traced upon the Surface of the Sphere. 



393 



that is, & = K, or generally 6 = n i rr + K (1.) 



(where n is a whole number.) 



which gives the value of corresponding to the perpendicular from the pole 

 upon the tangent. 



Recurring then to (XL. 1), we have 



cos 



sin & sin (f)' cos <' 



tan K = 



sin & f- + cos & sin d>' cos d>' 

 a u 



cos & ^y sin & sin (J)' cos <f)' 



* d( i>' * M 



sm CfHry 1 + cos u sin m cos i 



cos K=r-h 



(2.) 



tan A = - 



sin *<p' 



cos X = + 



sn 



/~ d(f>' 2 



V sin^' + ^j 



(3.) 



The last equation of set (3) agrees with (XLI. 1), as indeed it ought to do, 

 being, in fact, an expression for the same line in functions of the same 

 quantities. 



When from equations (2), (3), and the equation of the particular curve 

 under consideration, f((j> 0) = 0, we can eliminate 0' and &, we shall obtain 

 an equation in terms of K, \, and the constants which enter into/(0 0) = ; 

 and this will be the equation of the curve traced out by the foot of the per- 

 pendicular on the tangent. We shall hereafter give an example or two. 



