f 382 J 
The latter part of this theorem, though flowing from 
the general demonftration, may be alio demonftrated, 
immediately, in the following manner ; Let BP ng. 5. be 
we have MO 
ad x 
a — x 
7cc* X -f a * 
__ L/. 
,and CO- 
5 a — 
- la? dx -f 2 &axd x 
CO 
# a 
— 2 x X -f x x 
neglect- 
ing the dx) and a 05 — CO 
2 aaX dx 
OL—V 
; Therefore 
the effeCt of the ffream upon OM, which is 
rt$vv 
120 
. (aci 
* \ c P . n(Svv , 
CO/ — will become — — — (20^ — 
CP 
2 xdx) whofe integral is 
? 0 V V 
I iO 
(2ux — xx) where- 
in puttting x = « — /, we have -//) for 
the total effed of the ftream upon the wheel, which 
is the fame as that of a Tingle float-board A B in a 
vertical pofition. „ , , , - . 
k VIII. This theorem will alfo hold true for the 
cafe of § V. wherein we have fuppofed the height 
of the float- boards very fmall, in companion of the 
radius of the wheel; we havefeen that the effeCt of a 
{ingle float-board placed vertically was — nabw 
f 2 r -V- a ) ; the demonftration of the preceding § wi 
be applicable here after the fame manner, and will 
{hew that whatever be the number of float- boards, 
the effeCt will be ever the fame. 
It does not however follow that the number of 
float-boards fhould be indifferent; for the wheel 
coming to turn the float-board, its lower part, which 
received the perpendicular impulfe, will no longer 
n receive 
