M E 0 H A N I C S. 
663 
wheels, or wear their teeth. But, as it is' impoffible in prac¬ 
tice to give that perfect curvature to the faces of the teeth 
which theory requires, a quantity of friction will remain 
after every precaution has been taken in the formation of 
the communicating parts. 
2. The fecond mode of a£tion is not fo advantageous 
as that which vve have been confidering, and fboulil, it 
poffible, always be avoided. It is reprefented in fig. 95. 
where A is the centre of the pinion, B that of the wheel, 
and A B the line of centres, as before. It is evident from 
the figure, that the tooth C of the wheel a'£ts upon the 
leaf D of the pinion before they arrive at the line B A ; 
that it quits the leaf when they reach this line, and' have 
afl'umed the pofition of E and F ; and that the tooth c 
works deeper and deeper between the leaves of the pinion, 
the nearer it comes to the line of centres. From this laft 
circnmltance a conftderable quantity of fri&ion arifes, be- 
caufe the tooth C does not, as before, roll upon the leaf D, 
but hides upon it; and from the fame caufe the pinion 
foon becomes foul, as the dull which lies upon the acting 
faces of the leaves is pufhed into the interjacent hollows. 
One advantage, however, attends this mode of aftion : it 
allows us to make the teeth of the large wheel rectilineal, 
and thus renders the labour of the mechanic lefs, and the 
accuracy of his work greater, than if they had been of a 
curvilineal form. If the teeth C, E, therefore of the 
wheel BC are made rectilineal, having their furfaces di¬ 
rected to the wheel’s centre, the ailing faces of the leaves 
D, F, &c. mult be epicycloids formed by a generating 
circle, whofe diameter is equal to the radius Bo of the 
circle op, rolling upon the circumference mn of the pi¬ 
nion A. But, if the teeth of the wheel and the leaves of 
the pinion are made curvilineal, as in the figure, the faces 
of the teeth of the wheel mult be portions of an interior 
epicycloid formed by any generating circle rolling within 
the concave fuperficies of the circle op, and the faces of 
the pinion’s leaves mult be portions of an exterior epi¬ 
cycloid produced by rolling the fame generating circle 
upon the convex circumference mi 2 of the pinion. 
3. The third mode of aftion, which is reprefented in 
fig. 96. is a combination of the two firll modes, and con- 
fequently partakes of the ad vantages and difadvantages of 
each. It is evident from the figure, that the portion eh 
of the tooth ails upon the part be of the leaf till they 
reach the line of centres A B, and that the part ed of the 
tooth acts upon the portion b a of the leaf after they have 
palled this line. Hence the acting parts eh and he mud 
be formed according to the diredtions given for the firft 
inode of adlion, and the remaining parts ed, b a, mult have 
that curvature which the fecond mode of adlion requires ; 
confequently eh lliould be part of an interior epicycloid 
formed by any generating circle rolling on the concave 
circumference mn of the wheel, and the correfponding 
part b e of the leaf lliould be part of an exterior epicycloid 
formed by the fame generating circle rolling upon b E O, 
the convex circumference of the pinion ; the remaining 
part ed of the tooth Ihould be a portion of an exterior 
epicycloid, engendered by any generating circle rolling 
upon eL, the concave fuperficies of the wheel ; and the 
correfponding part b a of the leaf Ihould be part of an in¬ 
terior epicycloid defcribed by the fame generating'circle, 
rolling along the concave fide b E O of the pinion. As it 
would be extremely troublefome, however, to give this 
double curvature to the adling faces of the teeth, it will 
be proper to life a generating circle, whofe diameter is 
equal to the radius of the wheel BC, for deferibing the 
interior epicycloid eh, and the exterior one be ; and a ge¬ 
nerating circle, whofe diameter is equal to A C, the radius 
of the pinion, for deferibing the interior epicycloid b a, 
and the exterior one e d. In this cafe the two inte¬ 
rior epicycloids, eh, b a, will be ftraigbt lines tending to 
the centres B and A ; and the labour of the mechanic will 
by this means be greatly abridged. 
