TERRESTRIAL MAGNETISM. 203 



We see further that P' has the form 



R-^ P' = a cos M + /3 sin u cos X + 7 sin ti sin X, 



where a = - I cos u^ r'^ d /m, /3 = — / sin u^ cos \^ r° d ^a, 7 



= — I sin u sin X,^ r° d fi. Therefore, according to the expla- 

 nation laid down in page 13 of the Intensitas Vis Magneticce, 

 — a, — /3, — 7j are the moments of the earth's magnetism, in re- 

 lation to three rectangular axes, of which the first is the axis of 

 the earth, and the second and the third are the equatorial radii 

 for longitudes and 90°. 



The general formulae for all co-efficients of the series for — may 



be assumed as known ; it is merely necessary for our purpose 

 to remark, that in relation to u, X, the co-efficients are rational 

 integral functions of cos u . sin u cos X, and sin u sin X, and of T" 

 of the second order, T'" of the third, &c. It is the same as to 

 the co-efficients P", P'", &c. 



The series for -, and for F, converge, so long as r is not less 



than R, or rather, not less than the half diameter of a sphere, 

 which includes all the magnetic particles of the earth. 



18. 

 The function F being composed of — / —, satisfies the fol- 

 lowing partial differential equation : 



_ rcPrV d^ V dV 1 d"- V 



" - dr'' + du" + ''''^'*- du + ilin? • ^aJ' 

 which is only transformation of the well-known equation 

 _ <PV cPV d^ V 

 - dec' + dif + dz~' 

 where x, y, z signify the rectangular co-ordinates of O. If we 

 substitute the value of V, 



j^ R^P' R'^P" R^ pi" 



^ =—;a + —;3- + ~;a- +> &c., 



it is clear that for the several co-efficients, P', P", P"', &c., there 

 will Ukewise be partial differential equations, of which the general 

 expression is 



au- du smu^ d\^ 



