610 Method of solving Problems in Conduction of Meat. 



6. It remains to discuss the roots of the equation 



F(«) = o-cos olcl sin /jloc(o^- a) -f sin ua cos jjlolQj — a) = 0. 



From the graphs of 



y = o cot eta 



and . y = — cot/xa(/> — a) 



it is clear that there are infinite number of" real roots, and 

 the position of the same can be determined. 



Also F(a) is an odd function of a and the real roots may 

 be denoted by 



0, ±«i, + « 2 , 



By examining F'(a), it will be seen that these roots are 

 not repeated. 



Also it is clear that F(a) has no pure imaginary root. 

 We have now to show that it has no roots of the form i; + in. 



Consider the functions U^ U 2 defined as follows: — 



Ui = sin olx, < x < a 



TT smuxCb'— x)' . / /i 



Uo= . ; 7 (sinaa, a<x<b, 



sin /*«(/> — a) 



here a is a root of F(a)=0, 

 id (r=K 1 /K 2 >, P = \/(ic\\k^). 



Then we have 



Also 



d 2 JJ 



~r^.r + « 2 Ui =0 , < x < a ; . 



•ax* 



• • (i) 





• • (2) 



U! = 0, when # = 0. . . .-■-.- 



• • (3) 



u 1= =u 2 -, 





dUj T . dU 2 [ when «=«• ■■■■ • 



1 dx ~^ 2 dx J 



• • W 



U 2 = 0, when x — h. . . 



• • (5) 



And 



Let /3 be another root of F(a)=0, and V 1? V 2 the corre- 

 sponding functions. 



