528 Dr. J. W. Nicholson on the Bending of 



Vanishing of the derivate of an exponent. 

 It has been shown that 



This may be reduced to a dependence on series of the above 

 type for all harmonic terms of high order, by the use of the 

 asymptotic expansion for the zonal harmonic, valid when 9 

 is not small. But this process is not justifiable until it is. 

 proved that the harmonic terms of low order do not contri- 

 bute substantially to the magnetic force. Some other 

 expression must therefore be used, and the Mehler Dirichlet 

 integral formula is at once suggested. Writing 



P(V) - 2 f cosmcft^ 



and defining the action of an operation g (6) upon any 

 function co by 



Then 



(ff\ _ 2 ' sm2 # d C 9 *d4> /QQ . 



#JW " *«V dp\ ViV(cos^-cos^)- ' (38> 



yp = g(6)X m(R n n nr )h(e l *»-*™ + e l **-*™ + 2Xn ) cos m<l>, 

 and 



yp = g(6)f u(e tV * + e iv * + e"* + e lV +), 

 i 

 where 



Vi = (/>« — <j>nr + m<f> 



V 2 = 0n — <j>nr-4>n — ™</> f" (^ 9 ) 



l? 3 = (j) n — <j) nr + 2^ w + ??l<£ 

 ^4 = <£n — <£nr + 2% ra — m(j) 



It is now necessary to determine all possible cases in which 

 the exponents of these series can have zero derivates' with 

 respect to as (the case v' = of the summation formulae). 



Let n-\- 4 = m =2 sin a as before. Then when z— m is of 

 higher order than 2*, a being between zero and \tt, 



(j) n = z cos a. — \nir 4- ma, 

 taking the main terms of cj> n only. 



