314 PR3CEEDINGS OF SECTION A. 



A more useful form of integral may be obtained by changing the 

 variable from 9 to r by means of the relation r dr =. — ab sin.0 dO; 



Also since x z=. h •\- a cos. 6, 



r'^ _ l^— ^^r-' - h-' J^ a"-) 



Hence we obtain the general integrals — 



= Jff'iF{,-)dY (8) 



b - a 



§ 3. Let F [_r) — r" and yl = O 



[ r"- F (r) dr - f r" ^ - f/r = "—ll 

 J ^ ^ J n + 3 



= ?^^ U + a) '- 3 r/,^ + 3) « - 6] 



5 (n 4- 2}(n + 3)^n + 4) ' ^ ^ LV ^ > J 



+ (^, _ „)'-3[-(,,^ 3)« + ^] j (10) 



This expression takes the form — if « ;= — 2, or — 3, or — 4. 

 For these particular cases we have — 



*^ o - a ^ 



= 2 7rff ^ — r ' log- 7 (11) 



h — a 



JJJ [11 + 3)(« + 4) 



