solvit! [i Problems in the Conduction of Heat. 609 



where 



F (a) •= a cos aa sin fia{b — - a) + sin aa cos //.a (b — a) , 



and these integrals are taken over the standard path (P). 

 The second integral reduces to 



Vq i V cos ota sin fia(x — a) 4- sin aa cos fia(x — a) e~ KiaH , 

 2 f7rj cr cos aa sin /ia(/j — a) + sin aa cos yita(6 — a) a 



over the path (P). . .: . (8): 



The values v ± and v 2 given in (6) and (8) satisfy all the 



conditions of our problem : for, from the way in which they 



have been built up, they obviously satisfy the differential 



equations (1) and (1'), and (2), (4), and (5). 

 Further, putting x = b in (8), we have v = v Q . 

 We shall prove below that the roots of the equation 



F(a) = a cos oca sin//,a(/> — a) -f sin aa cos fia (b — a) =0 (9) 



are infinite in number, all real and not repeated, and it is 

 clear that to each positive root there corresponds an equal 

 and opposite negative root. 



Assuming this to be the case, the same argument as in § 2 

 will show that the initial conditions (3) and (3'j are satisfied. 



Finally, the solution is obtained as an infinite series. 

 For we have, from (6) and (8), 



v Tsin olx e-^ aH 7 



^ij 



v (Vcos aa sin yua{x — a) + sin aa cos //a(#— a) <?~ Kia2 * 



F(a) ~* d *> 



over the path (Q) of fig. 2. 

 Therefore 



Vl ~ crn{b-a)+a + °7 F 7 ^ ^" ' 



<rfi(a; — a)-\- a 

 — a) +a 



cr cos a„a sin //,«„ (a? — a) -f sin a n a cos /za„U 1 — a) e~ K i tt * H 



v 2 = v 



074(0 — a) +a 



+ 2r 2) 



i -b (a„) a K 



the summation being taken over the positive roots of (9). 



