of certain differential Expressions, &c. 
239 
dx 
V(»- 
i+e' 
~ i"— ma y transformed into 
or mto 
V(l-w' l J (»“ 
!+<') ( 1 + 0 ") (!+*'") 
_ or i l+g,) ( 1+g,,) du " 
'») ’ 2.2 ‘ /(i2_m" 2 ) (1 -e n u"*) » 
. I+«W rfu(”) 
. 2.2 .... 2 * ^/( 1 — mW 1 ) (I—. e (") 1 «W Z ) 
And, similarly, putting U / = v /(i-^w' 2 ) (1 — e'* u'%) 
U"=v/(i — w"*} (i-«"*k"*) &c. 
(A + Bx 1 ) . d* 
\/( 1 — (i — e 1 x % ) 
f2A+B\ du' B j .. Be 
• Tp r~ . du' 4 - 
U 2 p * 
2 /. 
may be transformed into 
u'^.du! 
tp"' XT ’ 
/2A+ B 
Be 1 A' \ 
1 2 P 
2 P 3 B' ) 
or, to render the last term like the original form, into 
r-5*'+ w( A '+ B '“"),^- 
And, into a form exactly similar may (A ; + B' u /2 ) . 
be transformed. 
Hence, to transform df or dx J ( ^ — -f* ^ - T , 
’ ^ V ^ 1—X r ! yp_# a ) (X— e z x z ) 
A = 1 , B = — e* ; consequently, 
^ • -^ + 4 ( * ^ 
or, since ,/(i— <*) = -^r and T = *TT7" 
similarly, df' = -C-. (i+t")*"- (-^J; + -7+7“ • d S" ’ 
df"= (].+<?'") du" 1 — &c. 
The utility of this transformation will appear, by observing that 
the quantities e', e", e 1 ", &c. continually decrease; thus, 
e’ — -i=y(.L±f!L — _g * g 
i + V['~e z ) [i + Vi'—e*)) 2 ’ (i + V(i-^)) 2 ’ 
Hence, if e be a fraction, e'= aa fraction ; consequently, 
e 1 is z. e ; similarly, e 1 ' is z. e', e"' z. e" &c; hence, if the series for 
fdx J [—7 ~t) does not converge quickly, transform df as above, 
I i 
MDCCCIV. 
