Astronomical and Nautical Collections. 131 



meeting at right angles in the middle of the needle, on one of which 

 the sphere must be placed, in order to destroy the effect of the 

 other, and on the other, in order to imitate it ; its distance from the 

 compass depending, in both cases, on the magnitude which we may 

 choose to assign to it. 



I shall conclude this extract with the formulas relating to the 

 action of a sphere magnetized by the influence of the earth, from 

 which we may deduce, by very simple calculations, the results cotL*- 

 tained in the second paragraph of my memoir, which have already 

 been explained. 



Let a be the radius of the sphere ; r the distance of its centre 

 from the given point on which it acts ; x, y, and z, the three co- 

 ordinates of that centre, referred to three orthogonal axes, passing 

 through the given point ; a, |9, and 7, the results of the action of 

 the sphere reduced to those axes ; and X, F, and Z, those of the 

 action of the earth expressed in the same manner ; we shall then 

 have: 



fee - ^^■^'^ - ^^y^ — ^y^^ \ 



\ r2 r2 r« / 



( — ^y^^ ~ 3gt3:z _ 2/gyg \ 

 \ r" r^ r- J 



The quantity represented by k being a fraction depending on the 

 nature of the sphere, and apparently differing little in general from 

 unity. When the sphere is hollow, we must substitute for this 



quantity, v "" H + J in ^hich b is the interior semidia? 

 ^ ^ (l + k)a3 - 2k^b^ 



meter of the shell ; this factor also differing very little from unity, 

 except when the thickness, a — b/is extremely small in comparison 

 with a. 



In order to compute the deviation of a needle by these formulse, 

 we must consider its middle point as the beginning of the co-ordi- 

 nates, and neglecting its length in comparison with r, the values of 

 the forces X, Y, and Z, with regard to the two poles, may be con- 

 sidered as equal, and having contrary signs. 



K3 



