238 Mr. Woodhouse on the Integration 
df , df", df', &c. being similar differential expressions, df may be 
resolved into mdP-\- a, . df-\- (3 . df, (dP being a perfect differen- 
tial, m, a , ( 3 , &c. constant coefficients,) in like manner, 
df may be resolved into m'dP'-\-ot! . df-\-( 3 ' . df", 
df into - - m"dP"-\-d' . df'-\-( 3 ". cf iy , 
&c. 
and, consequently, df may be resolved into 
m.dP^-ccm' . dV-\- a a! m' m" dP" -f- &c. 
-\- au ' u " V" &C. df &c ‘ -f &C. 
This resolution depends on a very simple, and, if I may use the 
term, natural substitution, in the form dxj ( of which, 
to the best of my knowledge, M. Lagrange is the author. 
Let y =x J(± 2 LJ) ; 
then, a; 2 
e*y 2 +i 
dx 
fv/(l+2 («■— 3) /+«*/)• 
dy dy 
an ^ V( i—x 2 ) (i—e* x 2 ) i—2x 2 +e 2 y 2 V(i + 2 . (e 2 —z) e*y*) ' 
Now, if p = 1 -f v/(i— e 2 ), p*z= 2— 
if ^ i e"), 2— e 2 — 2^/(1— e 1 ), 
and 1 + 2. y) . (i — q* y 2 ) 
£?*• tfy du! 
• • (I-C** 1 ) (■-«*?*) = , 
putting y—j-, and putting X = e’, 
is transformed into du 
V[x— X 2 ) (i-e 2 x 2 ) (i_ 
similarly, putting//— i-f f(i — e' 2 ), e"— y 
and u"=p'W J 
i + V(i-/ 2 ) 
du' 
du" 
(i —/ 1 m' 2 ) p ‘ * -/(»— m" 2 ) (I* 
Hence, since e 
W (£—«*) 
i + Vl 
-—Typ — 1 + et )”T+ 7” 3 
