Harmonics oj the Second Type. 

 •expansion because the integral 



a/ — 1 " 



exists, and in fact can be determined readily. 

 For we know that if p is any integer, 



Moreover 



J' 1 P P ( / X)^ = 0. 



i: 





*Q» • 



Near /*= + !, — ^ is infinite like 



Thus 



dfju 



P,.Q ) 

 1-V 



or like 



P»(D 

 1~> 2 



Q„(/a) d/z, = — — ? — —4- = if n is 



even. 



^(71 + 1) 



Accordingly, reverting to the definition of Q M , 



if n is odd. 



i 



t/-i 



P M (»log — '-^dfl 



= if n is even. 



if n is odd, 



'This is equivalent to 



P»(/0 QoO*) ^ = 



j: 



_2 



n(?i+l) 



or zero, 



and enables us to write down the expansion of Qo(aO at once 

 in the form 



-QoW = J72 Pi(a*) + o Ps(m) + O P ' (/i) + • • • • ( 2 ) 



This result is quite elementary. For the general expan- 

 sion, we return to the differential equation, and can show 

 readily that 



(m — n)(m + n + l)\ T n {fju) Q hi (aO dfi ■ 



J-i 



= [(l-^((tP/-P.Q«')] 1 ', (3) 



where the only terms in the bracket, not obviously zero as 



B 2 



