130 Professor Baker, On a property of focal conies, etc. 



where 



ip (x) = 2 {Ix^ + mx+ n), iJjq (x) = 2 {l^x'^ + m^x + n^). 



With this notation we find, for the Cayley separation of the two 

 extreme points, 



„ = tanh-(^)-ta„h-(^^), 



leading in particular, if r be the distance of these points, to 

 r = I — Iq, and to 



r^{d-X,){d~f.,){e-v,) 



2 tanh w 



Q + ipQ (d) tanh w ' 



The character of the symmetric functions of the places 

 (A), (fx,), (v), regarded as functions of w, along any portion of the 

 curve, seems eminently worthy of investigation. 



And it appears that the total value of tv, along any closed 

 portion of the continuous curve, is expressible by an aggregate of 

 the periods of the integral 



Q [ {x-p){x-q ) 



2{d-v){d-q)] {x-d)y ^' 



where y^ ^ F {x), with integer coefficients; these will then be un 

 altered by any continuous small deformation of the arc of the 

 curve. This remark appears to lead to all the known results. 



In conclusion I should like to refer the reader to a most inte 

 resting note by Mr A. L. Dixon, Messenger of Mathematics, xxxii, 

 1903, 177. 



