ON MAGNETISM. 



Now forming the values of N, W, C, and remarking 

 that (as the integral with respect to fyt applies only to 

 elements entirely independent of u, X, and r,} the dif- 

 ferentiations with respect to u and X can be performed 

 under the integral sign, and the differentiation with 

 respect to r will be entirely external to the integral 



1 dV 



sign; the general term of N or . -*- will be 



T Ct/U 



B n+ * dP^ 

 r^'~du> 



1 fJV 



that of TFor = . will be 



r sin u d\ 



1 fi* 1 * dP n 



sin u ' r n ** ' d\ ' 



that of (7 or -j- will be 

 dr 



Also it is to be remarked that T is 1, and therefore 

 / T 9 S/i is (because the total amounts of red and of 

 blue magnetism are supposed to be equal), and there- 

 fore P l is 0. And, if our needle be on the earth's 

 surface, r = R. Thus we obtain 



du 



sin 



C=+ 2P X + 3P, + &c. + (n + 1) P n + &c. 

 where r w+2 .P = -r.r n . 



