DEFINITE INTEGRAL OF THE VALUE OF AN EXPEESSION. 103 



Now, from the deliiiitious of <l> (x, m, n) aud qj (x, m) we have — 



cp Çc, m, n) ^ (fi (^x+ 2 > "^ + « } — « <p \x + "— 1, m-\-nJ+ 



+ (— 1) ^n~r I r V (■'■-' + IT ~ ''' "' + ") + •■•• +'"" ^^ '^ y' " 2' '" + ") 

 Heuce by (iii.) — 

 f ^<l> (.r. m, n) </x = y> (r + '-M^> '« + n + l) - y. (.'-■ + ''-^, m + « + 1 ) 



_« |,^. (,, + 'A+i_l, ;« + «+ l) - r/. (.,■ + '^1 _1, ™ + « + l) }+ 



+ {-\)\-r^\ 9 (■^■+'^4-^ - '■' '" + « + 1) - 'p (■'■+"^ - '•. "^ + «+ 1 ) I+-- 



+ (— 1) \ <p {.i- + " 2 — /(, JH + /( + l) — '/^ (■'• + '*-^- n, m + /(. + 1 j ( 



= </^ (.-;+ 'i-±-\ m + « + 1 ) - (M + 1) .p {.v + 'l+J^- 1, m + n + 1 )+ 



'' I M + 1 / ?t + 1 \ 



+ ( - ^) ln+T:^r,^r <^ (■"+ "V " ''' "^ + « + + 



= <^ (.(.■, /)(, ;i + 1) (v) 



If, theu, we obtain any fuuctiou which is equal to <p {x, 0), where </> (./;, 0) may have 

 the more extended meaning last given to it, we can, by the aid of (ii) aud (i), obtain a func- 

 tion which will be eqvial to (p {x, m), and then by the aid of (iv) and (v) a funi'tion equal 

 to <p {x, m, n). 



Now the expression — 



^^ f_ cp (V) dv + i 2^ /_ ^ ^ (0 cos "^^EÂE^ do 



where ^ represents summation with respect to «, is known to bo equal to <p (x), for all 

 values of .-c between the limits — / and /, except where any discontinuity occurs in <p (x), 

 and there the expression is still finite. 



Hence, when x is between the limits — / aud /, we have — 



( I 

 V (-i-, 0) = ,- J dc + — ^ J cos ^-^ du 



= i- + — - 2: — Sin — H ' -\- —- -5, — Sin 



"I Inn I I Inn I 



, , ^. ■' 2 . (2n— 1) TTx 



1 (2 ft 1) TT I 



