[ 379 ] 
confider all the points of the float-board AB as 
affedted with the fame velocity z. 
n b 
Let C P fig. 3. = x, we fhall have — (*u— z) % f. xdx 
nb 
for the eftedt of the portion AP, and — zj* 
X - - for the effiedt of the whole float-board AB. 
2 
This quantity mu ft be made equal to d?r, and then, 
juft as in the foregoing cafes, fuch a value of 2; be 
fought, that the weight sr and its velocity may be the 
greateft pofiible ; that is, the differential of z (v — z) z 
muft be made — o, which gives z ■== 1. v. There- 
r , ' tiabvv . . , . 2/Odw — naabvv 
fore d 7 T = ( 2 r~\-a ), and r— 7 , 
270 1 ' onabvv 3 
and the velocity of the weight tt will be =* 
2 n cl a b v v 
V V, 
8lod 7 r + aav 
§ VI. We have feen that the calculus was much 
Amplified by fuppofing one of the velocities conftant 
for all points of the float-board. For this velocity 
being c, the effedt of the whole float-board will be 
fimp'y r 0 ( vc y 
acc 
■ff 
It will therefore not be 
unufeful to inquire what this velocity c muft: be, that 
the effedt of the float-board may be the fame, as 
fuppofing, as we have hitherto done, a variable 
velocity, and proportional to the diftances from the 
axle, we have only to make ^ — c (— — ) 
n@/ , rjr . « 3 — f* 
-rJ ivviza-fD-^vz—— 
2 
~ 4 -/ 
4 ‘ 
CC / 
IV.'lbut the equation whence we got the value of 2; 
C c C 2 (S IV.) 
