MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
607 
Reduction of the General Equation to its Simplest Form. —§§ 253—255. 
253. We have seen that the number of principal spheres is the same as the number 
of roots of the discriminating quintic H(<^> -j- 0. Thus in general there are five 
principal spheres, and we have seen that these cut orthogonally, and it is clear that if 
we take these spheres as our system of reference, we can express the equation of the 
cyclide in the form # 
ax 2 + by 3 + cz~ + did 1 + ev 2 = 0; 
and we might expect, perhaps, that this equation would still be the simplest form, 
when two or more roots of the equation —0 are equal. But five orthogonal 
spheres cannot be all real, one must be imaginary; and we shall find that if one of the 
principal spheres corresponding to the unequal roots is imaginary, then this form is the 
simplest form of the equation; but if all the principal spheres corresponding to the 
unequal roots are real it is not the simplest form. 
254. Let us suppose that two of the roots of H=0 are equal; then, taking for our 
system of reference the three principal spheres (x, y, z), say corresponding to the unequal 
roots, and any two other spheres (w, v ) forming with them an orthogonal system, we 
can at once reduce the equation of the surface to the form 
ax~-{-by 2 + cz 2 + dw z + ev 2 + 2nvw—0 ; 
and the discriminating quintic is 
K = (a+k)(b+k)(c+k){(d+k)(e+Jc)-n*} = 0, 
which can only have equal roots provided d—e, n= 0 ; fie., supposing n is real. Thus, 
supposing one of the three spheres (x, y, z ) to be imaginary, so that (tv, v ) must be real, 
the equation can be reduced to the form 
axr -}- by 2 -j- cz~ + dw z + dv 2 = 0, 
or since the equation of the absolute is 
x 2 + y 2 -f $ +w 2 + v 2 = 0, 
the equation of the cyclide can be put in the form 
ax 2 + by 2 + cz 2 — 0 ; 
and we see that each of the points common to (x, y, z) is a node ; thus the surface has 
two nodes ; and, moreover, any two spheres which with ( x , y, z ) make up an orthogonal 
system may be taken as principal spheres. 
Similarly, if the sphere (x) be imaginary, and the discriminating quintic H = 0 has 
two pairs of equal roots, the equation can be reduced to the form 
ax 2 -\-b(y°-\-z 2 )-fd(w 2 -\-v 2 ) = 0 ; 
* [The equation of a cyclide was first given in this form by Casey (1871) (' Phil. Trans.,’ vol. 161, 
P 600).—October, 1886.] 
