52 Colorado College Studies. 



•problem. To wit, if P = be the equation of the polar of c, s, in 

 respect to the new circle, then P" + k {oc^ + y' + 2ax + cr — ?'^) = 

 must be identical with equation (1). This furnishes, as before, 

 six conditions to determine four unknowns, which in this case 

 are A, H, B and A"). 



A comparison of equations (1) and (5) shows that the latter 

 is derived from the former by giving to A, H, B, the values 

 r" — s\ as + cs, and v' — a' — 2rtc — c: respectively. Then if 

 the same letters be replaced by these values in equation (3), 

 the result will be the equation demanded by the present prob- 

 lem, viz., 



[2a (c' + s'-') + c ic' + s' + rr)] x + s (r + .r — a") y 



+ (c^ + s')(«' + 2rtc + c- + s- — 2;--) = 0. ^ 



It is desirable to notice at what point the line whose equa- 

 tion has just been written, intersects the line which joins the 

 intersections of the two circles, /. e., the axis of X. If for this 

 purpose we put ?/ = in the equation, and solve for x, replacing 

 at the same time c~ + s^ by Br, we have, as the abscissa, m, of 

 the point in question, 



K' {a- + 2«c + P- — 2r-) „ 



This expression does not contain s; hence if we regard the 

 two circles as fixed in position but the point c, s as moving, 

 (subject of course to the condition that it remain upon the cir- 

 cumference of the circle or + y' — P"), we may discuss the 

 effect upon the position of the intersection m, 0, by supposing 

 the value of c alone in the formula to be varied. The equation 

 (7) is linear in c and in m, so that for any value which may be 

 assigned to one of these quantities a single real value would be 

 obtained for the other. But not all values of c are possible 

 consistently with the restriction of the point c, s to the circum- 

 ference of a circle; c, in fact, cannot be greater than P or less 

 than — P. Hence the values of m which would be obtained by 

 assigning to c any value transcending these limits are equally 



