956 DR JOHN DOUGA.LL ON AN ANALYTICAL THEORY OF THE 



the order in a increases by tivo units from term to term of the series. The terms 

 which involve z-derivatives of P(2) of negative order (so to speak) are just those 

 which come from the permanent terms of the point source solutions. 



If V{x, i/, z) is a rational integral function of 2, the series (5) terminates, and we 

 obtain a particular solution in finite terms for Z]<.z<Z2* To bring out this particular 

 solution explicitly we only need to expand the functions U^/D^, etc., in ascending 

 powers of ^. 



In the case when P has an unlimited number of successive ^-derivatives it may 

 happen that E„ tends to zero as n increases indefinitely. For this it is easy to get 

 a simple sufficient condition. 



Thus in Il„, the last line of 39 (4), the first part may be written 



U,„ 1 _fcil)-] 



e a 



and the /3-summation can be replaced by 



dz', (7) 



~ -\e~ a dfS, (fourth path of Art. 6) (8) 



Now consider the power series in the variable ^, 



2a^|p(.)r (9) 



Suppose that the radius of convergence of this series is ^0, where 



^o>V^-t (10) 



We shall show that this condition ensures the vanishing of R„ for n infinite, and 

 the consequent convergence of the series (5) and its term-by-term integral, with 

 respect to the accented variables, over a cross-section. 



For, by (10) we may take a positive number b such that 



^<^<^, (H) 



to 



Then the "fourth path" in (8) can be taken to be a semicircle of radius h about 

 the origin as centre, along with those parts of the imaginary axis for which | ^ i> h. 



From the consideration that | h//3 \ does not exceed unit)' at any point of the path, 

 the integral is easily seen to be a finite multiple of 1/6", say M/6'\ where M is finite for 

 n infinite. 



Hence 



(7) = (-])"J'm|^|>(^')cZ.' (12) 



Here the integrand is a finite multiple of the general term of (9) for ^= i/b, that is, 

 by (11), for a value of ^ less than the radius of convergence of (9). Thus (7) vanishes 

 for n infinite ; and similarly with the second part of R„ in the last line of 39 (4) ; the 

 statement after (10) is thus verified. 



* (Jf. a method due to E. Almanki, PJm-nMo}). d. Math. IViss., iv. 25, I3d. 

 t Here h^ means the least moduhis of a zero of D,„. 



