123 
The fifth crown fhall be entitled toa 
gold medal, as a prize, on which the 
zame of the viétor fhall be ftruck.. On 
the exergue this infcription fhall be en- 
graved: NON GMNIS MORIAR. 
There triumphal crowns fhall be ad- 
judged by the order of government, and 
in open public aflembly. The talents of 
the victor wiil thus be configned to im- 
mortality. - 
On the zeal and laborious efforts of the 
artifts toacquire eminence in their pro- 
feffion, will entirely depend their title 
zo the acceffits of encouragement, which 
are to afcertain their fedulity, and to re- 
ward their fervices. 
An acceifit of encouragement fhall be 
entitled to a premium of 300 livres. 
Seven acceiiits of encouragement fhall 
obtain, asa recompence, a firft crown, 
and a penfion of 2<0 livres. 
Five new acceflits of encouragement 
thall receive a recompence of a fecond 
crown, anda fecond penfion of 250 livres: 
‘Three new acceffits or encouragement 
fhall entitle the receiver to a peniion of 
400 livres. 
Thus, an artift who fhall raife himfelf 
to diftinétion, in the fubaltern claffes, 
wil be entitled to receive fifteen acceflits 
Mathematical C. orrefp ondences 
[Feb, 
of encouragement, three crowns of ap 
probation or applaufe, and a penfion, 
or life annuity, of goo livres. 
The payment of the penfions fhall be 
under the guarantee of government ; 
the Odéon, however, defraying the 
charges out of the income arifing from 
the reprefentations. 
The Odéon thall poffefs a literary jour- 
nal, the objeét of which fhail be, to an- 
nounce to the public the pieces which are 
to be reprefented in it: this work is des 
figned to be a valuable vade-mecum for. 
fuch perfons as are not in the habit of de- 
ciding on the merits of theatrical perfor- 
mances. It will analyfe the fubje& of 
every piece, and illuftrate the art ac- 
cording to which it fhall be conduéted. 
It will, moreover, contain judicious cri- 
tiques_ by the beft appreciators of literary 
talents. The artifts will there difcern, 
at once, the encomiums which they fhall 
have merited by the truth, the gran- 
deur, and the beauty of their acting, and 
the cenfures which they may incur, in 
confequence of the negligences, or in- 
voluntary errors, they may happen to 
fall into, in the courfe of their repre- 
fentation. 

- MATHEMATICAL CORRESPONDENC E. 

To the Editor of ibe Monthly Magazine. 
SIR, 
ATELY examining Mr. Simpson’s {clution of the 37th problem, in his fecond volume of 
Fluxions, I perceived that the contents of article 482, being the firft corollary of that 
problem, were applicable to the inveftigation of the cafe contained under the following fup- 
pofitions : " . 
If a flender rod AC revolve round the point C, as a centre, with a velocity fuch that the point 
D (whofe diftance from C is one foot) moves in the quadrant DE with the uniform velocity m, 
the plane of the rod’s motion being perpendicular to the horizon: and if a ring of iron, whoie. 
mafs is r, be connected by a ftring, pafline over a pully at C, to a weight B, hanging perpendi- 
cularly, and whofe mafs is 4; what will be the nature of the curve defcribed by the ring, the 
firft pofition of the rod being parallel to the horizon; and fuppofing alfo, that the ring, inde-— 
pendent of the other forces which aé on it, is refitted by fri€tion, the force of which is every 
where to the centrifugal force aiifing from the paracentric velocity of the ro¢, inverfely as 
7 to b? 
Let AC==d be the firft pofition of the rod, and A the place of the E 
ring. Alfo, let IFC be any other pofition of it at the times; and put Ee: 
EF any variable {pace paffed by the ring == x, AF being the curve de- 
fcribed by it ; v =the velocity down IC at F; x==are DG; y= GH 
the correfponding fine ; and 2g = 32! the force of gravity. 
The forces acting on the ring are evidently thofe arifing from the pa- 
racentric velocity of the rod; the refitting force arifing from the friétion ; 
and the compound forces of gravity on the ring and body B. 
mm? d— 
r+ 0 
= 
~ 
a 
— 

ee 
p 

ae ene te 
Now the ‘firft of thefe forces is evidently = ; the fecond 
Oo 

bm?.d—x att ryt-5 ‘ = cone 
ae and the third 2¢x seals Of thefe forces, the two firft always act in oppofition te 
reo r 
the laft, fuppofing that, at the commencement of the ring’s motion, rdm® is lefs than 2g. 
Again 3 

