2 56 Mr. Woodhouse on the Integration 
j-— ■ _JL_f 1 _ ) . 
“ i+a/H V(i + 6) J ’ 
which conclusion agrees with Legendre's, obtained by a different 
process. See Mem. de V Ac ademie , 1786, p. 665, 
The foregoing method may be continued at pleasure; thus, if 
thenu"=y[-^f), and »'*= }; and, 
putting this value = m'\ x 2 must be determined from the equation 
o: 2 = \{ l ~ m e > ; anc j, similarly must the process 
be conducted, if u iy , or u y , or u VI =i. 
These results, applied to an ellipse, cause it to appear, that right 
lines can be assigned, respectively equal to the difference between 
an arc and half the quadrant, between an arc and one-fourth of 
the quadrant, between an arc and one-eighth of the quadrant, &c. 
Here may again be remarked, the connexion between the arti- 
fices of computation and the properties of curves; for the series 
expressing fdx^/^ (c ceteris paribus) converges more 
quickly, the less x is; consequently, the whole integral is more 
commodiously calculated by the theorem /(1) = 2 f(x— 
— 1 + 6, than if x were put = 1 , in the form of the expansion of 
dx (— - t~t) ; still more commodiously, by the theorem 
/(*) . = ^j\x=:a)— 2(1-^) + Vby, 
where i - -^7^ } is less than and so on. 
It has been already observed, that the methods of determining 
/, by/', and/", or by/', /",/"', or by/",/'", &c. as Legendre 
has done, or by the regular form which the indefinite reduction of 
/, into /(*”*),/(”), assumes, are, aufond 3 the same methods; and 1 
purpose now to show that the substitution, which is to be considered 
as the base and principle of the method, is the same, although dif- 
