to® Mr. Woodhouse on the Independence of the 
Hence, to answer the equation = -^==r- 9 
i x may == z — fi — 7 
J t . z.-j 1 1.2. 3.4.5 
*'5 
4 - - 
* 1.; 
1.2.3 
or 
or %" — 
;i.2.3 
r >«3 
1 . 2.3 
- &c. 
- ic. 
•2 3-4-5 
z "s 
! __ _ &C. 
• 1 . 2-345 
z" , &c. representing 2 0+£, 30~fi%, &c. 'j>. 
Suppose now it is necessary to deduce 2; a/, a", -&c. in terms 
of x and its powers, by reversion of series. What does the 
reversion of series mean ? Merely this ; a certain method or 
operation, according to which, one quantity being expressed in 
terms of another, the second may be expressed in terms of the 
first. Hence, in all similar series, the' operation must be the 
same ; consequently, the result, which is merely the exhibition 
of a formula, must be the same; so that, whatever is the series 
in terms of x, produced hy reversion in 
-f — &c. the same must be produced 
% — 
1.2 3 
Ty reversion in x — z' — - 
in x = -z" — 
&c. 
z' 3 
1 . 2-3 
I.2.3 
+ “ 
‘ i-z-3-4-5 
+ &C. 
— &C. 
0 0 
The series produced by reversion in these cases is, x -j — 4 " + 
3- 2 
J£ 
5.8 
T See. Hence it appears, that we know, a priori, that must 
happen which D'Alembert considers as a paradox to have 
happened. Why this paradox found reception in the mind of 
this acute mathematician, I have stated, as my opinion, one 
cause to have been, an inattention, from geometrical considera- 
tions, to the real origin and derivation of certain expressions that 
appeared in the course of the calculation. Another cause ! ap- 
prehend was, the want of precise notions on the force and 
