102 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



In fact, in the first problem if we use z to denote a given constant (which may 

 be = 0), then taking a, a' and i(z — a") for the co-ordinates of the centre and for 

 the radius of one of the given circles ; and similarly b, b\ i(z — b") ; c, d, i(z — c") 

 for the other two given circles ; and £, S',i(z — S") for the required circle ; the 

 equations of the given circles will be 



(* - a) 2 + (y - a') 2 + (z - a" J 2 = , 

 (x-bf + tu-h'f + (z-b')*=0, 

 (x-cY + iy-t'f +(z-c"Y = o . 



and that of the required circle will be 



{x-sy + (tf-sy + (z-s-f = o. 



Tn order that this may touch the given circles, the distances of its centre from 

 the centres of the given circles must be i(S'—a"\ i(S'—b"), i(S'-c') respectively; 

 the conditions of contact then are 



(S-a) 2 + (S'-a'/ + (S"-a"f = 0, 

 (S - bf + (£'- by + (S" - h"; = , 

 {8- cf + (S' - cf + {S"~ c"f = 0, 



or we have from these equations to determine S, S', S'. But taking (a, a', a"), 

 (b, b', b"), (c, c', c") for the co-ordinates of three given points in space, and 

 (S, S\ S") for the co-ordinates of the centre of the cone-sphere through these 

 points, we have the very same equations for the determination of (S, S', *S"), and 

 the identity of the two problems thus appears. 



I will presently give the direct analytical solution of this system of equations. 

 But to obtain a solution in the form required, I remark that the equation of the 

 cone-sphere in question is nothing else than the relation that exists between the 

 co-ordinates of any four points on a cone-sphere ; to find this, consider any five 

 points in space, 1, 2, 3, 4, 5; and let 12, &c. denote the distances between the 

 points 1 and 2, &c; then we have between the distances of the five points the 

 relation 



0, 1 



1 



1,0, 12 2 , U 2 , U 2 , 15 2 

 1 , 21 2 > , 23 2 , 24 2 , 2o 2 

 1 , 31 2 , 32 2 , , U 2 , 35 2 

 1, 4l 2 > 42 2 , 43 2 . , 4o 2 

 1, 51 2 > 52 2 > 53 2 , 54 2 , 



= 0; 



whence taking 5 to be the centre of the cone-sphere through the points 1, 2, 3, 4, 



