6(32 
MECH A N "I C S. 
or drift upon the centre ; and the wear of the whole will be 
equable. 
It would be eafy to (how, did.the limits of this article 
permit it, that, when one wheel impels another, the im¬ 
pelling power will fometimes aft with greater and fome- 
times with lefa-force, unlefs the teeth of one or both of the 
wheels he parts of a curve generated after the manner of 
an epicycloid by the revolution of one circle along the 
convex or concave fide of another. It may be fuliicient 
to (how, that, when one wheel impels another by the ac¬ 
tion of epicycloidal teeth, their motion will be uniform. 
Let the wheel C D, fig. 91, drive the wheel A.B by means 
of the epicycloidal teeth mp, nq, 0 r, afting upon the in- 
finitely-fmall pins or fpindles a, b, c ; and let the epicy¬ 
cloids mp, nq. Sec. be generated by the circumference of 
the wheel A B, rolling upon the convex circumference of 
the wheel C D. From the formation of the epicycloid, it 
is obvious that the arch a b is equal to m n, and the arch 
ado mo; for, during the formation of the part nb of the 
epicycloid nq, every point of the arch ab is applied to 
every point of the arch mn ; and the fame happens during 
the formation of the part co of the epicycloid or. Let 
us now fuppofe that the tooth mp begins to aft on the 
pin a, and that b and c are fuccefiive pofitions of the pin a 
after a certain time; then, nq, or, will be the pofitions 
of the tooth mp after the fame time; but ab—.mn, and 
a c~ mo ; therefore the wheels AB, CD, when the arch 
is driven by epicycloidal teeth, move through equal fpaces 
in equal times, that is, the force of the wheel CD and 
the velocity of the wheel AB are always uniform. 
In illuftrating the application of this property of the 
epicycloid, which was difeovered by Olaus Roemer the 
celebrated Danifli aftronomer, we (hall call the fuiall wheel 
the pinion, and its teeth the leaves of the pinion. The line 
which joins the centre of the wheel and pinion is called 
the line of centres. There are three different ways in which 
the teeth of one wheel may drive another, and each of 
thefe modes of aftion requires a different form for the 
teeth. 
x. When the aftion is begun and completed after the 
teeth have pafl'ed the line of centres. 
2. When the aftion is begun and completed before they 
s'eacli the line of centres. 
3. When the aftion is carried on, on both fides of the 
line of centres. 
1. The firft of thefe modes of aftion is reprefented in 
fig. 93. where B is the centre of the wheel, A that of the 
pinion, and AB the line of centres. In this and the 
three following figures, B is fuppofed to be placed at the 
centre of a large wheel, the upper part of which only is 
(hown, and A at the centre of a pinion w orking in it. It 
is evident from the figure, that the part b of the tooth a b 
of the wheel does not aft on the leaf m of the pinion till 
they arrive at the line of centres A B ; and that all the 
aftion is carried on after they have pafl'ed this line, and 
is completed w hen the leaf m comes into the fituation n. 
When this mode of aftion is'adopted, the afting faces of 
the leaves of the pinion (hould be parts of an interior epicy¬ 
cloid, generated by a circle of any diameter roiling upon 
the concave fuperficies of the pinion, or within the cir¬ 
cle adh; and the faces ab of the teeth of the wheel (hould 
be portions of an exterior epicycloid formed by the fame ge¬ 
nerating circle rolling upon the convex fuperficies odp of 
the wheel. 
But, when one circle rolls within another whofe diame¬ 
ter is double that of the rolling circle, the line generated 
by any point of the latter is a jlraight line, tending to the 
centre of the larger circle. Therefore, if the generating 
circle above-mentioned (hould be taken with its diameter 
equal to the radius of the pinion, and be made to roll 
upon the concave fuperficies adh of the pinion, it will ge¬ 
nerate a ftraight line tending to the pinion’s centre, which 
will be the form of the faces of its leaves ; and the teeth 
of the wheel will be exterior epicycloids formed by a ge¬ 
nerating circle, whofe diameter is equal to the radius of 
the pinion, rolling upon the convex fuperficies 0 dp of the 
wheel. This reftilineal form of the teeth is exhibited 
in fig. 94.. and is perhaps the molt advantageous, as it re¬ 
quires lefs trouble, and may be executed with, greater ac¬ 
curacy, than if the epicycloidal form had been employed, 
though the teeth are evidently weaker than thofein fig. 93. 
