BOUNDARY PROBLEMS AND DEVELOPMENTS. Ill 



it Is found by precisely the same method that V = U. In consequence 



we have for the Green's timet ion 



( ( <& I 



(99) G(x, t, X) = Y(x) \\ or \ + A- 1 W a Y{fl) (s£) 



-A- 1 W b Y(b)(6%) r Z(t), 



where the upper form is to be chosen when t < z, and the lower one 

 for t > x. 



By means of this formula G(x, t, Xj may be explicitly represented bj 

 choosing as the solutions Y(x) and Z(<) a pair which are analytic in A 



and have the forms 



obtained in sections V and VI. It should be observed that G(x,t,\) 

 is unique (see page 70), despite the fact that if n ^ 2 the choice of 

 Y(x) and £(/) thus determined upon changes from anj one to any 

 other of the sectors within which no quantities li\K\yJ.r) — y/xj\\ 

 change sign, and despite the fact that the values of o, t and o,] change 

 from any one to any other -eetor v^. 



From (95 we have, upon denoting by S m (x) • the -um of those terms 

 of the formal development of F(x)- which correspond to the charac- 

 teristic values enclosed by the circle C in question, 



b 



(100) SJx>- ='f-l G(x, t, \) R(t) F(t)'dt rf\. 



C a 



Substituting in this the value of G(x, t, X) as found above in (99) it is 



4 



seen that S m (x)- = £ S)2(x)", where the quantities S«(«)- are given 



t-i 



by the relation- 



