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LXVII. Analytical Solution of the Problem of Tactions. 

 By Arthur Cayley, Esq."^ 



IT is well known that the eight circles, each of which touches 

 three given circles, are determined as follows : — viz. consi- 

 dering any one in particular of the four axes of similitude of the 

 given circles, and the perpendicular let fall from the radical 

 centre (or centre of the orthotomic circle) of the given circles, 

 there are two of the required tangent circles which have their 

 centres upon the perpendicular, and pass through the points of 

 intersection of the orthotomic circle and the axis of similitude, 

 or in other words, the axis of similitude is a common chord or 

 radical axis of the orthotomic circle and the two tangent circles. 

 This suggests the choice of the radical centre for the origin of 

 coordinates ; and the resulting formulae then take very simple 

 forms, and the theorem is verified without difficulty. 



Take then the centre of the orthotomic circle as the origin of 

 coordinates, and let the radius of this circle be put equal to unity ; 

 then if («, /3), («', /S'), (a", /S") are the coordinates of the centres 

 of the given cii'cles, the equations of these will be 



a;'^ + y^ + l-2a x-2j3 y = 0, 

 x^ + y^ + l-2u'x-2l3'y = 0, 

 a^s + j,2 + 1 _ 2x"x - 20"y = ; 



and the radii of the circles will be v/a^ + /3^— 1, v/a'^ + /8''^— 1, 

 '/u"^ + /3"'^~l. It will be convenient to write 



y' = ± ^«'2+y3'2_i^ 



/= ± ^a"2 + ^"2-l, 



where the three several signs + are fixed once for all in a deter- 

 minate manner. If, however, all the signs are reversed, the 

 result is the same, so that the system is one of four (not of eight) 

 difi'erent systems. The coordinates of a centre of similitude of 

 the second and third circles are 



aY-c"y' /3'y<'-/3"y' 

 y" — yl ' 7" — y ' 



And forming the corresponding expressions for the coordinates 

 of the centres of similitude of the third and first circles, and of 

 the first and second circles, the three centres of similitude lie on 

 a line which will be an axis of similitude : to find the equation, 

 write 



* Communicated by the Author. 

 2M2 



