DETERMINATION OF THE ELEMENTS OF THE SUN'S SPIN. 271 



Then considering the nearly righl angled triangle PQl J \ we have 



ô'-ô P'Q PP' cos f PP'N + NT'S J 



PP' cos PP'N cos z'-PP' sin PP'N sin a' 



.'. (V — (5 =X. cos z'—Y. sin x' 



if we assume 



X -PP' cos PP'N, Y PP' sin PP'N. 



It A and 1 can be found from a series, or from many combined series 



of observations, then 



PP' y/X^+ï , tan PP'N ^. 



Bu1 



X PP' cos PP'N PR ( N-NJ' sin I 



)' PP' sin PP'N= P'R=l-I' 

 lli-iire we have the following formula: giving the true elements 



/ I' + Y 

 N N'+P.cosecl. 



I he above treatment gives ô'— <5 .V cos z'—Y sin t! as the 

 equatious of conditions to lind A and Y. It is a tremendous piece of 

 work to apply this equation to each individual sei of observations of 

 spots. Hence it is necessary to consider in what way these quantities 

 A' and Y can most advantageously he found from the whole ac- 

 cumulated mass of observations. Since each series of observations 

 yields a certain number of observed values of ô' corresponding to 

 obsen ed \ allies of a', we may a>>um<' i he following series of equations. 



Ô„'— Ô=-X eus ô — Y sin it 



ôi' — ô-^X cos b — Y sin h 



fj / — <5 = X cos c — Y sin c 



By successive subtractions, we get 



Oft — b'a—X fcos b — cos a J — Y (sin b — siu a) 

 ôc'— ô\ = X (cos c—cos bJ—Y (sin c—sin b) 



