40 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



if (pyS), &c, denote (/8 — 7) (7 — S) (S — 3), &c, become 



a : b : c : d = a(J3yd; : — /3 'ybaj : y(iafl) : - d'a(3y 



which are convenient formulae for the case in question. 



90. If the points A,B,C, D, are on a line, then taking this line for the axis of 



x, we may write A°= (x — azf + y 2 — a" 2 z 2 , Sec. It is to be remarked here 



that we can, without any relation whatever between the radii of the circles, satisfy 



the equation 



aA° + 1)B° + cC= + (1D° = (i : 



in fact this will be the case if we have 



a + b + c + d = 



aa 4- bi + ec + to — , 



a (a 2 - a"*) + b(b 2 - b" 2 ) + c(e* - c" 2 ) + ,1 V- - ,/" 2 ) = 0, 



equations which determine the ratios a : b : c : d. In the case where the circles 

 reduce themselves to the points A,B, C, D, these equations become 



a + 1 > + c 4- d = , 

 art + bb + ec + dd = , 

 aa 2 + ]jr- + cc a + dd 2 - 0, 



giving 



a : b : c : d = (bed) : — (odd) : (dab) : — (< 



if for shortness (bed), &c. stand for (b — c)(c — d)(d — b), &c. ; and for these values, 



we have 



a A + bB + cC + dD = . 



91. A very noticeable case is when the four circles are such that the foregoing 

 values of (a, b, c, d) also satisfy the equation 



aA° + bB° + cC° + dD° = ; 



the condition for this is obviously 



;v ,"- + W* + c<" 2 + MT* = 0; 



or, as it may also be written, 



a" 2 b" 2 c" 2 dT 



(a-b)(a-c)(a-d) ' (b- c)(b-d) (b-a) (c - d)(c- a)(c-b) (d- a) (d-b) (d - < 



= 



On a Locus connected ivith the foregoing Properties. — Art. No. 92. 

 92. If, as above, A, B, C, D are any four points, and A, B, C, D are the squared 

 distances of a current point P from the four points respectively, then the locus 

 of the foci of the conies which pass through the four points is the tetrazomal curve 



a J A + b Jb + c ^C + d Jd = . 

 In fact the sum aA + bB + cC+dD has, it has been seen, a constant value for 



