262 OX THE DETEKMINATION OF THE TOTAL 



and applied to deflect another substituted in its place, the moment 

 of its force to turn the latter is 



mm 17', 



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 integrals 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 ', r/, and a', for m, 11, and a. Hence the equa- 

 tion of equilibrium is 



Y cos i)' - X cos of sin r{ = m U ; (2) 



the quantity m' disappearing from the result. The magnetic 

 moment of the deflecting needle, m, is eliminated from equations 

 (1) and (2) by multiplication ; and we thus obtain a single re- 

 lation between the intensity of the earth's magnetic force, the 

 observed angles a, TJ, a', rj', and the quantities ', r, and TJ. Hence 

 the magnetic intensity will be determined when these are known. 



There are three obvious cases of these formulas, 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, and a = 0, the left-hand members 

 of (1) and (2) are reduced to mR sin (0 - ij), R sin (0 - r{) ; R 

 denoting the total force, and the inclination. "Wherefore, by 

 multiplication, we have 



R 2 sin (0 - n) sin (0 - ,,') = Uwr. (3) 



2. When the planes in which the needles move are per- 

 pendicular to the magnetic meridian, or a - 90, and a = 90, the 

 left-hand members of (1) and (2) become, respectively, w*Fcosj, 

 FCOSTJ' ; whence 



Y- cos j cos i{ = Uicr. (4) 



3. Finally, the equilibrium may be produced, in both cases, by 

 turning the instrument in azimuth until the free needle stands ver- 

 tically. In this case >, = 90, r,' = 90, and the left-hand members 

 become - mX cos a, - X cos a ; whence 



X- cos a cos a = Vur. (5) 



Thus we may apply this principle to the determination of the 

 total intensity, or to that of either of its two components. 



