246 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



y = ?/Lzl/^_VLzA^/ (68.) 



and from (69.) we have 



y = f^^,. . . • • • • (69.) 



which two equations, (68.) (69.), equated, give, after simple reduction, 



and hence also ^ (70.) 



Again, we have, 



1 K - j «, - ^^^^ - ^) ^ - (a, - «,)y J '^V' ib, - ^.,)^ - (a, - a,)y / I 



_ {(«, - a;f + {b„ - b,n { b, x' - a,yr \^^^'^ 



{(5,~z.,)y-(«,-ar)yr * -^ 



And in the same manner we obtain 



R TT2 — {(^„ - «y)^ + {b„ - b,y} {b,,x'-' a,it/) .^^ . 



{{bu-b,)a:>-{a„-a,)yr ^ ^^ 



Also 



TN2=(«^-^)2+(5^-y)2 . (73.) 



UN2 = (a,-a^)2 + (^^^-y)2 (74.) 



But by the property of the magnetic curve, expressed in equation (44.), we have 



RUUN^ 

 R T — T N3' 



or 



b.^- a,I/' _ r (a, - ^r + {b, -1/y I i- 

 b„:^-a,^-\{a,-a:'r + {b„-2/rS ^^^'^ 



This is the equation of the curve of contact of the tangent from O to the magnetic 

 curve, whose poles are T and U, with the curve ; and in rectangular coordinates is, 

 like the magnetic curve itself, of the eighth order when freed from fractions and ra^ 

 dicals. 



Having now eliminated x^ from the preceding equations in which they appeared, 

 we may drop the distinguishing accents from o/y in (75.) and reduce the fractions 

 and radicals. We thus obtain 



(b, X - a,yf{(% - xY + {b, - yY)^ = (^, x - a,yf {{a, - xf + {b, - yf}^ . (76.) 



And, as the intersection of this curve with the circle gives the points concerning 

 which this inquiry is instituted, we may write the circle at once, its radius being r, 



x^ + y^=:r^ (77^) 



