MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
5G5 
167. To determine the equation of the tangent ( i.e ., tangent great circle) to the 
curve at the point ( xy'z'w'), we must determine Jc in equation (184), so that the 
equation may he satisfied by (WTc ^& 4 ) the coordinates of the circle at infinity. Thus 
the equation of the tangent is 
l dx' 
00 
00 
3 dz' 
00 
d\fr\f dcf) , 00 , 00 , 00 
w‘ 
aa;' +: ' / 0y' +z 5i/ + "s«' 
(185) 
168. The circle given by equations (183) or (184) will touch the curve at the point 
x y z w ) it 
00 . 0ip 00 00 00 ,0-0 00 ,00 
0a?' 00_00 00_0Y 0/_00 00. 
00 , 00 00 , 00 00 , 00 00 , 00 
a? 00 00 7+ ^ 00 00 00 00 
he., if & satisfy the quartic equation 
0 2 
0 2 
0 2 
0 2 
00 
dxdy 1 
dxdz’ 
0,r0w 
0 2 
0 2 
0 2 
0 2 
dxdy’ 
df’ 
dydz 5 
dydw 
0 2 
0 2 
0 2 
0 2 
dxdz’ 
dydz’ 
0?’ 
dzdw 
0 2 
0 2 
0 2 
0 2 
dxdw’ 
dydw’ 
0200 
00 
H(0+h0) = O, 
I:\fj) — 0 , 
(186) 
where H(w) denotes the Hessian of w. 
We infer then that there are, in general, four systems of bitangent circles, each 
circle belonging to a particular system cutting a certain circle orthogonally; the 
coordinates of these four circles being proportional to the minors of the constituents of 
any row of the above determinants, corresponding to the four values of Jc. 
169. If the coordinates of a bitangent circle satisfy the condition 
?t_r_7, ^4.7- 
lc i 0£ ^ ~r L 
?±_ 1 _i d±-o 
3 0 £ +% 0 « ’ 
the circle is a great circle; there will, in general, be eight such great circles, two 
belonging to each system of bitangent circles. 
