48 ilf?' Watson, Bessel functions etc. 



B (ii). The function 



g,{d) = 2s (7 + 3c) (/)-^ - 3 (3 + 2c) (1 _ c)^ <^ + f s (1 - c? 

 is not positive. 



To prove B (i) we observe that 



^ [g, (6)1(85 + 163c + 84c-^ + ISc^} 



= - s- (1 - cy (644 -I- 416c + 60c-)/(85 + 163c + 84c- + 18c--)- 



The denominator may be written in the form 

 I873 + 3O72 + 497 - 12 

 where 7 = 1 + c, and so the denominator changes sign once on 



i 



when < ^ < TT, say a,t 6 = ^. Hence 



g, (^)/(85 + 163c + 84c^ + 18c0 



decreases from to — czd and then from + 00 to tt as ^ increases 

 from to yS and then from /3 to tt. .Hence gi(6) cannot be 

 positive. 



To prove B (ii) we observe that 



^ [15^5 (0)1 [s (7 + 3c)|] = (1 - c) g, (0)/{2 (1 + c) (7 + dcf} ^ 0, 



by B (i) ; and so g^ (6) ^ g^ (0) = 0, as was to be proved. 

 To prove the main theorem, we have 



where g (6) = {(3 + 2c) f ^ - s (1 - c) (f)^ (1 - c)-\ 



Now g' (6) = - ^a (0) f " (1 - c)- ^ 0, 



so that g(0)^g (0) = 64/6i 



and so // (^) ^ 0, as was to be proved. 



(C) To prove that f(<j>) = (f>i sin 6 . (1 -cos (9)-i is a de- 

 creasing function, we have 



^^ = i<^-ni-cos^)-^3(^), 

 where g^ (0) = sin ^ (1 - cos e)-S(d- sin 6). 



Since g/ ((9) = - 2 (1 - cos ey we may use the arguments of (A) 

 to prove the truth of theorem (C). 



