and the Dynamical Theory of Diffraction. 417 



Writing c=a—ib we readily find, as others have found 

 before for this equation, 



t n f . nn + l n(n 2 -l 2 )n + 2 n(n 2 -l 2 )(n 2 -2 2 ) ) 

 | =C 4 1 ^2^ +A 2^-^- 2.4.6 + «^ 



The series ends after n + 1 terms; and so we can put it in 

 the form 



- nil <cl-%n% °y + &c - } • 



The following properties of the functions G n are useful. 

 Writing c=a—ib, we have 



2n + l *^ _1 ~~^ n+I ' ; 

 ^ =c |_C„+^ 1 [(n+l)C„_ 1 +nC tt+1 ]}= C O„_ 1 -C„0 + c) 



C. +1 =C„(l-^±l) + i^. 



\ cp / c dp 



Putting cp = cp\/l + 2s, we have 



Extending the notation so that we should write G n (p) in the 

 place of G n as above, we have, if we put^>c5=^, 



C oe «p=^|Co+fCo+ 3 ^0 1 (p) + T -|-3C 2 ( i >)+&c.}, 



pr ) ^=^Si{^(i>) + fC M+1 ( i >) + I ^a +2 ( P ) + &c.}. 



These expansions are useful in many calculations from the 

 equations, and I believe they are given here for the first time, 

 together with the differential expression from which they are 

 derived. 



Hence we have obtained a complete solution of one of the 

 equations. Now the general value of V~( n +i) is 



d n 1 



By varying the axes h }J h 2 , A 3 , &c, the value and form of 

 V-(n+i) will change, and can thus have many values while C 



