58 



vity,* will enable us to determine the product of the earth's 

 total magnetic force into the moment of free magnetism of 

 the needle ; and the ratio of the same quantities may be ob- 

 tained (as in the case of the horizontal component) by removing 

 this needle, and employing it to deflect another substituted in 

 its place. 



Let us suppose, for generality, that the needle moves in 

 any vertical plane, inclined to the plane of the magnetic meri- 

 dian by the angle a', and let R denote the earth's magnetic 

 force, X and Fits horizontal and vertical components, and m 

 the magnetic moment of the needle. Then, the effective mag- 

 netic forces are mXcosa, mY; and their moment to turn the 

 needle is 



m{Y cos Tj - X cos a sin jj) ; 



in which n denotes the actual inclination of the needle to the 

 horizon. This moment is opposed by that of the weight. Let 

 this be applied in the manner adopted by Mr. Fox, namely, 

 at the circumference of a light pulley, whose centre is on the 

 axis of the cylindrical axle. Its moment is in this case in- 

 dependent of the position of the needle, and is equal to the 

 weight, W, multiplied by the radius, r, of the pulley at whose 

 circumference it is applied. Accordingly, the equation of 

 equilibrium is 



ni{Y cosri - X sin ri cos a) = Wr. (1) 



There are two cases which deserve consideration, — namely, 

 that in which the plane of motion of the needle coincides with 

 the magnetic meridian, and that in which it is perpendicular 

 to it. In the former case a = ; and substituting for Xand Y 



' The principle of this method appears to have been first suggested by 

 Mr. Christie, for the relative determination of the intensity ; and it has been 

 since applied, under different modifications, by Mr. Fox and myself, to the 

 same purpose. Mr. Fox's mode of applying it, although not the simplest in 

 practice, is undoubtedly the best. 