It is recommended both by De la Hire and Camus as par¬ 
ticularly advantageous in clock and watch work. 
. The attentive reader will perceive, from fig. 93, that, in 
order to prevent the teeth of the Wheel from afting upon 
the leaves of the pinion before they reach the line of cen¬ 
tres A B, and that one tooth of the wheel may not quit 
the leaf of the pinion till the fucceeding tooth begins to 
aft upon the fucceeding leaf, there muft be a certain pro¬ 
portion between the number of leaves, in the pinion and 
the number of teeth in the wheel, or between the radius of 
the pinion and the radius of the wheel, when the diftance 
of the leaves A B is given. But in machinery the number 
of leaves and teeth is always known from the velocity 
which is required at the working point of the machine. 
It becomes a matter therefore of great importance to de¬ 
termine with accuracy the relative fadii of the wheel and 
pinion. 
For this purpofe, let A, fig. 94, be the pinion, having 
the afting faces of its leaves Itraight lines tending to the 
centre, and B the centre of the wheel. AB will be the 
diftance of their centres. Then, as the tooth C is fuppofed 
not to aft upon the leaf A m till it arrives at the line A B, 
it ought not to quit A m till the following tooth F has 
reached the line A B. But, fince the tooth always afts in 
the direftion of a line drawn perpendicular to the face of 
the leaf A m from the point of contact, the line C H, 
drawn at right angles to the lace of the leaf A m, will de¬ 
termine the extremity of the tooth C D, or the laft part 
of it which (hould aft upon the leaf Am, and will alfo 
mark out CD for the depth of the tooth. Now, in order 
to find AH, HB, and CD, put a for the number of teeth 
in the wheel, b for the number of leaves in the pinion, 
c for the diftance of the pivots A and B, and let x be the 
radius of the wheel, and y that of the pinion. Then, fince 
the circumference of the wheel is to the circumference of 
the pinion, as the number of teeth in the one to the num¬ 
ber of leaves in the other, and as the circumferences of 
circles are proportional to their radii, we (hall have 
a : bzzzx : y; then by compofition (Eucl. v. 18.) a + b : 
b=.c : y (c being equal to xfy), and consequently the ra- 
c b 
dius of the pinion, viz. y— -then, by inverting the firft 
a-\-b 
analogy, we have b : a—y : x, and confequently the ra¬ 
ay 
—; y being now a known 
dius of the wheels, viz. xz 
number. 
Now, in the triangle A H C, right-angled at C, the fide 
A H is known, and likewife all the angles (H A C being 
1 36° t 
equal to —-j ; 
the fide AC, therefore, may be found by 
plane trigonometry. Then, in the triangle A C B, the 
/CAB, equal to H A C, is known, and alfo the fides 
AB, AC, which contain it ; the third fide, therefore, viz. 
C B, may be determined ; from-which D B, equal to H B, 
already found, being fubtrafted, there will remain C D 
for the depth of the teeth. When the aftion is carried on 
after the line of centres, it olteu happens that the teeth 
will not work in the hollows of the leaves. In order to 
prevent this, the fC BH muft always be greater than 
half the The is equal to 360 degrees di¬ 
vided by the number of teeth in the wheel; and CBH is 
eafily found by plane trigonometry. 
If the teeth of wheels and the leaves of pinions be form¬ 
ed according to the directions already given, they will aft: 
upon each other, not only with uniform force, but nearly 
without friftion. The one tooth rolls upon the other, and 
neither Aides nor rubs to Such a degree as to retard the 
1 wheels, 
* 
