628 Mr. W. Sutherland on the 



different planes with their centres at distance r apart. Through 

 the centre of C D draw a b parallel to A B, and let /x 12 be the 

 cosine of the angle between C D and a b, and \ p \ 2 the cosines 

 o£ the angles made by A B and C D with r, then the action of 

 A B on CD consists of forces R, H 1? and H 2 given by the 

 equations 



R = 0*i2 — 5\ 1 \ 2 )3m 1 7?2 2 /r 4 , 

 Hi =X 2 3m 1 m 2 /r 4 , H 2 = \ l 3ni i m 2 jr 4: ; 



and also of two couples one of which acts in the plane of a b 

 and CD with a moment sin (HjH^whmgA* 3 , where (HiH 2 ) 

 is the angle between H x and H 2 which the couple tends 

 to increase, while the second couple acts in the plane of R 

 and H 2 and tends to diminish the angle between these 

 directions with a moment cos (RH,) sin (RH 2 ) 3m, m 2 /r 3 . 



The more important standard case for our present purpose 

 is that in which the two magnetic axes are in the same plane 

 with the join of the middle points of the magnets. This can 

 be further simplified for the discussion of a typical case by 

 assuming the two magnetic axes to be parallel with one another 

 and making an angle 6 with the join. Then the forces reduce 

 to a central repulsion 



~(2cos 2 6>-sin 2 6')3m ] m 2 /r 4 , .... (1) 



and a component 2 sin cos 6 . 3m 1 ??2 2 /r 4 acting at right angles 

 to the join. These latter rotational forces and the couples 

 equilibrate one another if the two magnets are part of a rigid 

 system. We shall neglect them for the present, and confine 

 our attention to the central force. When = this becomes 

 — 6m 1 m 2 l )A i the minus sign denoting that it is an attraction; 

 and when d = 7r/2 the force is repulsive of amount 3m 1 m 3 /r 4 . 

 This well-known case of the attraction in one standard position 

 being double the repulsion in another might lead to an 

 erroneous conception of how attractive force might prepon- 

 derate over repulsive. For example, we might determine the 

 average force acting between two magnets as one moves in a 

 quarter circle of radius r from # = to d = 7r/2, namely, 



_2p 3 



2 msa- 3m i m » . l 



- 2 (2 cos 2 9- sin 2 0)dd 



and imagine that, if a number of magnets or electric doublets 

 direct their axes to parallelism, they will exert forces on one 

 another which are pre pond eratingly attractive. It is im- 

 portant to expose the fallacy of this incomplete reasoning, 

 because in doing so we can touch upon a matter germane to 

 the present inquiry. 



