EMCH: THEORY OF COMPOUND CURVES. IO3 



The co-ordinates of the center of the first circle are 



a— bk— bl'- i+k^ 

 m^= 



b + ak + al/i-4-k2 

 n== 



(10) 



and of the second 



a — bk + bl i+k3 



(II) 



b + ak— al i-f-k^ 



2 

 The radii of these circles respectively are 1 ni^-j-n^ and 



1 ' m'2-(-n'-. It is easih' verified that 



(m— m')2-j-(n— n')2^ms+n3-fm'2+n'2. 



This, however, is tlie condition that two circles are normal to 

 each other. Hence: 



Tlic tiuo circles fonui Hi:; the locus infer sect each othei' at rij^ht angles. 



From this follows, that the points P, Q, O, M, T in fig. 2. all 

 lie on the same circle with the line Py as a diameter. 



3. The normal pencil of circles of the pencil (4) is obtained by 



X — a 



considering the normal to the straight line (2), :=k, which is 



y-b 



xk-f-y — ak — b=o, 



and the zero-circle at the point (a,b) as two circles of the pencil. 

 The required normal pencil is therefore given by the equation 



(x — a)2-|-(y — b)'' — 2A"(xk-p-y — ak — b)=o. (12) 



For a fixed value of A" the co-ordinates of the center of the cor- 

 responding circle are 



a" = a-fA"k, 



/3"=b+A", 

 and p"=A"l/i+k2. 



