Infinite Series. 165 
between two fpheroids, of which the axes in the one do not 
differ much from correfpondent axes i ft the ocher, in which 
cafe the fluents found of the following fluxions may be 
ufeful. 
■/- 
1. 
* . 
“T 
V— 3 S ' 4 
-*V 2 (. 2 -.vV 2 
V — 5 
2 /' 2 
(/ 2 -*V 
» — 4 . « — 1> 
J. - - = -■= = — X 
• »-3 • «“5 • ” 7*° 
x n — 4 . 72 — b . n — 8 
7 + a 
71 — 3 • n — 5 > 7 • » — 92 s 
X 
\ q 
2 
« — 4 . w — 6 . w — 8 ^ ^ ^ 
»— 3 . 72 — 5 . «— 7 . 72—9 - • . 22*~ 3 * — 
72 — 4 . 72 — 6 . 72— 8 . . . 3 . I C X *c 1 J 1 
=— — — — x / - — - if n be an odd 
n—2.n — 5 . 7 . »~9 • . 2< 3 ~J t —x 
num- 
ber; but 
i. r— — ==L-x — y ._-- 
V 7— B -3‘ a .. ,Y— »-3-«-S' 4 
\LZ- 1 «- 3 < a n-Z.n-st y=J 
, H-4.K — 6 „ £ _L 
4“ ■ ,■ r ■ - X \ n 7 • • • • • T 
n ~3 • «-5 4 n "l t 
y— 7 • • • 
— x 1 ) 2 
72 — 4 . 72 — 6 . — 8 . « — 10 . . . . 4 , 
if # be 
72-3 . WH5 .”7 . »-9 71 3 . ( t 2 ~- x 2 )i 
even. 
Thefe principles may be applied to the finding approxi- 
mations in very many philofophical problems. 
Cor . 1. From hence may be deduced the fubfequent arith- 
metical theorems. 
2m- I 
+ 
am— 2 
am— 2 . am — 4 
am— 1 . am— $ am — 1 . 2772 — 3 . 2/72 — 5 
am — a . am — 4 . 2/72 — 6 
am — a . 2m — 4 . am — 6 . . . . 
2/72— I .2*72 — 3 . 2772- 5 . 2772 — 7 2772- 1 . 2ff2— 3.2772-5.2772— 7 ...3. I 
! » I ; or, which is the fame, % ===. -f 
1 * 3771— 3 2772-3 . 2/72-5 
2W - 4 
