MECHANICS. 
686 
fpeak more properly, it will not receive the impulfe.of the 
wind fo advantageoufty as thofe parts of the fail which have 
a lefs degree of velocity. In order, therefore, to counteract 
this diminution of force, we mull make the wind act more 
perpendicularly upon the faii, by diminilhing its obliquity 
or its angle of weather. But, fince the velocity of every 
part of the fail is proportional to its diftance from the 
centre of motion, every elementary portion of it mu ft 
have a different angle of weather, diminishing from the 
centre to the extremity of the fail. The law or rate of 
diminution, however, is ftill to be difcovered ; and we are 
fortunately in poffeffion of a theorem of Euler’s, after¬ 
wards given by Maclaurin, which determines this law of 
variation. Let a reprefent the velocity of the wind, and c 
the velocity of any given part of the fail; then the effort 
of/the wind upon that partofthe fail will be greateft when 
the tangent of the angle of the wind’s incidence, or of the 
/ 9 c c 3 c 
fail’s inclination to the axis,is to radius,as-v H--f-—-to i. 
4«za 2 a 
' In order to apply this theorem, let us fuppofe that the 
radius or whip E D of the fail a/Soy, fig. 25. is divided 
into fix equal parts ; that the point n is equidiftant from 
E and D, and is the point of the fail which has the fame 
velocity as the wind ; then, in the preceding theorem, 
we fhall have c— a, w hen the fail is loaded to a maximum ; 
and therefore the tangent of the angle, which the lur- 
face of the fail at n makes with the-axis, when a—i, will 
O'? 
2-U--p-=23’56i — tangent of 74 0 19', which gives 
4 2 . 
15 0 41' for the angle of weather at the point n. Since, 
at | of the radius c = a, and fince c is proportional to the 
diftance of the correfponding part of the fail from the cen¬ 
tre, we will have, at A of the radius sm, c — at t of the 
2 a 4 a 5 a 
radius, c=z — ; at f, c = —, at c — — ; and at the ex- 
3 3 3_ 
tremity of the radius, c—ia. By fubltituting thefe dif¬ 
ferent values of c, inftead of c in the theorem, and by 
making <2=1, the following Table will be obtained, which 
exhibits the angles of inclination and weather which mult 
be given to different parts of the fails. 
Parts of the lta- 
dius from the 
Centre of Mo¬ 
tion at E. 
Velocity of the 
Sail at thefe 
Diftances —or 
Values of c. 
Angle madewith 
the Axis. 
Angle of the 
Weather. 
11 
a 
Deg. 
Min. 
Deg. 
Min. 
3 
63 
26 
26 
34 
1 
6 
2dZ 
3 
69 
5+ 
20 
6 
1 or \ 
a 
43 
, 
*9 
is 
4 
~d 01 f 
3 
77 
20 
32 
40 
A 
6 
5f 
3 
79 
27 
IO 
33 
I 
2 a 
81 
O 
9 
O 
Mr. Smeaton found, from a variety of experiments, that 
the common practice of inclining plane fails from 72 0 to 75 0 
to the axis, was much more efficacious than the angle af- 
figr.ed by Parent, the effect being as 45 to 31. When the 
fails were weathered in the Dutch manner, that is, when 
their furfaces were concave to the wind, and when the an¬ 
gle of inclination increafed towards their extremities, they 
produced a greater effect than when they were weathered 
either in the common way, or according to Euler’s theorem. 
But, when the (ails were enlarged at their extremities, as 
represented at «. &, fo that cs /3 was one-third of the ra¬ 
dius ED, and «D to D /3 as 5 to 3, their power was greateft 
of all, though the furface ailed upon by the wind remained 
the fame. If the fails be farther enlarged, the effect is not 
increafed in proportion to the furface; and betides, when 
the quantity of cloth is great, the machine is much expoled 
to injury by fudden fqualls of wind. In Mr. Smeaton’s 
experiments, the angle of weather varied with the diftance 
from the axis; and it appeared from feveral trials, that the 
moft efficacious angles were thofe in the following Table : 
Parts of the .Radius 
ED, which is di¬ 
vided into 6 Parts, 
Angle with the 
Axis. 
Angle of Weither. 
I 
72 
18 
2 . * 
- 7i 
1 9 
. 3 
72 
18 middle 
4 
74 
l6 
5 
77l 
I2i 
6 
83 
7 
If the radius E D of the fail be 30 feet, then the fail 
will commence at a ED, or 5 feet from the axis, where 
the angle of inclination .will be 72 0 . At |- E D, or 10 feet 
from the axis, the angle will be 71°; and fo on. 
The following maxims deduced by Mr. Smeaton from 
his experiments, contain the moft accurate information 
upon this fubjeff. 
Maxim 1. The velocity of wind-mill fails, whether un¬ 
loaded or loaded, fo as to produce a maximum effect, is 
nearly as the velocity of the wind, their ffiape and pofition 
being the fame. 
2. The load at the maximum is nearly, but fomewhat 
lefs than, as the fquare of the velocity of the wind, the 
ffiape and pofition of the fails being the fame. 
3. The effects of the fame fails at a maximum, are- 
nearly, but fomewhat lefs than, as the cubes of the velo¬ 
city of the wind. 
4. The load of the fame fails at the maximum is nearly 
as the fquares, and their effects as the cubes, of their num¬ 
ber of turns in a given time. 
5. When fails are loaded fo as to produce.a maximum 
at a given velocity, and the velocity of the wind increafes, 
the load continuing the fame : iff, The increafe of effeft, 
when the increafe of the velocity of the wind is fmall, 
will be nearly as the fquares of thofe velocities ; adly. 
When the velocity of the wind is double, the effects will 
be nearly as 10 to 27,^5 but, 3dly, Whc-n the velocities 
compared are more than double of that where the given 
load produces a maximum, the effedts increafe nearly in 
the fimple ratio of the velocity of the wind. 
6. In fails wdiere the figure and pofitions are fimilar,. 
and the velocity of the wind the fame, the number of 
turns in a given time will be reciprocally as the radius or 
length of the fail. 
7. The load at a maximum that fails of a fimilar figure 
and pofition will overcome at a given diftance from the 
centre of motion, will be as the cube of the radius. 
8. The effedts of fails of fimilar figure and pofition are 
as the fquare of the radius. 
9. The velocity of the extremities of Dutch fails, as 
well as of the enlarged fails, in all their ufual pofitions. 
when unloaded, or even loaded to a maximum, are con- 
fiderably quicker than the velocity of the wind. 
A new mode of constructing the (ails of wind-mills has 
been recently given by Mr. Sutton, and fully defended 
by Mr. Hefleden of Barton, in a work exclufively devoted 
to tiie fubjedt. Mr. Sutton gives his fails the form repre- 
fented in fig. 26. and makes the angle of weather at the 
point M, equidiftant from A and B, equal to 22 0 30'. The 
inclination of the (ail at any other point N of the fail, is, 
an angle vvhofe fine is the diftance of that point from the 
centre of motion A, the radius being the breadth of the 
fail at that point. The figure (hows the angles at the dif¬ 
ferent points of the fail; and the apparent and ab.folute' 
breadths of the (ail at thefe points. Mr. Sutton’s mode 
of regulating the velocity of the fails, and of bringing, 
them to a ftaie of reft, is particularly ingenious. 
Of 
