traced upon the Surface of the Sphere. 329 



Now, whether R be greater or less than -=, we have, when (f> :r R, the 



length = ; for that is the point which we have taken as origin. Hence 



c = cos R ; 



and we have, finally, 



s (f) cos R> 



If, now, instead of considering the curve as referred to polar co-ordi- 

 nates, we transform it into a rectangular equation referred to the equator ; 

 then we shall have (20) converted into 



. /sin (f) sin R\ r 



-+- I 5 I =L (21.) 



V cos R / 



If, also, instead of considering the origin to be at the point of contact of 

 the epicycloid with the tropic, we had taken it as the intersection with the 

 meridian : then when </> = 0, we should have L = , and hence 



c = 0, 



and the length of the curve would simply be 



sin <p 



cosB. 



= L (22.) 



The length of the curve estimated from the equator, then, varies with 

 the sine of declination of the corresponding point in it ; a property so 

 singular in the estimation of BERNOULLI, that he did not think another 

 curve could exist having that property*. 



7T 



The equation (3), on the hypothesis of r p ^, is transformed into 



.. co&d) 



6 sin R = cos- 1 -- - sin R cos- 1 cot <p tan R ............... (23.) 



This curve, then, is generally a transcendent upon a spherical surface, 



* Vide BERN. Op. om. iii. p. 233. ; or Mem. de 1'Acad. 1732, p. 234. The statement 

 above of the property is not exactly in form the same as that given by BERNOULLI. He 

 calls the " abscissa" the versed sine of the arc of the rolling circle which has already been 

 in contact. This versed sine is in the plane of the circle itself, and therefore is to the 



