DEFINIT!-: INTEGRAL OF THE VALUR OF AN EXPRESSION. 



105 



Adopting this notation, but (unitting' sporific mention in each case, of the limits of 

 integration of the extraordinary integrals, wliiih will, throughont this paper, be between 

 the values zero and inlinity of the variables, e(juation (viii) becomes 



fp (.-■, Ill, n) :^ I 



I'V n m +1 « „ 



--' V^' — •>) ., " ,, ros (x — ._, tt) (sill .f ) 



— 1- H- "^ / ,"'^, "^ ilH 



m + K ' TT </ III -r II + 1 



w 



for it is obvious from the way in which the terms 



n if III -^11 — 1 



^ (■'■-.) .1,1-1 



\ ni+n—\ ' ^ ' — ■■■ '^'^■ 



were introduced, that they are the earlier terms in the expansion, in ascending powers 

 of X, of 



ros {.r H — ,-, tt) 



\ . H^ " 



It now remains to deduce from equation (x) the approximate value of 'P (.f, »i, n), 

 when ». is large and in a small integer. We shall suppose in what follows that x is of the 

 order \/i^, and we will put 



vSubstituting this value of ./: in the integral occurring in (x) and writing 2 '^ for ^ 

 we get 



qj (.1-, m, n) = ^ 



I m + n 



+ 11 



- ^ JM + 1 - . lit, 

 + ' / : ^ - dO 



'> "• TT I III + II -i- I 



H 



.>' --TT W 



n H" 



Eut sin H — H f '■• ^1 — ^^^^ — ko.^ so that 



+ 11 



Cj){.V, m, 11) := \ 



^ (■'■ - ^ 1 .- COS (r v/« fi - ''^-'^) ■ ' 



n «- 

 (J 



(1- 





If in this last equation we change ^ into -y^, we obtain 



(fj (x, m, 11) 



J Cv - ^) 

 m -\- n 



, ,1 111 -t- 1 s 



ros (r H — — I — TT) 



I- ^',/H 



6- 



, cos (/• ^ — — r, — tt) — (j 

 fi 



(xi) 



180n -^ n'""-" J 



The series of extraordinary integrals in (xi) proceeds in inverse powers of n. and 



Sec. HI., 1885. 14. 



