TO THE THEORY OF MAGNETISM. 101 



In the finite integrals just given, the increment of * is 1, and 

 the corresponding increment of 7 is - = dy (because a is in- 



finite), the finite integrals thus change themselves into ordinary 

 integrals or fluents. In fact (Mec. C6L Liv. 3), U always 

 satisfies the equation 



and as 6 is infinitely small whenever V has a sensible value, 

 we may eliminate it from the above by means of the equation 

 ad = v, and we obtain by neglecting infinitesimals of higher 



f\ 



orders than those retained, since - = 7, 



dv 2 vdv 

 Hence the value U ] is of the form 



seeing that the remaining part of the general integral becomes 

 infinite when v vanishes, and ought therefore to be rejected. It 

 now only remains to determine the value of the arbitrary con- 

 stant A. Making, for this purpose, = 0, i.e. v = 0, we 

 have 



Ur = (a- by and ^ ^ = 4* : hence (a-bj 





By substituting for A and r their values, there results 

 Z7/V" = ( - 5)' (a - 6 + )^ cos 



