28 Mr. A. Cayley on the Porism of the 



I, e. 



dyjr dd 



V'm — n cos S-^/r v'm + w -v/l — A;-siii^^ 



2> 



where F= — — has the same value as before. Hence the rela- 



m + 71 



tiou between 0, & is 



dd d& 



which is identical with that between and <^' ; and in fact the 

 equation between Q, <p is 



tan tan cf) = -jj, 



which, if <^ = amM, gives ^ = am(K— m). 



The differential equation contains only a single arbitrary para- 

 meter; hence the same differential equation might have been 

 obtained from different values of a, E, the parameters which 

 determine the circle enveloped by the moveable chord. The 



condition for this of course is — ?- = — , i. e. 



n n 



2aR 2a'll ' 



i. e. {aa!-W) (a - a') - R(a';-2 - ar'^) = 0, 



which implies that the enveloped circles intersect the other circle 

 in the same two points, or that all the circles have a common 

 chord. 



Suppose for '>/^ = 0, ■\}r' = fi, then it is easy to see geometrically 

 that 



Let the corresponding value of 6' be 6' = u, i. e. suppose that for 

 6z=0, we have d' = u, then 



tan «_ ^^_^^2_^.2 ^2 



or, what is the same thing, 



R a 



sec a = 1 — . 



;• r 



And a having this value, we have for the finite relation between 0,0', 



