[ 376 ] 
It follows from our laft remark, that the greateft 
weight with which the ftream can conftitute an equi- 
librium, will be = but then the wheel will 
3 I20d 
not have any motion, nor confequently the weight 
7 r : If the float-board be increased, or the weight 
diminifhed, from that inftant the wheel will begin to 
turn, and the fwifter as the float-board is greater, or 
the weight lefs ; but in moft machines, it is required 
that the weight may be the greateft poflible, as alfo 
the velocity with which it is raifed. A queftion 
therefore here offers itfelf, whofe folution is of much 
importance. What muff be the velocity of the float- 
board whofe dimenfions are given, that the product of 
the weight by its velocity fhall be the greateft poflible ? 
§ IV. The velocity of the weight tt is - z feet in a 
fecond, which being multiplied by the value of 
±vv — ivz + ±zz) will give the pro- 
dudl vz — ~2 VZZ t z ) which muft be a 
maximum ; for which purpofe make i. wdz 
— +vzdz -f- ^ z % dz = oj whence we have 
z ~ 8 ~ ^ 10 = 0,5 3 75 2^ : this value of # being 
fubftituted,make the equation '-*uv 
II 4 - 2 -v/ io 
zz 
/3ax ” 
8 i 
11 + 2 \/lO 
4860 
vv, fo that we have the equation 
d nr 
nvv 
+ 
rt dw 
200,520 , 
' ' nvv * 
which expreffes the dimenfions of the float-board 
where the effed will be the greateft poflible. If the 
float-board be plunged no deeper than to CC, as we 
ihave at firft fuppofed, the molt advantageous value 
of 
