SECTION I. 



USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 ARE EMPLOYED IN THE THEORY OF TERRESTRIAL MAGNETISM. 



1. IF r be less than unity, and if 

 be expanded in a series of ascending powers of r, then 



where P , P l} &c. are functions of 6 only, and are known as Legendre's 

 Coefficients or as Zonal Surface Harmonics. 



When r is greater than unity, 



F=-(l- 2 cos0+ 

 r r t 



= P - + P + 4-P -- 4- "? 



r o r T f-i r -, -i- . . . -t- -r n . ^ n+1 -r ^ 



Putting cos = 1*, we get from Laplace's equation 



2. Let p. be a variable which changes between the limits 1 and 1, 

 and let /(/A) be a function of p. of n dimensions, such that when multi- 

 plied by any function of lower dimensions the integral of the product taken 

 between the limits - i and 1 shall always vanish, so that 



^O, &c., 

 the limits being 1 and 1 in each case. 



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