240 
Mr. Woodhouse on the Integration 
da' 
and the series for-^ r , df ', converge more quickly ; but, if not with 
sufficient rapidity, again transform df, and the resulting forms 
du" 
■jyr, df, may be converted into series of still greater convergency ; 
so that, by this method, we may proceed with certainty to the 
computation of fdx J (— 7 ) • if we stop at the first transfor- 
mation, there results a series for f, the same as is given in a 
very able memoir of Mr. Ivory’s, inserted in the Edinburgh 
Transactions, Vol. IV. p. 178. 
Thus, df=‘- 1 (1-K) du'- U=fL . . df, (a) 
or (!+*') du'+ ^ (A) (B) 
Now, u'—px^J which quantity is at its maximum when 
x — 7- , and then u'— 1 ; consequently, whilst x from o be- 
comes 1, u' from o passes through its maximum (1) to o again; 
consequently, J-^jr from x=o to x=i, — 9 -J~xj r from W= o to 
u'— 1 . 
Now, /A . /B . 
U' 
— /- A ih< / i-l — d 1 1 . e'* m /2 -J-d 2 i~ i . e'* u 1 * — &c. } 
_ f B *' a • du .' ( i-Z — Di + v 2 i-l . e'+u'*— &c. j 
But, by a preceding form, page 227, 
+ 
2W 1 2tt (2M — 2) 
' — i) (aw — 3) •••■ 5 
- &c. } + 
■_± r_f_ 
• 2 j Vi -u'* ’ 
2tt . (2« — 2J .... 4 
put u'= 1, and all the terms vanish, except the last; consequently, 
from u'—o to u'— 1, 
/*?'*-* 
Zi'*” rfz/ 
(2H— l) (2W — 3) .... 5 ■ 3 ■ I 
2« . (2«— .2) .... 6.4.2 
