MECHANICS. 
the fame way, only, indead of twice D*/, the diameter of 
the circle mull be twice F fe When any other mode of 
action is adopted, the teeth are to be formed in the fame 
manner that we have pointed out for common wheels, 
with this difference only, that different epicycloids are 
lieceffary for the parts 'F and D. It may be fuflicient, 
however, to find the form of the teeth at F, as the re¬ 
maining part of the tooth may be fhaped by direfting a 
ftraight rule from different points of the epicycloid at F 
to the centre A, and filing the tooth till every part of 
its acting furface coincide with the fide of the ruler. The 
reafon of this operation will be obvious by attending to 
the fliape of the tooth in fig. ioo. When the fmall wheel 
CD, fig. ioi, impels the large one D E, the epicycloids 
which were formerly given to C D mull be given to D E, 
and thofe which were given to D E muft be transferred 
to C D. 
The wheel reprefented in fig. 103, is 'fometimes called 
a crown-wheel, though it is evident from the figure that it 
belongs to that fpecies of wheels which we have juft been 
confidering; for the afling furfaces of the teeth both of 
the wheel MB and of the pinion EDG are directed to 
C, the common vertex of the two cones C M B, CEG. 
In this cafe the rules for bevelled wheels muft be adopted, 
in which AS is to be confidered as the radius of the 
w heel for the profile of the tooth at A, and MN as its 
radius for the profile of the tooth at M 5 and the epicy¬ 
cloids thus formed will be the feflions or profiles of the 
teeth in the direction MP, at right angles to M C the 
furfaces of the cone. When the vertex C of the cone 
MCG approaches to N till it be in the fame plane with 
the points M, G, fome of the curves will be cycloids and 
others involutes, as in the cafe of rackwork, tor then the 
cone CEG wiil revolve upon a plane furface. 
Of the Wipers of Stampers , and the Teeth of Rackwork. 
In fig. 104. let A B be the wheel which is employed to 
elevate the rack C, and let their mutual aMion not com¬ 
mence till the afting teeth have reached the line ofcentres 
AC. In this cafe C becomes as it were the pinion or 
wheel driven, and the afiing faces of its teeth muft be 
interior epicycloids formed by any generating circle rolling 
w ithin the circumference p q ; but, as pq is a liraight line, 
thefe interior epicycloids will be cycloids, or curves gene¬ 
rated by a point in the circumference of a circle, roiling 
upon a ftraight line or plane furface. The aMing face op, 
therefore, will be part of a cycloid formed by any genera¬ 
ting circle; and m n, the acting face of the teeth of the 
wheel, muft be an exterior epicycloid produced by the fame 
generating circle rolling on mr, the convex furface of the 
wheel. If it be required to make op a ftraight line, as in 
the figure, then mn mult be an involute of the circle mr, 
formed in the manner reprefented in fig. 97. 
Fig. 104. like wife reprefents a wheel depreffing the rack 
C when the third mode of ait ion is ufed. In this cate 
alfo C becomes the pinion, and DE the wheel; eh there¬ 
fore muft be part of an interior epicycloid formed by any 
generating circle rolling on the concave fide ex of the 
wheel, and he muft be an exterior epicycloid produced by 
the fame generating circle rolling upon the circumference 
of the rack. The remaining part cd of the teeth of the 
wheel, muft be an exterior epicycloid deferibed by any 
generating circle moving upon the convex fide ex, and 
ba muft be an interior epicj'cloid engendered by the fame 
generating circle rolling within the circumference of the 
rack. But, as the circumference of the rack is in this 
cafe a ftraight line, the exterior epicycloid he and the in¬ 
terior one l>a will be cycloids formed by the fame gene¬ 
rating circles which are employed in deferibing the other 
epicycloids. Since it would be difficult, however, as has 
already been remarked, to give this compound curvature 
to the teeth of the wheel and rack, we may ufe a gene¬ 
rating circle whole diameter is equal to Dj;, the radius 
©i the w heel, for deferibing the interior epicycloid ek, and 
VoL.Xiy. Mo. 1003. 
