ig2 Mr. Woodhouse on the Integration 
nd f+jdvj (^) =/(i) 
Now, 
, z Z1 = o, both when v is o and when v is 1 ; conse- 
V(i — e* v*) * 
quently, there is an intermediate value of v , with which g 
V(I 
is a maximum. Such value of v, investigated, appears to be 
Vi'.+ VI .-«»;) — also x ; consequently, 2/ =/(i) + 1— /(!-«■) 
=/(!)+ 1 -b* 
Now, from this property of the integral of dx J 
i i 
7['+b)’ Z ~~ Vb ’ 
the whole integral be computed ; for, since x 
consequently,// 1) = s f — i -|- b 
= i +b (for, putting * =~=, .{-J±££ ) = i) 
— 2 { Dl~l Dl~* .D 2 i~ H 4 -fD 2 1“' . I) 3 1“ \ b 6 -f &C. } 
i + V(i + W 
{ 
V(i+*) 
} 
V(i-ffc) ’ 2 
2D1" * 
/Ji 2.i 
V(i + *) l 4 4 • 2 / 
1 5 | jn_ __ _j_ ^ 1 
i + L ^ 6 . 4 "T" 6 . 4 . 2 J 
( 3 ) 
2D* 1 — 2 
c .J 6 3 
VI 
&C 
This form is, in fact, the same as what is given by Legendre, 
Mem . de V Acad. 1786 ; and, if the integral had been taken by a me- 
thod a little different from the above, a series exactly coinciding 
with Legendre’s would have resulted. Thus, 
7 r j J ( i 4- b* z 1 ) dz 
since df —dz ■ 1 ^ 
(1+2* 
— — r( i?+Dii& 2 %*+D 3 i£& 4 £ 4 ~f &c. ) 
1 +2 a ]T L C J 
* I have, in a succeeding page, deduced this theorem of Fagnani from the general 
y i —e z x z 
' — 1 — Id-' 
