCLES AND SPHERES. 
517 
57. The power of the straight line 
ax + % + cz-\-clw= 0 , 
and the circle (£ y, £, w) is given by, 
*K=f|f+ij| fc +#+.| t 
b 0\ 1 'fyu 1 *dv 1 op 
— 3 /y/ ^'-•(a^H-5i7d-c^d-c7w) 
(85) 
And the loci represented by the equations 
ax+by -f- cz + dw = 0, 
a x-\- h'y -\-cz-\- d'w= 0, 
intersect at the angle (f>, given by 
cos (f>=—± 
,3¥ . ,,3^ . ,3¥ , 7; 0^ 
a 3» +6 8t +c 3c- +( % 
v 7 T"(«, 6, c, <7).''P(a', b', c', cl') 
( 86 ) 
58. The coordinates of the line at infinity are k x , k. 2 , k 5 , k A ; hence the equation to 
the line at infinity is 
d\lr 0-vp d\Ir 0ip 
x 3k+^ + % +w dFr 0 - 
The Point .—§§ 59-61 
is 
59. The power of the point ( xyzw) with respect to the circle 
ax + by -j- cz -f- dw = 0, 
ax + by + cz -f dw 
ak x -f- bho -t~ cfc . 2 A dk A 
But if the equation represents a straight line, then we see that the perpendicular 
on it from the point 
= (ax+by+cz+dv)) a/^ .(87) 
60. The power of the two points {xyzw), {x'y'z'w') is given by equation (76) 
