150 ON MAGNETISM. 



purpose we must so express these factors as to shew how 

 much depends on the position of the ship's keel and how 

 much on the position of the particle in the ship. 



Now # 2 = r 2 . cos 2 1 . cos 2 (A + a) 



= - r 2 cos 2 b . [1 + cos 2 A . cos 2a sin 2 A . sin 2a}. 



But, as a is the azimuth of the particle measured from 

 the ship's head or from the line of the keel, there will 

 be as many particles with a positive as with equal a 

 negative. The term sin 2a will therefore vanish : and 



the sum of all the terms 5 will be (putting 2 to 

 express the summation) 



^ Sim . cos 2 b ^ A ^ 3m cos2 & cos 2o> 



2 + cos 2 A . 2 - o -- . 



r r 



Then xz = r 2 . sin b . cos b . cos (A + a) 

 = r 2 sin b . cos b . (cos A . cos a sin ^4 . sin a) : 



. . , . ,. r ,, Glmxz 



which in the same manner gives for the sum of 



6lm . sin b . cos & . cos a 



r 2 



And xy = ~ cos 2 Z> . sin (2^4 + 2a) 

 25 



r 



= ^ cos 8 6 . (sin 2 J. . cos 2a -f cos 2^4 . sin 2a) : 



. , Slmxy 



which gives for the sum of ~ , 



o A \> . cos 8 b . cos 2a 

 sm 2A . 2 -- o -- 

 r 



