of certain differential Expressions, See. 241 
3 • 14159+ 
= =+=D*i“* x ± . e f2 \~ = whether n be 
du' 
JY l—l . D^ 1 1 '~'^e f%n .~ 
C C 2 
even or odd, jd* 1 j 2 e ,2n . ; 
similarly, f± d* i~K e ,2 \ u /2 "+ 2 . y (i _ m . 
Hence, putting for n the several values o, 1, 2, 3, 4, &c. the sum 
of the integrals from u'= o to u'= 1 
= + (Di-i)*e'*+(D'i-ij-«" + &c. } 
+ — ^-{nl — D“ l - * e"-|- D“ 1 — 4 D 1 1 — » c" -f- &C. } 
or, putting for A and B, their values — ‘ the integral 
+ 2D 1 
( 
and, generally, the coefficient affected with e' 2n is 
' ' + 2D*~‘ l~i . S’ l _i + (D* I"*)' = { D”-' 1“*+D' l~i }*, 
I 
f 1 1 
W ) . _X 
= 7 TTT(T+ 7 r) 1 + 2D1 
V I 
+ (di-») : 
. D 2 1' 
c 
+ &c. 
D 1 
1 c 
but — (2ft— -l) D" if = Y 1 ~* r 
and 2 ft . d” if = D”~ 1 i”* 
D" 1 = jy’ 1 - 1 1“* 4 - D” I~i 
£ (T 1 C 
Hence, the coefficient affected with e' 2n is jD”if) 2 ; and, conse- 
quently, the integral from u' — o to u'~ 1 
= 1 + ( Dli )‘- e "+ (?* !i )‘- «'*+ (?’ »*)*• «'*+ &c - } 
the double of this is the integral (/) from x = o to x — l, 
or fdx J (-— “r) from x — o to x = 1 
= T^+ 7 'F { 1 + ( D1 ^) 4 - *'*+(?* **)*• e ' 4 + (? 3 ^) 2> e ' 6 + &c -} 
or, developing the symbols Dif &c. 
I i 2 
