106 U. S. COAST AND GEODETIC SURVEY. 



on integration this becomes 



logesin ^-loge COS ^ = n loge sin ~-n loge cos ^ 



— n loge sin^ + n loge cos ~^y 



or 



loge tan ^ = 71 loge tan ^ — n'loge tan ^> 

 or, on passing to exponentials, 



The constant which enters into the expression for tan 

 ^, denoted by tan ^, is determined by the fact that the 

 straight line parallel is to have the colatitude p^. When 

 p is equal to Pq, p' becomes equal to 2 and r= 00 . In the 



further discussion we shall consider Pq>2 and reckon p 



and p^ from the North Pole. That v^U throw the straight- 

 line parallel into the Southern Hemisphere. 



The angles are everywhere preserved 'except at the 

 poles; in order that they may be preserved also at these 

 two points, it is necessary that we should have n equal 

 to unity, and then we have the stereographic projection 

 upon the horizon of the place of the central meridian 



TT 



which has the latitude (Po = Pq — 2- 



CAYLEY'S PRINCIPLE. 



This puts us in position to explain what is sometimes 

 called Cayley's prmciple.* Since in the stereographic 

 projection n must equal unity, the meridians in the hori- 

 zon projection are simply the same arcs as those of the 



* See Cayley's Collected Mathematical Papers, Vol. VII, p. 397. Also mentioned in the 

 ninth edition of the Encyclopfedia Britanniea, Vol. X, p. 203, in which place some aston- 

 ishing mathematical analysis is given in explanation of the principle. 



