analytical and geometrical Methods of Investigation, i n 
from the beginning in first series, is b” * i™ 1 . B ,s 6" **' c* ; 
which, n even and m = -I— = b w . c 
J 2 6 
#• n odd and m = — 1)82+1 i"”" 1 . m 4 - 1 h. 
Z 6‘ * 
At these terms the series terminates ; all the succeeding terms 
being equal o,- since d ? ”+ 1 6'”, d^+ 2 5 to—1 , are respectively 
= M.?n — i in — 2 ... 3. 2,1.0 = m — i m— 2 ... 3 . 2.1.0 and = 0. 
Hence, the series written in a reverse order is {n even) 
L D 
” c 
D 1 
c 
W+l — 1 C ' m — 1 J_ J> W 4* 2 1 T \ m — 2 r ? n — 2 
* £ ° T* r 
1 -B b T .C ... B 1 
c £ 
D"!- 1 
c 
(w odd) 
D" + ‘ l“‘ . D” 6”+*. c * + D”+ 8 j- . D ”- 1 b" +, .c"- i + &C 
Now, ‘ = — = 1 (k even) or = — l (» odd) 
and .*. the former series becomes 
=*= b n =i=Db n ~\c=t= B a b n ~ z . c‘r±z &c. and consequently, the term 
affected with x n in (2 + b x) (1 + bx -f ex 1 )- 1 
is 
2 b n 
b n 
2 d b n ‘~ i , c 
d 6*— 2 .be 
2D a i> n - 2 .r a =+= See. 
jf b n ~~3.bd=fz &c. f 
or =t= 6” : 
n— 1 
d b 
n — 1 
.C 
n 
C r • -in, m r n — m — 1 
J for, since B b 
TP.® l n — m 1 n — » — 1 
l'' ±8 ? b =r=D b 
n — 2 
x 6 
x& 
d 2 6”- 2 .r a 
n — 3 
d 3 6”~3. c 3 db &c; 
« — 2 7 ?Z jjW “ 
n — m c 
n 
, 1 b >11 
B 0 
} 
/i~ »2 C 
Hence, «- + /3* = b n - ^ d^" 1 + i-j d 6 b “ 2 — &c. c being 
= u @ = 1 . 
The law of the series is truly and unambiguously represented, 
by means of the symbol or note of derivation d ; but, if it is 
required to express the law numerically, in terms of n, since 
