traced upon the Surface of the Sphere. 399 



^ ^ 



Let sin 6, 6 -! + cos 6 t 6 sin 0, cos 0, + cos 2 0, tan = ... (1H.) 

 dd t 



be the equation of the normal (vid. XL. 5.) 



This by (17) is rendered a function of 0, or 6, only ; and hence if we dif- 

 ferentiate it relative to 0, or 6 t (as the case may require or best admit of), 

 and between these results, eliminate that quantity, we shall have an equation 

 between Q and the constants of (17), which will be that of the evolute. 



The involute is also found in a manner precisely analogous to that em- 

 ployed in the investigation of plane curves ; and requires no remark, as 

 there is no principle employed in the process when performed on the sphere, 

 which is materially even a variation of that which is employed in the in- 

 vestigation of plane involutes. 



XLV. 



The Polar Equation of a Cone of the Second Degree and its trace up- 

 on the surface of a Concentric Sphere. 



Let B be the vertex of a cone of the second degree, and CED any cir- 

 cular section of it. Let the perpendicular BA fall from B upon CKD. 



Refer the circle to the polar co-ordinates A and AK. Put DO = a, 

 AB = c, AO=6 ; then AC=o c, AD=+c. Put BA=r, EAK=0, 



