Dr, Waring on the 
to 
; and the feries i 4- qx + + sx 3 + + &c=r i -f- - #4- 
■> 1 A 
6CA— 2B* , , 18CAB-8B 3 3 26C 2 A 2 -8B + . 
X -f- — . — TT 7 -x 3 + ■*— : — - — ~ x* + 
1 . 2 A" 
1.2. 3A 3 
1 . 2 . 3 . 4 A 4 
i8oC 2 A 2 B-i2oACB 3 +i6B s s 2i6C 3 A 3 4-2i6A a C 2 B 2 - 288 ACB 4 4 - 64 B< 
Af + 
1 . 2 . 3 . 4 . 5 . 6 A 6 
.V 
i • 2 • 3 • 4 * 5^ 5 
•+ &C. 
The terms of thefe two feriefes can eafily be deduced by the 
fubfequent method. Let K#” - 2 4- L”“* 4 - Ma;% be fuccefiive 
terms of the feries Ax4-Bat 4-Gv 3 4-&c., and KV^ + LV -1 
fuccefiive terms of the feries 1 -f qx + rx 2 + sx* 4- tx* + &c. ; then 
will M = = — — and L = . 
n . n— I x A A 
Cor . 1 . Let B = o, and the two feriefes Ax 4- Bx* 4- Cx 3 4- Da: 4 4- 
&c. and 1 -j- qx + rx z -j- &c. become refpe&ively Ax 4- Cx 3 4- 
— - . J — x -*’ + ‘ ‘ ~ - 
2 . 3 . 4 . 5 A 
2 3 . 5 3 C 3 7 
■J v V J. - — . 
2 . 3 . 4 . 5 . 6. 7 A 2 3 . 4 . 5 . 6 . 7 . 8. 9 
X ^3 x 9 4- &c., and i 
2 C 2 
“ X X +- 
2 A I 
z z 
o * o 
r>z 
- X r .,f+ 
4 A' 
2 3 . 3 3 
—7 X — 4- &c. 
. 2 . 3 . 4 . 5 . 0 A 3 
If in thefe feriefes for A be fubftituted 1, and for C be fub 
ftituted - — - , there will refult the feriefes x - — + 
2 . 3 ’ 2 . 3 J 
.rS 
2 • 3 • 4 • 5 
— &c., and 1 • — 1 
A- 
- &c. which give the 
1 . 21 . 2.3 
fine and cofine in terms of the arc a 1 . 
Cor . 2. Let C = o, and the above-mentioned feries. A# 4- Ba 2 4- 
&c. becomes Ax-\ Bx z — 
1 . 2 
2 - 3*4 
B 3 4 
*A 2 * 
I . 2 . 3 . 4. 5 
X 
B 4 5 , 2 6 
■ — x * 4 — _ 
A 3 1 . 2 . 3 . 4 . 5 . 6 . 7 A 3 
B & 7 
x — x x 4“ 
B? 
x 
1 , 2 . 3 . 4.. .8 A 6 
■X # 
2*3*4 
10 
B 9 10 
X 7 T* ~ 
A 8 ■ 
, 10 
B 
xo 
S . 2 . 3 
X Tqf 1 4- &c. The law of 
. . 11 A y 
this 
