5* 
Mr. Ivory on the Attractions of an 
pansion of ^ = - - ' ■ . z . 1 
v f (r a + « l )Y V 1 r a +fl 2 
and, by the bino- 
mial theorem, the term containing p/ will be = *•*'* •"' Zl ~ l x 
= • whence it is 
(r a +a a )i-f| 1.2.3... i r i+ 1 , wnence ir is 
plain that A^= h 3-5 •••• 2 *~ 1 . consequently, 
c (i) = 1 -3-S - «-i f i_ i(i-i) . f_ 2 , i(i-l) (i-g) (i~3) 
*- 2 -3 ••• * t' 2(21 — 1 ) ^ T” 2.4-(2i— 1) ( 2 z — 3 ) 
p.* - 4 — &C. | 
If we take the fluxions rc times successively in the last for- 
mula, we shall obtain 
rf”C (0 I - 3 S 2 * — 1 f ,, 1 — « z — m.z — «■ — i z — n 
= 1-2.3 i-« * “ ZiT-T? • ^ 
— n — 2 
+ 
z— zz . j—n — 1 . i—n — 2 . i—n — 3 
2.4 . 21 —- 1 . 21— 3 
&c. }. 
When i — n is an even number, —• will contain a part 
a n 
equal to 
±2.4.6... i — n > 
independent of ^ ; and when i — n is an odd number, the same 
quantity will contain a part, equal to 
1.3.5 - '+«+ 1 
i-3 5 •— i+n 
— 2.4.6.... i—n—i ’ P* 
multiplied by p, only: these two parts of the value of d ~- 
we shall afterwards have occasion to refer to. 
4. It is proposed to investigate the fluent of 
(*' 
/ , 2 \n d n C (0 r> 7 
(l p- ) P . dp, 
d\x 
between the limits p = — 1 and p, = 1 ; supposing P to be a 
rational and integral function of p,. 
