Mr. Herschel on various points of Analysis. 
452 
a 
\ *2 * 9 9 • X 
— I 
v/— I + I log ‘I 
: 0 
Now we have 
D(/log t = 
(/log-): ^ = 
V^-l . 
(\/— 1 + 6^)3 
1 ^" (/ log“^): ^ = 
°A . + * A . ,(-^- 0 ^ + . . . -^—lA . 
(v/: 
■1 -f . 
+ * 
In order to determine the numerator of this fraction, vve shall 
adopt the elegant artifice used by Laplace * on a similar 
occasion. 
”A . + . . . . '-‘A . *' = (v'lTi + 
= _ (V'_i + 6')-'+‘ . D-^ {s— — 
V~ . s— *— s“^* + Vim. &c. } 
= (— (v'_i + £*)'^+' . 
{ 1". r‘- vm . V. 3' - 3 '+ &c. } 
Now, as this equation is rigorous, and the first member con- 
tains only positive powers of the negative powers in the 
second must destroy each other, and may therefore be neglected. 
Expanding then (s^ + V — 1 )‘*^'*"* in powers of multiplying 
together the two series, and retaining only positive powers 
of s\ we find 
• Laplace. Mem. de I’Acad. 1779. Sur I’usage du calc, des diff. particlles dans 
la theorie des suites. 
