{n-m)(n-m-l)(n-m-2)in-7n-3) n-m-i \ m 



204 C. F. GAUSS ox THE GENERAL THEORY OF 



From this equation^ combined with the remark in the pre- 

 ceding article, we obtain the general form of PW. If we repre- 

 sent by P"/"* the following function of u, 



/ n-m /n-m)(n-m + l) n-m-2 



( COS U — ^-j^ TT COS u 



\ 2 (2 n — I) 



n—m—4 

 '-'J 

 — -nns'i 



2-4{2n-l) {2n—3) 

 then P^") has the form of an aggregate of 2 w + 1 parts, 



pin) _^«,0 pn,0 ^ (^n,l ^qs \ + h""'^ sin X) P"'^ + 



(^"'2 COS 2 X -I- A'*'2 sin 2 X) P»'2 +, &c. + 

 (^"•« COS wX + A**'" sin nX) P"'% 

 where «"''', ^"'^ A"'^, ff^^'^> &c. are determinate numerical co-effi- 

 cients. 



19. 



If the magnetic force at the point O be resolved into three 

 forces perpendicular to each other, X, Y, and Z, of which Z is 

 directed towards the centre of the earth, and X and Y are tan- 

 gential to a spherical surface concentric with the earth, passing 

 through O, X directed northwards in a plane passing through 

 O and the axis of the earth, and Y directed westwards in a 

 plane parallel to the equator of the earth, then 



Z-_— Y= ^^ Z=- — • 



"" r d ?«' r smu dX' dr' 



consequently, 



X--~ T-^ ^— — ^ ^^"' &c^ 

 ~ /"^ \ du ' r ' du r''' ' du ' ' J 



7^&mu\d\ ^ r ' d\ '^ r'^ ' d\' ^' ) 

 ^= p (^2P' + — ^ + -^^-, &c.j 



On the sm-face of the earth X and Y are the same horizontal 

 components which we have designated above by those letters ; 

 Z is the vertical component, which is positive when directed 

 downwards. 



The expressions for these forces on the surface of the earth 

 are, then, 



/rfP' </P" dP'" , \ 



X = — \ -r- + -j — + —J — +, &c. ) 



\du du du I 



