[ 332 ] 
two following Problems, which I fhall fubjoin as 
Examples thereof. The firft is ; 
To find the Time of Ofcillation in the Arch of a 
Cycloid , in a Medium refifting according to the 
duplicate Ratio of the Velocity. 
Let A denote the whole Arch of the Semi- 
Cycloid, or the Length of the Pendulum, a the 
the Arch defcribed in the whole Defcenr, and x 
any variable Part thereof defcribed from the Begin- 
ning of the Defcent ; and let the Denfity of the 
Medium be, every-vvhere, as£: Then the Fluxion 
of the Time will be found — 
, 2 a 
*-i+ /7 x 
x 2* 1 , 4X 3 8x 4 
2 IJb “T* 2. y 3 - 4- 5 h * ^ C ‘ 
1 
2 - 
T2\ ^xx* ' x * : which being compared with 
a—bx n +dx w -±ex^ &c^Xx^' 1 ^ vide Corol. 2 .) 
we fhall, in this Cafe, have »= 1 , mm — 1, f—b 
/ 2# d 2. e if 2 f 
«— a > 7~^T=— &c. 
Alfo 
— -» — 30 ' — — 9 / 22 ' _ 45 - 
p+m _ P 
-c ./v 
v r -3-S-7 (M) * I- 3- 5- 7 _ "71" , and 
* 2. 4. 6 . 8. 10. 0+ w ) V 
_ i M" 1 -** t 
I A I 
^+ 1 . ?-F 2 - 
j»+ OT + l - ^ 1 . j p-4-’W-i~2. £ 
2 = I +^T7)+ 
\zb 2 h x 
* The Inveftigation of this, and the Fluxion in the following Exam- 
ple, are both given in my E flays. 
