of certain differential Expressions, 
Hence, 
d f=T+?- du '+ UT7' i(.+,T -^" 
3 43 
i + e' du' , 
2 + 
£_ 
i + 
i+e" i 
du." 
i+e' * U" 
[ du' t i dti” \ , i 
1“+ i + e' [T* 
(i+e') (i + e") 
— 7+7' • du ’+ (7+77(7+77 *' [r+TTfl+oTT+Tq &c * 
du 1 " 
, i + e' du' 
r • “if 
du! 
i+e * 
2 
du!’ 
i du!' 
r+7*“ 
i 
i+e" 1 
z (i+e) i+e" 
~{ir+ tt7'‘9- + (i+o (.+<") • ^ + &c -l 
U=r + &c - 
‘ (*+«') (i + O &c. 
XT _ du' i+e" du" ( i + e") (i+e'") du 1 " 
MOW, — — 2 • — — 2.2 * "U 5 "' 
= ( I + g H 1+e ) . / I+e) ( g} v> v, representing the last terms 
of series, e', e", e'", &c; u', u ", u &c ; U', U", U'", & c.), 
8 _____ e g . / e". &c. (i+e') (i+e") .... (i + e) 
(i+e) ( i + e") ... l+s 
E a K* dv 
4 . 4 . 4 . &c. 
let p=(i+0 (i-K') ..... (i+0; 
and,df(")= 
then, 
<*/= -f . ( x +,') .*+ . *-+ 
-f &c. (d£w') 
+f{* +wf + t+ft. 77F + ^ + &c '}- r 
M° + 
7- + 
(1 + er ( 1 +eT ' P 5 
2 a . 2 " (i + e) 
i+e 
consequently, 
dj=dzu' + l r .4- 
2 1 e' 
(i + e') z (i+e’) 
8* * 
Z”p . V 
P f 2 e' i 2 1 e" i o 1 
~Fl(7T7r+(.+7F(~r + &c */ 
_L £ * . 
-r p • y * 
and, consequently, since = iTpFT.T? 
&c. 
