DKFINITE INÏl-XIJRAL OF TlIK VALUK OF AX FXPRKSSIdST. 109 



1_., = — 81 /— 1- (S /•' — 10 /■•' + 5) £ - 



These values ayree with Camhy's. 



Now it is easily shewn, bv repeated intégrations of J- . , . between the limits x — h 

 and x + h, that , ~ 



II " +* 



X J " 2 .r. .n . X- rr ii _ 1 



" ~[»+4 ~ 48 "*" ~wr + iTT . 144 ~ 16 ' âW) 



= (4)' |^-^rff-tJ..=-à| ,i„«.= '^ 



Differentiating with respect to x or — ^j— we get 



71+3 



|«+3 ^ •" ^ 12 



n + 2 

 ^ . Ms 



So i^" :L 2^ =(i/3^il„ 

 [n±2 ^' ' 12 " 



|w+l — i ■ -2" • " 



n 



n 



If we substitute these values, and those of I„, .^ j, &c., iu equation (xi), which may- 

 be written 



m 4- » 



we obtain the following approximate values for (p (-^ ^/^m, ■») that is for 



_ 1 1 ( ^ " + "' ,^ 1 n + ,„ , „ 4. „, 



^„+.. • | »,-f-n j (/• n + «) — ~ • (,-„ + „_2) + "^J*— " (;-rt4-,j_4) — ... [ ' 

 the series to be continued only as long as the quantity raised to power // + in is positive, 