665 
the exterior one be-, and a generating circle whofe dia¬ 
meter is equal to the radius of the rack, for deferibing the 
interior epicycloid ah, and the exterior one de-, a b and 
eh, therefore, will be ftraight lines, and be will be a cy¬ 
cloid, and de an involute of the circle ex, the radius of 
the rack being infinitely great. 
In the fame manner may the form of the teeth of rack- 
work be determined, when the fecond mode of aftion is 
employed, and when the teeth of the wheel or rack are 
circular or reflilineal. Bur, if the rack be part of a cir¬ 
cle, it muft have the fame form for its teeth as that of a 
wheel of the fame diameter with the circle of which it 
is a part. 
In machinery where large weights are to be raifed, fuch 
as fulling-mills, mills for pounding ore, &c. or where 
large piltons are to be elevated by the arms of levers, if. 
is of the greateft confequence that the power fhould raife 
the weight with an uniform force and velocity ; and this 
can be effefled only by giving a proper form to the wiper. 
Now, there are two cales in which this uniformity of 
motion may be required ; and each of thefe demands a dif¬ 
ferent form for the communicating parts. 1, When the 
weight is to be raifed vertically, as the pifton of a pump. 
Sec, 2. When the weight to be raifed or depreffed moves 
upon a centre, and riles or falls in the arch, of a circle, 
fuch as the Hedge-hammer in a forge, See. 
1; Let A H, fig. 105. be a wheel moved by any power 
which is fufficient to raife the weight MN, by its extre¬ 
mity O, from O to e, in the fame time that the wheel 
moves round one-tourth of its circumference. It is re¬ 
quired to fix upon its rim a wing O B C D E H, which 
(hall produce this effetft with an uniform effort. Divide 
the quadrant O H into any number of equal parts O m, 
mn, Sec. the more the better, and Oe into the fame num¬ 
ber oh, he, cd, Sc c. and through the points m, n, p, H, 
draw the indefinite lines A B, A C, A D, AE, and make 
A B equal to Ai, A C to A c, AD to A d, and A E to 
At; then, through the points O, B, C, D, E, draw the 
curve O B C DE, which is a portion of the fpiral of Ar¬ 
chimedes, and will be the proper form for the wiper or 
wing O II E. It is evident that, when the point m has 
arrived at O, the extremity of the weight will have ar¬ 
rived at b, becaufe A B is equal to A b \ and, for the 
fame reafon, when the points n, p, H, have fuccefiively 
arrived at O, the extremity of the weight will have ar¬ 
rived at the correfponding points c, d, e. The motion 
therefore will be uniform, becaufe the fpace deferibed by 
the weight is proportional to the fpace deferibed by the 
moving power, Ob being to Oc as O m to On. If it be 
required to raife the weight MN with an accelerated or 
retarded motion, we have only to divide the line Oe ac¬ 
cording to the law of acceleration or retardation, and di¬ 
vide the curve OBCDE as before. 
2. When the lever moves upon a centre, the weight will 
rife in the arch of a circle, and confequently a new form 
muft be given to the wipers, or wings. Let A B, fig. io 5 . 
be a lever lying horizontally, which it is required to raife 
uniformly through the arch BC into the polidon AC, 
by means ot the wheel BFMEH, furniftied with the 
u'ing B N O P, which ails upon the extremity C of tiie 
lever; and let it be required to raife it through BC in 
the fame time that the wheel B F ii moves through one- 
half of its circumference ; that is, while the point M 
moves to B in the direction M F B. Divide the chord 
CB into any number of equal parts, the more the better, 
in the points 1,2, 3 ; and draw the lines 1 a, a b, 3 c, 
parallel to A B, or a horizontal line pulling through the 
point B, and meeting the arc CB in the points a, b, c. 
Draw the lines CD, «D, bO, cD, and B D, cutting the 
circle B F M E H in the points m, n, o, p. 
Having drawn the diameter B M, divide the femicircle 
BF M into us many equal parts as the chord CB, in the 
points q, s, u. Take B m, and let it from q to r; take B n 
and let it from s to t; take Ba and let it from u to vt 
8 G and* 
