of certain differential Expressions, &c. 
269 
since F(i) = (i+cf')(i+*") .... f = P . 
A p ( ifabe put=i)= T ^=(i + 0 {!+«") «*• 
Ao-ain R — 1 + e " zF ^) __ i±£ 2 /(0 . 
Again, .D e >. I+e '' w e' • v » 
but /( 1 ) = T-( l -«Q) P > 
... B = 
2P 
:^-o 
0+0 
2P r .4 1 , e' • e " . e ' . e " . 
or ’ = T^lT + 777 + 7777 
&c.} s 
or =.(1+0 (1+0 (1+0 &c.{ z+Lf + L^C. + &c - } 
which agrees with the result given by Mr. Ivory, Edinburgh 
Transactions, Vol. IV. p. 187. 
If, instead of the series used for F(i), /(1), we employ the 
series ■ 
• hyp- log. ( ^, 
/iJS±2 
l 2 2 
b .'b . (1 +'&) 1 ■+» & 
+ &c - ) + hyp- log. 
2.2.2 
we shall obtain expressions for A and B, which, in certain values 
of b, are more commodious for computation than the preceding 
expressions. 
In like manner, if 2w-fi= —5, 
A _ 8-K-J-fQ ft , ^ __ . 2.F( P 
3. |af_ 5 ? 7 3 . (a— 6) ary ( ^ 
a 4 + 14a 4 Z> 4 +-Z> 4 
6 ’ 3. (a 1 — 6 4 ) 4 (fl-j-6) 
b= 4-. 7. H f:7 4 7,^ /(i) 
a (a 4 + Z) 4 ) 
3 . b [ a 2 — b 2 } 2 («+&} 
F(i). 
Since N = . (a . ( 1 ) 2m + l fR zm + 1 . cos. . c?0 ; by what 
has preceded, N may always be determined by a direct process, 
and independently of the preceding terms. For the purposes of 
computation, however, it is commodious to deduce N from the 
