Now, if m be an even number, the powers of z in 
the former part of the equation will be the even ones, 
and those in the latter the odd ones: but if m be an odd 
number, then, vice versa. 
In the first case our equation may be wrote thus, 
f -r + 1 b2 4 +z*...+z* 
z 
Where, since ~ f z—a, ~ f z x — s 4 — 2, 
— 6 3 —3i, -f z* — s* — 4s* -J- 2, &c, we shall 
by substituting these values in each series (proceeding 
from the middle both ways) have 1 f s* 2 + 
s* -— 45 + 2 4- &c. = c into s -f ^ —~3s + &c. 
But, in the second case, where m is an odd num 
ber, and the even powers of z come into the second 
series, we shall, by the very same method, have 
- 5s 1 + + &c. zi c into l 4- 
2s + a- 4 — 4a 4 + 2 &c. 
In both which cases the terms are to be so far conti 
nued, that the exponent of s, in the highest of them 
Thus, if /?, the given number of terms, 
be 3, then being = l, the equation be 
longs to case 2, and will be s — c, barely. If n — 5, 
then m — 2: and therefore l -f s 2 — 2 r «, or 
s 2 — 1 — cs, by case 1, If n be 7, m will be 3; and