7797.4 Mathematical Corre[pondence. 129 
| ae 
Again; taking the two firtt forces from the laft, we get as ar m?.d-—x for the true ac- 
am 
celerating force on the ring down the rod at F; and’ fice = === 1, we get 2 £ ara ys yet 
; m 
° sie . Ps Fi 
But es, fuppofing % to be conftant, as in the corollary alluded to. Hence 




meV. 
in z 
5 haa tank h for 2¢-— dp f 
sae doeK-=-, which, “ubftituting # for 2g -———, and / for 27——— — d, 
tae m #it% 7 —pab m2,¢—-b 
gives Ns te hy gem fi? =o, an equation fomewhat fimilar to that arifing from the confidera- 
tions contained in the 4th corollary of the fame problem. - 
To find the oo or « mi terms i By a the feries Az?-L_Bz3-C24-+Dz5, é&c. for x, 
7 
cry &c.3 and, by going through the 
23.4.5 2.3.4.5.6.7° 
ploper eps, and equating ile nomolozohs terms, we get 


and fubftitute for y its value str 

red 
i es igual ee 
ae 
we — 23 ay x 
ES sp lS GN Le ae Ee tc do 
Ic Ae ic my 2.3.4.6-6.7.8.9.10.11 
% 
* 
[lta is ty (ole : ~, therefore, by reftoring the values of 4 and 4, we get x, 


2 
GID SUS US, | Se ee 
b N?4N72 r N2—N-z. 
the {pace def cended by the ting = = 2g——--— I Xx ee x e as 
2? rob ai mm? yb 4. jy Bs 
: ee Loe rn 
which, when 2 becomes a quadrant, and y radius, 18 22> —~-@ X S28 —— ay by putting 
m9 m?.7-1-b 








N2-LN72 Ne—N-2 1 
a ~I==!, and ———=", 
pe 4 2 
From this conelufion it Sa that before the rod obtains a vertical pofition, the ring cannot 
5 rn 
fee eee 
SH rhe? r—--b Siti afm. rs 
if its value be equal to that quantity, the ring will arrive at the centre C juit as the rod becomes 
perpendicular. ft is allo manifeft, that if. My or the angular Me locity per fecond, be lefs than 
iia IGU NR HE 
2obs 2grn 
Ae al ——-= aS ————, the ring will arrive at the centre me the rod becomes vertical. 
ds-ld. robs ds ds-d. rhb 
- If it be greater, it is evident the reverfe will take place. 
Under the circumftances of the data being fuch that the ring arrives at the centre juft as the 
rod becomes vertical, the ab{ceffa and erate to the curve AFC are very eafily found ; for, put- 


have arrived at the centre, unlefs d be lefs than 2¢ ; and that 


wecc. == NY 

a aa eae ch A 







+ 2 : Nz .N.72 
So I ore 5 Thay Ce == es 
pepe tata, be. SN 
Sof =f — often 
ie 28 a 2.3.4.5" 
$ z zs z 5 
el es See 
P26) 2G Dane 
3 5 IZ Nz 
“ So a os og * i sac ues 
2.3 2-3:4.5 He, 
zs 5 
And —Z — ote wale xc 
ee Vee 
x3 z5 NZ—N™ 
Flence ale + ) 8c, a Be NEO 
° aa BeBe ki 2134-65 4 2 
But thefe feries, putting N for-that number whofe hyp. log. ig unity, are equal * to © 


i 
Ww 
ie 
Ei 
i 
oe 
n= 
Sake 
Es 
SR CT eo rena Fg 
men ie 
—— 
