524 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
77. It follows that if (A, /x, v, p ) be the coordinates of the normal at ( xy'z'iv) we 
must have 
00 00 00 00 
x dx' +IJ -^ +v 
and 
00r 00- 00 00_ n 
x 3^ + '‘3/ +I 'a 3 ' + ' > 0 l .y-°’ 
00 00 00 00 
X 3i 1 +' i &+’'s;-° - 
(& 1} /; 2 , & 3 , /%) being the coordinates of the line at infinity. Hence the equation to the 
normal is 
00 
00 
00 
00 
1—0. . . 
dx ’ 
3y’ 
03 ’ 
010 
00 
00 
00 
00 
0x' 5 
0/’ 
0Y’ 
0«y 
00 
00 
00 
00 
0.Y’ 
3/’ 
0*” 
0(0' 
00 
00 
00 
00 
0/.0 
0V 
0/0 
0/q 
■cm 
equation (1 
)l) that normals 
= 0, at its points 
of intersection w 
00 
00 
00 
00 
= 0. . . 
0x’ 
3//’ 
03’ 
0(0 
00 
00 
00 
00 
030 ’ 
3//’ 
03’ 
0(0 
00 
00 
00 
00 
0.F’ 
3/’ 
0fi’ 
0?o' 
00 
00 
00 
00 
0/0 
0/0 
3/v 3 ’ 
3*4 
(102) 
This curve is clearly of the second degree, but since it is satisfied by (Zq, k 2 , k s , Zq) 
the coordinates of the line at infinity, it represents a circular cubic. Hence, in general, 
eight normals can be drawn from any given point to the curve. 
79. In the case of a circle cutting the curve </> = 0, normally at the point ( x'y'ziv'), 
we shall have 
x 
x 
,00 
,00 
3£ 
,00 ,00 
+ Z 0f + W S 
,00 ,00 
+ *0 +W 00, 
=o, 
= 0 . 
