202 C. F. GAUSS OX THE GENERAL THEORY OF 



The foregoing investigations apply only to the horizontal por- 

 tion of the earth's magnetic force. In order to embrace the 

 vertical force also, we must consider the problem in all its gene- 

 rality ; therefore V must be regarded as a function of three vari- 

 able magnitudes, expressing the position in space of an undeter- 

 mined point O. We select for the purjDose the distance r from the 

 centre of the earth, the angle u which r makes with the northern 

 part of the earth's axis, and the angle X, which a plane passing 

 through r and the axis of the earth makes with a first meridian, 

 counted as positive towards the east. 



Let the function V be expanded into a series, decreasing ac- 

 cording to the powers of r, and to which we give the following 

 form : 



R'- P'^ R^P' R'^P" R^ P'" o 



V = + 5— -4- o + i — , &C. 



The co-efficients P^, P', P",'&c. ai-e here fiinctions of u andX ; 

 in order to see how they are connected with the distribution of 

 the magnetic fluid in the earth, let d fj, he an element of the 

 earth's magnetism, p its distance from O, and let r^, u^, X^, 

 signify for d fi the same as r, u, X for 0. We have then 



V = — I — extended so as to include every d fx ; further 

 ./ P 



p= v' (r^ — 2 r r9) cos u cos ifi -f- sin u sin iP cos {X— X^) +■ r° r^, 



and if - be developed in the series, 

 P 



J ^ i ( TO , 7^/ L , T".-~+ &LC.) 



p r^ ^ r r^ ' 



then R^ P^ = - rT° d fju, R^ P' = - fr' r" dfM, 



R'i P" = - Ct" r^ r^ d /u,, Sec. 



As y = 1, and as according to the fundamental supposition 

 with which we set out, the quantities of positive "and of negative 

 fluid are equal in every measm-eable particle in which they exist, 

 and therefore are equal in the whole earth ; that is to say, 



/d /i = 0, it follows that 

 po = 0, 



or the first number of our series for /^goes out. 



