522 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
also 
^7 Sa'+tj s i/'+^ 8z/ +^S^=0, 
dcp 
w 
dz! ’ dvf 
d(f) 00 
a7 s ‘ + dv/' 
+5^ V + K7 8 ®'— 
Hence we must have 
00 d\fr 00 00 
0£ drj 0f dco 
d(f) , 00 00 , 00 00 , 00 00 00 ’ 
ox' dx' dy' dij 0Y 02 :' 0w/ ' 0w/ 
(95) 
and every circle whose coordinates satisfy these equations must touch the curve 0=0, 
at the point (x'y'ziv). 
72. Let (X/uy>) be the coordinates of the tangent at the point (x'y'z iv'), then we 
must have 
00 
0A, 
00 , x.00; 
0a/ 0a/ 
00 
0/A 
0-0 
017 
0i y 
dp 
01 jr 00 , 00 00 , 00 
07/' 0Y ' 02/ dio' div' 
(96) 
but, by § 56, we must have 
7. </0 
Al 0\ 
+ &2 
i I 7. S 0_ A 
0/A^ 3 017 + ^0/7“°’ 
where (& l3 L, Jc 3 , Zq) are the coordinates of the line at infinity; hence, if k be 
determined by the equation 
_L\ 
dw'J 
(0+/j0) — 0; 
(97) 
then the coordinates of the tangent to the curve are given by (96), and the equation 
to it is 
( 0 0 0 0 \ 
