712 
M E C II A N I C S. 
Prop. XXV. Suppofe that a plane ABC, (fig. 20.) moving 
with a velocity and dpreBion reprcfented by b B, is aBed upon by 
a jluid zvhofe particles move with a velocity reprefented by DB, 
and in direBions parallel to that line ; it is propofed to determine 
the angle of inclination ABD, Jo that the effeEl of the fluid may 
be the greatefl pojfible. —Since a particle impinging on the 
plane at B moves through the fpace DB in the time that 
the plane itfelf would pafs from the polltion a be to ABC, 
it is manifeft that the distance Dt of the laid particle from 
the plane (produced) at the beginning of that time will 
be the meafure of the relative celerity with which the par¬ 
ticles of the fluid approach the plane in a direction per¬ 
pendicular to it; and confequently, that the force of the 
ltream in that direction will CC De 2 ; whence, by the refo- 
lution of forces, the efficacy in the propofed direction B H 
will CCDc 2 %, fin. ABKCXD« 2 X fin.a^H. Now the angle 
^BD being given, as well as the lides B b, BD, containing 
that angle, the remaining angle BfiD will be known, as well 
as the fide D b ■, of confequence, De being the fine of the 
angle D be to the given radius Db, the effect De 2 X fin. abli 
will be a maximum, when fin. 2 D£ex fin. ab- H is a maxi¬ 
mum; that is, when fin. fDbatn <zZ>H)=:f fin. BfiD ; whence, 
the difference being given, the angles themfelves will be 
known. The geometrical conltrufilion is very fimple : 
Shus, having from the centre b with any radius deferibed 
the arc mr, on rb (produced if neceffary) let fall the per¬ 
pendicular mp ; take p q—^mp, and draw qs parallel t opr, 
cutting the circle in s ; then bifefit the arc ms by the line 
b a e, and the thing required is done. For the fin. sv of s r, 
that is, of the difference of the angles Dba, abH, is ^ ot mp, 
the fine of the whole given angle B bD, as it ought to be. 
To obtain a general theorem expreffed algebraically, let 
the velocity b B of the plane be put —v, and that ot the 
fluid “V ; alfo let the angle DB£ be called B ; and having 
drawn BFL perpendicular to the plane, or to bFe, put bF=ox, 
and BF —jy. Then, becaufe F B and FL are tangents of 
the angles F^B, FbL, to the common radius fiF, it follows 
that FL=2BF—2_y ; whence, if LR and DQ be perpendi¬ 
cular to HQ, we have, by fimilar triangles, B b : BF :: 
BL : B R; that is, v : y :: %y : — 
V 
1/2 —7/2 
- = BR ; and confe¬ 
quently, £R=BR—B b— 
IT 
Likewife, Bb : bF 
BL : LR ; that is3 y 
fin. B, and £Q=V . cof. B 
triangles, D Q : :: LR 
3*7 
3 ^=lr. 
But DQ=V 
B —v 
v v 
tremes of this analogy, we obtain 
V . cof, B —v 
-v-, w'e have again, by fimilar 
: bR ; or V . fin. B : V . cof. 
Multiplying the means and ex- 
.3 f 
-v: 
-X 3 xy. 
V. fin. B 
Subftituting in this equation for v 2 its equal x s -fy 2 , 
completing the fqttare and reducing, we at length find 
col. B— v \ 2 3 (V cof. B- 
W-h-F- 
V fin. B 
-j 2 _: 
-v) 
2 V.fin.B 
An<f this equation manifeftly expreffes the natural tangent 
of the angle b B F, or the cotangent of the required anple 
Fb H. 
Cor. 1. If the given angle D B b be a right angle (as is 
the cafe when the wind Its ikes againlt the fails of a wind¬ 
mill), then is the fine B=i, and cofine B=o, the expref- 
ifion for the tangent of bB F (which is here equal to the 
Jo v* *2 ZJ 
angle of inclination ABD) will become 2+-- u —— 
0 V ‘ 4 v2 T 2 y 
This, if v be taken = o, or the plane be fuppoied at reft, 
will be barely 2, anl wering to an angle of 54° 44.'. 
