RECIPROCAL COUPLE OF TWO MAGNETS. 581 



The couple of the two magnets is again equal to the product 

 of this expression by a function / (L, /, w, 8) which should only 

 contain even powers of the lengths L and /. For if the sign of 

 one of the lengths be changed, the corresponding magnetic moment 

 changes its sign. As the couple D retains the same value to within 

 the sign, the function /should not alter. We may then write 



Mm 

 



\ F h* // 4 1 



2 cos to sin 8 -sin u> cos 8J I+ ^ 2 + ^ + '" h 



the numerators h^h^ ..... are respectively homogenous polynomials 

 of the 2nd and 4th degree in L and /, only containing even powers 

 of these two lengths. 



If the magnets are reduced to four poles, we calculate the re- 

 sultant couple .either by the actions Z and H of the first on each 

 pole of the second, or by the four actions in pairs. The expansion 

 has been carried by Lamont* to terms of the 4th degree for the 

 case in which the middle of the second magnet is in the principal 

 position in respect of the first that is to say, is situated on the 

 line of the poles or in the plane of the equator. If a and ft 

 are the angles which the magnetic axis of the bar 2/ makes with 

 the equator of the former, the moments A and B of the couples 

 relative to these two principal positions are 



2L 2 - 3/ 2 (i - 5 sin 2 a) 



2M/ COS a j 



R 3 



R 2 



+ 3 



L 4 -5L 2 / 2 (i -5 sin 2 a)+-5/ 4 (i - 14 sin 2 a + 21 sin 4 a)~| 

 ~^RT- J' 



Mm cos P f 3 L 2 - / 2 (4 - 15 sin 2 



B = 



<-!* 



R 3 



15 L 4 - 2 L 2 / 2 (6 - 23 sin 2 fl) + 8/ 4 (i - 42 sin 2 ft - 21 sin 



in*/?)"! 



It may be observed that, in the expansion of the couple A, the 

 terms of correction for the angle a are of the same form as for a 

 coil acting on a needle centred on the axis (746), as should be 



* LAMONT. Handbuch dcs Afagnetismus, p. 281. 1867. 



