the Intensity of the Earth's Magnetic Force. 537 



the radius of the pulley by which it acts ;* and the equation of equilibrium is 

 therefore — 



M {Y coat] — X cos asm-))) — Wr. (1) 



When the needle is removed, in the second part of the process, and applied 

 to deflect another substituted in its place, the moment of its force to turn the 

 latter is 



3IM'U; 



in which M' is the moment of free magnetism of the second needle, and U a, 

 function of the distance of the centres of the two needles, and of certain inte- 

 grals depending on the distribution of free magnetism in them. The moment 

 of the Earth's magnetic force, opposed to this, is of the form already assigned, in 

 which we have only to substitute M\ y', and a, for M, % and a. Hence the 

 equation of equilibrium is 



F cos >;' - X cos a' sin 1/ =MU; ( 2 ) 



the quantity M disappearing from the result. The magnetic moment of the 

 deflecting needle, M, is eliminated from equations (1) and (2) by multipli- 

 cation ; and we shall thus obtain a single relation between the intensity of 

 the Earth's magnetic force, the observed angles a, r/, a', ?;', and the quanti- 

 ties W, r, and U. Hence the magnetic intensity will be determined when 

 these are known. 



There are three obvious cases of these formula3, each of which suggests 

 a different method for the determination of the terrestrial magnetic intensity. 



1. When the planes in which the needles move coincide with the magnetic 

 meridian, or a=0, a = 0, the left-hand members of (1) and (2) are reduced to 

 3fR sin (6 — 17), R sin (e - 9/ ) ; B denoting the total force, and the inclina- 

 tion. Whence, by multiplication, we have 



E' sin (6 -n) sin {e-r]') = UWr. (3) 



• It is here supposed that the weight is attached to a fine thread passing round a light 

 pulley, whose centre is on the axis of the cylindrical axle of the needle, in the manner proposed by 

 Mr. Fox. If the weight be attached to the southern arm of the needle, at a fixed point, its mo- 

 ment is Wr cos 7. 



