449 
Mr. Hersctiel on various pomts of Anahsis. 
or, ^=S,{ A, /"./(//(")} 
foi since X and j/ vary throiigli all their values, these two stuns 
are identical. Equating then the values of o- 
S {A^/./(tr)) =G, [a,.F{flf)} 
= {¥ :t iP) (/ log~’) : log< 
Let = 1, and for h writing and adding or subtracting 
5 {A,. /'■»*«»- > }=tll”l4J;«:;2.(/i„g-.) : log (,„) 
II. On Logarithmic Transcendents. 
The equation we have just arrived at affords us a method 
of exhibiting, in a finite form, the sum of the series in its first 
member, provided we possess the means of obtaining the 
second; and it appears, by what we Iiave before remarked, 
that this can be performed, whenever F (/i^) + F is a 
rational integral function of D. This includes among the forms 
of F those remarkable functions denominated by Mr. Spence, 
“ Logarithmic Transcendents,’" and w'e shall now proceed, by 
the help of the general property demonstrated by that author 
in his ‘‘ Essay &c.” to derive from these principles the sum- 
«/ 
mation of one of the most extensive classes of series which has 
yet received discussion. 
Adopting Mr. Spence’s notation, we will represent the 
series 
by the symbol ”L (i + x). Tiie property then alluded to is 
as follows : 
