MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
605 
must satisfy an equation of the second degree, and thus we shall have a double tangent 
cone corresponding to each system of bitangent spheres, the vertex of the cone being 
the centre of the sphere which cuts the particular system of bitangents orthogonally. 
247. Again, it is clear that if by any transformation the equation of the surface 
becomes and the equation of the absolute 'P, then the same value of h which 
satisfies H(<^-f-£i//) = 0 must also satisfy H(4>-l-/v4 / ) = 0 ; and hence the coefficients of 
the powers of h in the equation H(</>-j-X'0) = O are invariants. 
Equation of Normal at any Point .—§§ 248-250. 
248. Let (£77 £ wet) be the coordinates of any sphere which cuts the surface 
<f)(xy z iv v) = 0 orthogonally at the point ( x'y'z'ivv ), we must have 
^' + %' + ^' +a ^- +z %P +h ^ ==0 ’ . <276) 
for all values of Jc. 
It follows that if (X y v p a) be any plane containing the normal to the cyclide at the 
point (xy'zw'v), we must have 
0(f) 00 0(f) 00 _ 
X S +/i 3/ + Si'' -0 ’ 
0i/r dyjr df- dfr ^f__ n 
x dx’ + i‘-d ! ,' +v a7 +p dw' +cr sP- 0 ' 
0-v/r dyfr dqjr d^ fy_ n 
^Si+^dkNaipPdk+^dk-- 0 - 
249. If we take as our system of reference an orthogonal system of spheres, radii 
(r x , r 2 , r 3 , r 4 , r 6 ), then, by forming equations to planes containing the normal at (xf z'w'v') 
and passing through the centres of the spheres of reference, we easily obtain for the 
equations of the normal, 
X, 
y> 
Z, 
w, 
V 
0(f) 
00 
dcf) 
00 
00 
dx'’ 
dif 
0? 
dw” 
dv 
x, 
y> 
/ 
z, 
w, 
f 
V 
1 
i 
1 
1 
1 
3 
r i 
3 
3 
r s 
3 
*4 
= 0 .. (277) 
250. Again, from (277), we see that if a sphere cut the cyclide normally at the 
point ( xy'zw’v ), we must have 
,0<f> ,00 . ,00 , ,0$ ,0<f> n 
*0f + ^ + ^ + ^ + ^ = °’ 
d^ J dr, 0£ + 0« + 0ra 
If then (£y £ cj ct) be chosen, so that 
