M E C II A N I C S. 
together, the refinance will be as V 2 X the denfities of the 
fluids X areas of the planes. 
Prop. XIX. If a plane move obliquely in a refiflihg medium, 
with an uniform velocity, and, after the refolution of the force - 
with which the plane jlrik.es the fluid, the whole of that part 
which ads perpendicularly to the plane take fed, the refin¬ 
ance perpendicular to the plane will vary as the ftquarc of the Jine 
of the angle of inclination. —-Let AB, fig. 18. Plate XX. be . 
the plane moving in the medium LMNO in the direction 
xByi draw AC parallel to xy, meeting BC perpendicu¬ 
lar to AB in C; arid let B D be perpendicular to AC. 
Now the quantity of fluid which AB has to oppofe by 
its motion, being that which is contained between AC 
snd xy, is manifeftly in proportion to B D, or to the fine 
of BAG; becanfe, AB : BD :: rad. = i : fin. BAD, 
or B AC; and, the firft and third terms being conftant, 
the fecond varies as the fourth. Alfo, as the plane'acts 
again ft the fluid at the angle CAB, let AC be taken to 
reprefent the whole force of the plane acting againlt the 
fluid, upon fuppofition that no part thereof was loft, which 
force would be uniform, the velocity of the plape being 
uniform ; then, by the refolution of motion, the force 
acting perpendicularly to the plane will be in proportion 
to B C, or to the fine of B AC ; for, AC : BC ;: rad. — 
1 : fin. BAG 5 where, the firft and third terms being con¬ 
ftant, the fecond varies as the fourth. Hence, the whole 
aftion of the plane againlt the fluid in the direction B C 
(being in proportion to the whole quantity of fluid which 
oppoles its motion, and its effect in the direction BC con¬ 
jointly) will be as fin. BACX fin. BAG, or as tin. BAG I 2, 
And, as action and re-aftion are equal and contrary, the 
aftion of the fluid againlt the plane in the direction C B, 
or the refiltance of the fluid, mull vary in the fame ratio. 
Prop. XX. The ref fiance of the fame fluid to oppofe the 
plane in the diredion of its motion, varies as fin. BAC’l 3 ,/«/;- 
pofing, after the refolution of the re adion of the fluid in the 
diredion C A, into CB and BA, the part BA to be entirely loft , 
and CB to take fed. — As the whole erfeit of the refifi¬ 
ance of the fluid upon the plane is that part which is per¬ 
pendicular to it, let C B reprefent that whole refiltance, 
and revolve it into CD, D B, then will C D reprefent the 
refiltance which oppofes the motion of the body ; now 
rad. = i : fin. DBC, or fin. BAG, :: CB : CD—CBx 
fin. BAG 5 but CB, as reprelent ing the whole refiltance 
in the direftion C B, varies ns fin. BAG) 2 (Prop. XIX.) 
hence, CD varies as fin. B AC.! 3 . 
By experiments on plane bodies moving both in airand 
water, we find that they are not refilled according to the 
laws here deduced. Part of the difference may probably 
be owing to the two latter cafes mentioned in the firft ar¬ 
ticle of this feftion 5 but it principally arifes from the 
force after refolution not taking effect as here fuppofed, 
that part which is parallel to the plane not being all loft. 
Prop. XXI. The famefuppofition being made, the reffance of 
the plane, in a diredion perpendicular to that of its motion 
varies as jm. BAC| 2 X cof, BAG. —For, by the laft Prop. 
DB will reprefent that part of the whole refiltance which 
afts perpendicularly to the direftion of the motion 5 hence, 
rad. = 1 : fin. BCD, or cof. BAC, :: CB : DB=CBx 
cof. BAC , but CB, as reprefentine the whole effective 
part of the force, varies as fin. B AC) 2 (Prop. XIX.) 
therefore, DA varies as the fin. BAC 2 )x cos. BAC. 
If, inftead of fuppofing the plane to move in the fluid, 
we fuppofe the plane to be at reft and the fluid to move 
againlt it, the action of the fluid againlt the plane will be 
juft the fame as its re-aftion when the plane moves. 
Hence, the laft Prop, will fhow the effeft which the wind 
has upon the fails of a wind-mil!, when at reft, to put them 
in motion, admitting our hypothefis rsfpefting the effica¬ 
cious part of the force, to be true. 
