Mr. Herschel on various points of Analysis. 4^7 
and of course 
V = a_ ^ V““ + a, 'vA^ + . . . . 
'V“ A^; (4). 
Thus we may always develope in a series containing 
only the successive orders of 'vA , such as VA , &c. 
If the developement of contain no negative 
powers of t, we have 
a 
z 
and consequently 
v'A, = [/rT':o.A^ + 
1*2 ••••• S0 
^111“ VA 
1 * ' 1.2 
VA^ + &c ( 5 ) . 
Let 'vA^ = A^^^ — A^, and we have = -L — 1, 
and (0 “ ^ ~h whence we obtain 
v'a,= a^ + 
A + 
4 - > 
[ / I ' ; I D* I / I / : 1 
LLL— aa, + -J— i— . A. 
1.2 
(6) 
for it is evident that when i = o, ^ 0 ] becomes 
^ particular case, let vA^ = A^r+i and 
v'A^. = ^x+iJ (t) = T> 1-^1 : l = z (z — l) 
(/ — X + i), whence 
= A_^ + 4 . A A, .+ • A’A^ + &c. ( 7 ). 
. = aA^ = A^^, ^ A_^. and 'vA^ 
Again, if we suppose vA^ 
= A , we sliall obtain from {5) 
A-+ I 
3 M 2 
