ARE EMPLOYED IN THE THEORY OF TERRESTRIAL MAGNETISM. 245 

 / (p.) is the coefficient of r n in the development of 



in ascending powers of r. 



3. Let ( I 2p,r + r 2 ) 2 = 1 rx, 



then 1 2/xr + ?~ = 1 2rx + rx- ; 



2p,r = 2rx + r~ (l xr), 



or fj. = x + -(l-x') ; 



dx 



so that X = =(l -2ur + rV- = V. 



dp lrx 



Since x = p. - ( I x"), 



we may develop x in terms of /A by Lagrange's theorem and get 



-( "-} l ( r \ S d 

 1 . 2 \2/ dp, 



Differentiating with respect to p, we get 



Hence the equation 



gives the value of the Legendre's coefficient. 



4. Several convenient relations may be found between successive 

 Legendre's coefficients, which are useful for determining or checking the 

 values of these coefficients. 



Thus 



log V= - log ( 1 2pr + r"). 



Differentiating we get 



I ^Z 



V dr '' 



