#83 Dr. BARKER’s MILL’ 
different from that of a fluid confined in the tube whenit 
begins to move, becaufe a particle at the extremity of the 
tube is not toreceive its whole circular motion there, but 
has gradually acquired it by a uniform acceleration during 
its paflage along the tube: fo that inftead of the ufual way 
of computing inertia by the centre of gyration, I muft in- 
veftigate a new theorem for the purpofe (at leaft new to 
me) which may be thus; 
Suppofe a particle P (plate a fig. 3,) * moving iniform= 
ly in the line and direGtion CA, while this line has a uni- 
form horizontal motion toward the pofition CB; thea P 
defcribes the common fpiral of Archimedes to Q, &&c. and 
the velocities in P and Q, in the direCtion of the circum-= 
ferences pafling through thofe points, are as thofe circum- 
ferences, or as their radii CP, CQ, &c. in which ratio 
are alfo the times of its moving from C to P, Q, &c. And 
fince the velocities are as the times of moving from C, 
(as is the cafe of a body falling from reft) the particie P 
muft be uniformly accelerated, in the direction Pn by a 
conftant equable force, like that of gravity; therefore its: 
reaction againft the moving line CA, by its inertia, muft 
be the fame in every point from C to A; hence the mid- 
dle point of the radius is to be confidered as the centre of 
reliftance in this cafe. 
Let x = CP, the diftance in feet of a particle P from 
the centre at any inftant. 
v = the velocity of P per fecond, in the direCtion: 
of the radius CA. 
c = 3. 1416; a, r, t and w, as before. 
Then the moving plane or particle P will be ax, and its 
2Cx 
weight ax lbs. as before, alfo its velocity= ——and the: 
t 
time 
® The velocity muft be uniform if the tube be prifmical; but the effect in this cafe will be 
the fame if it taper, and the water be accelerated; for the fame quantity in the fame time 
Tals through (and is a¢ted upon) by every part. Otherwile we fhould ufe the logarithmic 
pira 
