2 5 § 
Mr. Wood li ouse on the Integration 
form J ; which substitution I conceive to be more 
obvious, more easily suggested, and more analogous to ordinary 
algebraical substitutions, than the substitution of c — j 
for the sin. 29', or, of - sin ' - . for sin. i|/. 
Of this substitution of id for an ^ trarj sfor- 
mationof dx J (- ^~ ) into Adu'+dx'^^- ^ - * - -^ , &c. M.La- 
grange is, I believe, the original author. 
When x is called the sine of an arc 0, dx J j may be 
expressed by sin. % -^ y by - ^ (I _ g . sin , , e) . 
Lagrange, Fond. Analyt. p. 90, has treated of the integrals of 
these expressions ; as has Legendre, Mem. de l’ Acad. p. 663, and 
Lacroix, Traite du Catcul diff . Vol. II. page 454. 
The results obtained by these authors, may easily be deduced 
from the substitution of x\/ {- ~y c r^ ) = v J pa ). Some of 
these results may appear curious ; but I apprehend, what is chiefly 
necessary for the solutions of problems in physics and astronomy, 
into which the expressions {l _ ' e ~ x -y dx J ( " L ~i-^ 1 ) 
enter, is a method of approximating to their integrals. 
A certain method of approximating to these integrals, has been 
given in the preceding pages. In different applications, its expres- 
sion may be varied; thus, / is transformed into an expression 
involving/', /", where /', /", can be more easily computed than 
f ; express this transformation with reference to an ellipse, and 
it appears that the length of one ellipse may be estimated, from 
the lengths of two ellipses of different excentricity. Again, 
f-4r — Tw 5— in order to be computed, is transformed into 
J VC 1 — •* ) C‘- e * ) 3 tr ? 
