Dr. Waring on 
Of this cafe 1 (hall fubjoin a few examples, of which fome 
have been already publifhed in the Medit. Analyt. 
Ex i Let the fine and cofine of a given arc A be refpec- 
tively 5 'and c; then will the fine and cofine of an arc A + e, 
where * bears a very fmall ratio to A, be refpedively 
-£ 4 +&c. (the figns pro- 
ce 
C —1— — ■ 
J ‘ r 1.2 r 
e z - 
Cp J i 
i. 2 - 3*' 3 1 1 • 2 3 • 4 r 
ceed by pairs alternately negative and affirmative) - J 0 
— i xV + v , • 1 , ~'TT« x " &C 0 ' + c ( ~ ~ i .2 .'p + i. 2.3.4” S ' 5 
0 . , * „ 1 + l x se 2 + See. (the figns 
- &c.) ; and c - - e - p— — « + , . 2 . 3,3 v 
proceed as before by pairs alternately negative and affirmative) 
= < l - 
I a , _J 
777 ? e + i • 2 . 3 • 
- &c.) - i (* 
2 . V 
/ + 
_ <?5 _ 6cc.) 
2 This fete can alfo eaf.ly be derived from Newton’s feries 
by plane trigonometry, and will converge much fwrf.e, than 
, /• . a A 3 _L - & c. &c. if e bears a fmall 
the feries A - — - -~z + 2 . 3 . 4 . 5 > + 
Vl °E^Thete be a fmall quantity in proportion to A and the 
j _ i _ ELl£. i + x i — 
given fluxion - + - 777 (r 2 -KT (>' + '? 
t ott (r' + r’ — 2'X f 'r v 2 , 
r^xr+&c.»^- FT? y« (PTfor- 
( 2 7 +F- 2 2 A}t 3 , rpx. the co-efficient of a term 
x w-py *— Xf + uu ’ 
i x e » of this feries will be a fraSion whofe numerator is 
?P? w -m + i.-j x 2 2 x 7 x ?+r + ® + 2 - — • 3 
w +2 m-fi m m ~~ 1 
w— 1 
W —2 
Xj'xfxf + f -m + 3 
• — • 
4 
X 
