604 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
And thus the equation of any tangent sphere is of the form, 
dt +k ^) u,+ (st +k ^) v=<> - 
Hence the equation of the tangent plane at the point ( x'y'z'w'v') is 
\ <A 1 1 A, k A, k d A <W\( x <tk . v d A_, z d _± . w <A, v <h£ 
1 0.z' + ^00 + ^ 3 0*' +4 0z</ + »dv')X 0A + 2/ 00 + dz f + W dw ,+V dv' 
1 A,/ A L / A it A\l A , 00 , 0 0 , 00 d$ 
■ i dA k ^ + ^d-’ +li dw' + ^)\ x dA y d i i' +z ii +w ^ +l % 
244. The sphere given by the equation 
S 3 0 0 0 \ 
^ +y 37 +2 S +w s«/ + 7^ +fa W =0 ' 
A' 
will clearly touch the cyclide at the point ( x"y"z"iv"v") if 
00 . 8-0 00 00 00 00 00 , 00 00 .00 
dx' dx' _00 00_0/ 0A_0m/ 0a 1 ' _ Bv' 8b 
00 z 00 00 70 00 z 0 0 00 Z 0 ^ 0 0 i A ’ 
07T 77 00 7+ 0“7' 07 + 07 07 + 07 07 07 
in which case & must satisfy the quintic 
(274) 
H (0 + /i’ 0 ) = O, .. (2 75 ) 
where H(z^) denotes the Hessian of u. 
We infer, then, that there are in general five systems of bitangent spheres ot a 
cyclide ; i.e., of the whole number of tangent spheres at any point of the surface five 
touch the surface elsewhere. Moreover it is evident that the system of bitangent spheres 
corresponding to a particular root of (275) all cut a certain fixed sphere orthogonally, 
the coordinates of this sphere being proportional to the minors of the constituents ot 
any row in the determinant H(00/o/;). 
245. If the coordinates of a bitangent sphere satisfy the absolute, i.e., if the radius 
of the sphere be indefinitely small, the sphere may be considered as a focus. It is 
evident that the foci belonging to any system of bitangents will form a curve of the 
second degree on the corresponding orthogonal sphere. Such a curve will be a spheri- 
quadric. 
246. If the coordinates of a bitangent sphere satisfy the condition, for a plane, 
the corresponding equation will represent a double tangent plane; and its coordinates 
