52 Ellery W . Dains 



The equation of the companions to the first circle referred to 

 axes OX v , OY^ making an angle with the x- and y-axes is 



(a' + ia")- (a"—ia'Y 

 Thus the black hyperbola is 



2 ll 



*i _ y{ 



a' 2 a" 2 ~ 1 ' 



while the blue one is 



2 2 



{a'+a"f {a"-a'Y 



Changing the a's to b's gives of course the corresponding curves 

 for the other circle. 



In the particular case chosen for the diagram, the six curves are 



-y/ 2 =i, Vi 2 = o; 

 v' 2 V 



•\2 / "\2 — I > "^2 ^ > ^7? I • 



(1 4- 2*)- (2 —1>) 2 4 9 



There is the further condition that, if (x ll} y 1± ) belongs to the 

 first companion while (.r 12 , y 12 ) belongs to the second, then 



*i J\ I + * 3 — * 



x 2 y. 2 1 + 2* 5 



This method is equivalent analytically to expressing the coordi- 

 nates by a parameter = 0' -\- \6" . Thus, 



x = a cos 6, y = £ sin 6. 

 Whence 



#' = a ' cos 0' ch 5" + a" sin 5' sh 6", 

 x" =—a' sin 0' sh 6" + a" cos 0' ch 0", 

 |= (a' -fa") cos V ch 0" -f (a" — a') sin 0' sh 0", 

 3,' = ft' sin 1 ch 0" — &" cos 6' sh 0", 

 3," = b' cos / sh 0" — b" sin 0' ch 0", 

 - v = (&'_j_&") sintf'chfi" — (b" — V) cos0'sh0". 



52 



