100 Application of the Fluaxional Ratio, &c. 
It may here be noticed that the elements p, q, r, s, 8c. in the se- 
ries fr+-ph+qh? +rh?+W&c. represent the several functions of z 
together with their coefficients, in expanding the binomial (7-++h)". 
a— a, 
Hence fx + ph + gh? : rh® +sh' + &e. ao "tne 
z"-2h? +o i =a 2h? 4 Be, ines r+i for Zz. 
= 
becomes (z +4)" + n(x + yoke te aa yeh! hate - 
2 ere . 
y(a+i) 2h? 4+-4(2+7)""*h* + &e. 
The term (z+%)" being Ler form the equation, 
I pags + &c. 
(2+1)"=2" ns" i+ 7° * 
Put n—1 for n, and the aon -becomes, 
: n— = n—2 . 
(2-+1)" ta" 1 + ares rar ve +&c. - 
Put n—2 for n, and - —— i . 
: n— n—~2 n— 
(2piyr tats pe ap 
Put n—3 for n, oa te equation becomes . 
? 
2" 4i?-+8c. 
n—3 n—4 
(x+2)"- 37" 9 doce Sted ® aes gr 5424 &c. 
hence 2" i fer-tit fet pit gi 
Ss erfeen Radet bic —2 
ita iais Ohara ai teppei, 
. F n—2 ; wi Fees 
Ban-2 ago irae "3 2 artis =qtqitg 
a8 n—4 ; : / 
asp i 5 a-Si ibe Gt 
4 Stree 
ay ae" ~ 642 — 3-1 5/4-+-3//2? 
dnt 8 4p 3 tatéie 
za | 
therefore f((2 + 1)+h)=2" 4 ax" "t+ Ba"-2i2 4 (aa"-! pa 
.. n—i n— 
oe * 
2 . —3% 
2 3i2)h4 (Ba"-* 4 Ba" 
—3 Pat 
7) a” 442 )h? + &c.=fe+pr+qi+&c.+(p+pt+-p’? )h+ (q+ 
gfitg i? )h? + (r--ritri?)h? +(s+si+s%? hi +&c. (3.): 
