traced upon the Surface of the Sphere. 423 



ployed by VANDERMONDE for designating the squares of the chess-board* 

 appeared to me admirably fitted also for this purpose, and therefore I have 

 adopted it. 



The region of revolution and the region of polar distance are conjointly 



written thus ( m ) ' where n is the revolution, and m the polar distance. 



It signifies that the point is in the ( )th region. ThusY^ is the fourth 

 quadrant of the equator, and the third of the moveable meridian. Either 

 or both of these may be negative, as ( , j, or (" .V or ( , \ accord- 

 ing as the quadrants designated by these are reckoned on the positive or ne- 

 gative side of the origin. 



When the point is in one of the rectangular meridians, the revolution 

 has proceeded through a certain number, p, of quadrants. This will be 



(m \. 

 p j ; and when the polar distance has passed 

 2 ' / 



through r quadrants, we have this designated by ( r ]. When it is 

 at the end of p quadrants of longitude, and r of polar distance, we must then 



designate it 



Lastly, if we know separately the quadrant of revolution n, or of polar 

 distance m, but have not ascertained, or do not wish to express the remain- 

 ing one, we can write it( - ) or ( ) as the case may be. 



These will be fully adequate to express all possible varieties of combi- 

 nation of (f> and 6. 



Whether the reference to the equator in case of the equatoreal, latitu- 

 dinal, or longitudinal f co-ordinates, would not be better made by the upper 



* M6m. de 1'Acad. des Scien. 1771, pp. 566-73. The method was also applied by him 

 to space of three dimensions. 



f See the Mathematical Repository (No. 25.), for a discussion of these classes of 

 quantities and their properties. 



