io6 



JOURNAL OF THE 



In this paper an exact and thorough sohition of the 

 problem will be given, which is believed to be simpler and 

 more satisfactory every way than the approximate solu- 

 tions hitherto presented. 



Referring the curve above defined, to rectangular axes 



K' T 



X and Y (figure i), call the co-ordinates of L, x and j/, the 

 length of curve SEL being called s. At the point 

 {x + A;f, y + Aj) the length of curve from S = ^ + ^s; the 

 tangent at the last point making the angle a ^ i\a and at 

 {x^ y)^ the angle a with the axis of Y. Then by the defi- 

 nition of curvature, the curvature at the point (.r, y) is 



A« da 

 represented by limit — = — and this by the definition of 

 t^s ds 



the curve, must equal a constant (za) times s. 



da I 

 . •. — = - = 2 as (i); 



ds 



whence, 



as- 



■ (2), 



