traced upon the Surface of the Sphere. 347 



To find the equation of the developed gnomonic projection of the loxo- 

 drome. 



Here, we have cot <f> = % . {K K* }, from (xxxi. 7.) 



y = r cot <,... the condition of gnomonic projection ; 



- = 6,... the condition of equatorial contact. 

 r 



From which f H (x, y) = is readily found to be 



(9-) 



Again, To ascertain what curve on the sphere, gnomonically project- 

 ed on the equatorial cylinder, would give a straight line upon the de- 

 velopment. 



Our equations are now 



y = r cot (p, 



==* 



n cos a + y sin a = 0. 

 From which we get at once the equation sought, 



6 cot (p tan a, or 1 

 cot (f) = cot a . J 



It is obvious, however, that our inquiries in this way, though they may 

 often lead to interesting results in particular cases, are, analytically speaking, 

 exceedingly confined, on account of our confined power of solving equations. 

 To pursue it further here, would therefore be irrelevant. 



XXXV. 



The length of the loxodrome has been already often assigned; and, as 

 before observed, by methods which approximate very closely to our own. 



The expression 