If, in the three laft Propofitions, the area of the plane, 
the velocity and denfit.y of the fluid, be not given, then 
.(for the reafons in Prop. XVII. XVIII.) the refiltance 
will vary in the above ratios, and as the area of the plane, 
fquare of the velocity, and denfity of the fluid, conjointly. 
Prop. XXII. The fame fuppofition being made, let a cylinder 
move in the diredion of its axis, and a fphere of the fame diameter 
move in the fame fluid with the fame velocity ; then will the reflfi- 
ance to the motion of the cylinder be double that of the globe — 
Let AFE, fig. 19. be a diameter of the end of a cylinder, 
parallel and equal to BD a diameter of the fphere BFDG 
whofe centre is C, and CF, DE, BA, perpendicular to AE; 
draw QP parallel to CF the axis of the cylinder, and let 
it reprefent the force with which a particle.of fluid would 
aft perpendicularly at v, the end of the cylinder, in which 
cafe no part is loll. Now conceiving the fame particle 
to aft upon the globe at P, part of its effeft will be loll 
by the obliquity of the liroke ; draw PR a tangent to the 
fphere in the plane PCF, and QR perpendicular to it; 
draw alfo RS perpendicular to QP, and produce QP to m. 
Refolve the whole force QP into RP, QR ; then we here 
fuppofe that the part RP is wholly loft, and the part QR 
only to be effeftive. Refolve this into QS, SR ; then QS 
is employed in oppofing the motion of the globe, and SR 
(being perpendicular to it) can have no efteft in that re- 
fpect ; and moreover it will be deftroyed by an equal and 
oppolite force aftingat a point equidiltant from F, on the 
other fide. Hence, The force w ith which the point v at 
the end of the cylinder is retarded : the force with which 
the correfponding point P on the globe is retarded :: QP 
: QS :: (becaufe QP : QR :: QR : QS) QP 2 : QR 2 :: 
(by fiin. trian. PRQ, P»zC) PC 2 : Pm 2 :: PC : 
. P 77/ 2 \ 
(taking vn — — — PC : vn :: vm : vn-, confequently. 
The whole refiltance on the cylinder : that on the globe :: 
the fum of all the vm\ : the fuin of all the vn" s. Draw 
n r parallel to 1n C. Now vm : vn :: PC 2 : Pm 2 ; there¬ 
fore vm or CF : mn or Cr :: PC 2 or AF 2 : PC 2 — 'Pm 2 
or rn 2 ; therefore Cr varies asm 2 ; hence, the locus AtzCE 
of all the points n is a parabola. Conceive now this whole 
figure to revolve about FCG ; then will AE generate the 
end of the cylinder, and BFD will generate that half of 
the globe which is refitted; alfo, the lum of all the vm' s 
will be the folid generated by the parallelogram A E D B, 
and the fum of all the vn s will be the folid generated by 
the inferibed parabola ACE, which folids are as 2 to 1 ; 
hence, The refiltance of the cylinder : the refiltance of the 
globe 1:2:1. 
It appears by experiment, that this propofition is not 
true when bodies move either in air or water, the refiltance 
of the globe, compared with that of the cylinder, being 
lefs than that which the theory gives it. 
Prop. XXIII. The. fame fuppofition being admitted, if a globe 
whofe diameter is d, move in a refijling medium whofe denfity 
is n, with a velocity v, the ref fiance will vary as v 2 d 2 n._ 
For the refiltance of a globe is (Prop. XXII.) equal to half 
the refiltance of the bafe of a cylinder of the fame diameter, 
moving in the direftion of its axis with the fame velocity; 
therefore the refiltance of the globe varies as the refiltance 
of the cylinder; but, the end of the cylinder being a circle, 
whofe area is as d 2 , the refiltance (Prop. XVIII.) varies 
as v 2 d 2 n ; therefore the refiltance of the globe varies as 
v 2 d 2 n. 
If the reader wifli to fee any thing further on the motion 
of bodies in refilling mediums, he may confult the Vince’s 
Fluxions, p. 132, 224.. 3d edit. 
Prop. XXIV. As a body defends in a fluid, it continually 
adds more weight to the fluid until it has acquired its grealefl ve« 
locity ; at which time, the weight added to the fluid from the re° 
ffiance, is equal to the relative weight of the body. —For, as 
long as the velocity of the body increafes, the aftion of the 
body upon the fluid will continue to increafe; and, when 
the body has acquired its greateft velocity, the refiltance 
becomes equal to the weight of the body in the fluid, and 
the body then afts agai.nft the fluid with its whole relative 
weight. 
Prop. 
