emch: theory of compound curves. ioi 



x'--fy- — 2Axr=o, (3) 



(X — a)2-f (y — b)2 — 2A'(x~yk— a + bk)=o. (4) 



It is required that any of the circles (3) is orthogonal to as many 

 circles of (4 ) as is possible. Designating the co-ordinates of the 

 centers of any two circles by {a,/3) and (a',fi') and their radii re- 

 spectivel}- b} p and p', the condition for the orthogonality of the 

 two circles is 



(a— a')-' ^-{/3~/3')~=p-^p'"^. (5) 



Associating the values a. /3, p for a special value of A. with the 

 corresponding circle of the pencil (3) we have 



a=A, /3=:0, pi^A. 



In the same way we associate the values a, /?', p' with the pen- 

 cil (4) and have 



a'=a^A', /3'=b— A'k. p'=A' /(i^k-^). 



Substituting these values in equation (5) there is 



(A— a— A' ) " + ( A'k— b) •=X"' +A' 2 ( i +k2), 



or after some reductions 



2A'{ a — A — bk)=i2Aa — a" — b-, 

 whence 



2Aa-a-^— b2 



A'=^ . (6) 



2a — 2A — 2bk 



According to this condition, to each value of A belongs one and 

 only one value of A', i. e., taking any circle of the pencil (3), there 

 is one and only one circle in the pencil (4) orthogonal to that circle. 

 If we substitute in formula (6) for A' and A successively the values: 



h—13', 



A'^=a' — a, A=a, and A'=; — ,X.^^(i, 



k 

 we obtain the two expressions 



2aa — a 2 — b- 



a — a= , 



2a — 2a — 2bk 

 and 



kC2aa — a- — b") 



/?'— b=- -, 



2a — 2a-(-2bk 

 or 



a^ — b- — 2abk 



f 



a = , 



2a — 2a — 2bk 