But, if the velocity of the plane be fuppofed l, ■*-, 01 
of that of the medium or ltream, then the angle ABD, 
found from this theorem, will be equal to 58° 14', 6i° 27', 
63° 26', 66° 58', or 74° 19', refpectively : fo that, the 
greater the velocity of the plane, the greater alfo will be 
the angle of inclination. Hence it appears, that the fails 
of a windmill, in order that the effefit may be the greateft, 
ought to be more turned towards the wind in the ex¬ 
treme parts where the motion is fwiftgft than in the parts 
nearer to the axis of motion ; in fuch a manner that the 
tangent of the axle formed by the direflion of the wind 
and the fail may every-where be equal to the expreffion 
/ 9 v 2 -3 v 
y 2+ —rf, the velocity v being proportional to the 
dilfance from the axis of motion, and increafing till, at 
the extremity of the fail, it is foir.etimes equal to V, or 
even exceeds it. 
Cor. 2. If the angle D B A, which the direction of the 
ft ream makes with the plane, be given, inftead of the angle 
DBH or DB6; ir. will then appear tnat the effect will 
in that cale be a maximum when Sine ABH (the angle 
made by the plane and the direction of its motion) : fine 
DBA :: |BD : Bb. 
For, the force in the direction FB varying as De 2 , its 
effefit in the direction BH will OC De 2 X—— CC——— 
Bb B b 
Now DB, B b, and the angle D B E, being given, DE is 
thence given. And it is well known that the folid of 
the fquare of one part of a line into the other part is a 
maximum, when the former part is double of the latter. 
Confequently De mult be=2Ee; fo that Ee,orits equal. 
BF, will be iDE. 
But, fine B b F : radius :: B F (= ■§ D E) : B b, 
and radius j line DBA :: B D : D E; 
whence, fine B b F : fine DBA :: ■§ B D : B b. 
Cor. 3. The proportion in the preceding corollary can 
only obtain when B b is equal to or greater than |DE. 
For, when Bd is lefs than |DE, Ee (which is always 
lefs than B b) canqot be equal ro $ D E ; but will approach 
the nearelt to it when BF coincides with Bb, that is, 
when the angle F B H or ABH is of 90 0 ; and in this 
cafe the effect will be a maximum when the direction of 
the motion is perpendicular to the plane. If the given 
angle DBA be a right angle (which appears to be the 
molt advantageous, becaufe then DE —DB), it follows, 
that fine ABH will be to the radius as ^ of the velocity 
of the ltream to the velocity of the plane or litii. Hence, 
if the force of the wind be capable of producing a degree of 
velocity in a flip greater than ^ of its own velocity, it is evi¬ 
dent that the flap may run Jwifter upon an oblique courfe than 
when Jhe fails direBly brjore the wind. If the velocity be 
to that ot the wind as 1 to 3, and the courfe be 109 0 28', 
the force of the wind upon the veffel to promote its mo¬ 
tion will be greater than the force, in a direfit courfe of 
18o°, in the ratio of 1,2 to 3 4/27, or of 3‘i748 to 3. 
Of the Discharge of Fluids through Apertures ; and 
of Spouting Fluids. 
Prop. XXVI. If a Jluid run through any tube which is kept 
continually full, and the velocity of the jluid in every part of one 
and the Janie JeBion be the fame, the velocities in different feclions 
will be inverfely as the areas of the feEtions. —For, as the tube 
is always equally full, the lame quantity of fluid will run 
through every lefition in the fame time': but the quan¬ 
tity puffing th rough any fefition S with the velocity V in 
any given time manifeltly varies as S and V conjointly, 
or as S . V; and, in iike manner, the quantity palling 
through any other fefition s with velocity v mult vary 
as s . v in a given time: confequently we mult have 
S . V=i . v, and S : s :: v : V. It is fuppoied in this 
propofition that the changes in the diameters of the tube 
are continual, and no-where abrupt lo as to break the law 
of continuity in the fides of the tube ; for, if there be 
any angles dr confiderable linuolities in the tube, they 
will produce eddies in the motion of the fluid, and the 
propofition will not obtain. 
Prop. XXVII. If a fluid flowing through a very f mall 
orifice in the bottom of a veffel be kept cunjlantly at the fame 
height in the veffel, by being fupplied as JaJl above as it runs 
avt 