In order to find the relative diameters of the wheel and 
pinion, when the - number of teeth in the one and the num¬ 
ber of leaves in the other are given, and when the dif- 
tan.ee of their centres is alfo given, and the ratio of ES 
to C S, let a be the number of teeth in the wheel, b the 
number of leaves in the pinion, e the difiance of the pi¬ 
vots A, B, and let m be to n as E S to CS; then the arc 
7 60^ 
E S, or £ S A E, will be equal to——, and L D, or 
Z LBD, will be equal to 
confequently LD : 
360° 
But ES : CS=;i 
n ■, therefore (Enel. vi. 16 ). 
LC X « = LDx», and LC= £ 5 ^ 2 .. but LD is equal to 
3^0 fy O ^ 
-— therefore by fubftitution LC:= —-. 
a > am 
Now, in the triangle APB, AB is known, and alfo 
P B, which is the cofine of the angle A B D, PC being 
perpendicular to D B ; A P, or the radius of the pinion, 
therefore may be found by plane trigonometry. The 
reader will obferve, that the point P marks out the parts 
of the tooth D and the leaf SP where they commence 
their action ; and the point I marks out the parts where 
their mutual action ceafes. The letter L marks the in* 
terfedlion of the line BL with the arch era, and the let¬ 
ter E the interfeflion ot the arch b O with the upper fur- 
face of the leaf m. The letters D and S correfpond with 
L and E refpedlively, and P with I. A P therefore is the 
proper radius of the pinion, and B I the proper radius of 
the wheel, the parts of the tooth L without the point I, 
and of the leaf S P without the point P, being fuperfluous. 
Now, to find BI, we have ES : CS=ffl:», and CSs 
ESx# „ 360 
—-j but ES was fliown to be =:——, therefore, by lub- 
m. A ? * j 
ftitution, CS: 
_36oX n 
bm 
360 
Now the arch ES, or ^EAS, 
being equal to anti CS, or ,,/CAS, being equal to 
360 Xrc 
b m 
-, their difference EC, or the angle EAC, will be 
360 360X22 360 °X ’«—n 
equal to ~j ~——;-, or 
The ^EAC be. 
b m ’ b m 
ing thus found, the triangle E A B, or I A B, which is 
almolt equal to it, is known, becaufe A B is given, and 
likewife A I, which is equal to the cofine of the angle 
I A B, AC being radius, and A I C being a right angle ; 
confequently IB, the radius of the wheel, may be found 
by trigonometry. It was formerly fhown that A C, the 
radius of what is called the primitive pinion, was equal to 
cb ... 
, and that BC, the radius of the primitive wheel, was 
ACx<z 
If then we fubtradl AC or AS from 
a+b' 
equal to 
AP, we fnall have the quantity S P, which mull be added 
to the radius of the primitive pinion ; and, if we take the 
difference of BC (or B L) and D E, the quantity LE 
will be found, which mull be added to the radius of the 
primitive wheel. We have ail along fuppofed that the 
wheel drives the pinion, and have given the proper form 
of the teeth upon this fuppefition. But, when the pinion 
drives the wheel, the form which was given to the teeth 
of the wheel in the firft cafe, nuilt in this be given to the 
leaves of the pinion ; and the fhape which was formerly 
given to the leaves of the pinion mull now be transferred 
to the teeth of the wheel. 
Another form for the teeth of wheels, different from 
any which we have mentioned, has been recommended by. 
Dr. Robifon. He fliows that a perfedl uniformity of ac¬ 
tion may be fecured, by making the aft ing faces of the 
teeth involutes of the wheel’s circumference, which are 
nothing more than epicycloids the centres of whofe gene¬ 
rating circles are infinitely diftant. Thus, in fig. 97, let 
AB 
