(224) agrees with equation (214) in other respects, and in the condition of the ambiguous sign. It is moreover appa rent, from the form assumed by the modulus in this case and in that of the preceding article, that the greatest conomy of power is obtained by applying the moving and the working pressures on the same side of the line C,C, joining the axes of the drums. This is in fact but a particular case of the general principle established in Art. 168.

195. The initial tension T of the band may be determined precisely as in the former case (equation 217).

Representing by the angle sub

tended by the circumference which the band embraces on the second or driven drum, by P the maximum resistance opposed to its motion at the distance a, by the limiting angle of resistance between the band and the surface of the drum, and by t, and t, the tensions upon the two parts of the band, when its maximum resistance being opposed, it is upon the point of slipping; observing, moreover, that in this case

2(t,-t,) or 2t is represented (Art. 193.) by 2T

; then

substituting in the second of equations (220) this value for 2t, and P and a for P, and a,, and neglecting the exceedingly small term which involves the product sin. a, sin. , we have

Also, since a, represents the inclination of the two parts of the band to one another; since, moreover, these touch the surfaces of the drums, and that represents the inclination: of the radii drawn from the centre of the lesser drum to thetouching points, therefore 0-a,. Substituting this value of 0 in the above equation, and solving it in respect to T, we have

196. The modulus of the band when the two parts of it, which intervene between the drums, are made to cross one another.

M

If the directions of the two parts of the band be made to cross, as shown in the accompanying figure, the moving pressure T, upon the second drum is applied to it on the side opposite to that on which it is applied when the bands do not cross; so that in this case, in order that the greatest economy of power may be attained (Art. 168.), the working pressure or resistance P, should be applied to it on the side opposite to that in which it was applied in the other case, and therefore on the side of the line CC, opposite to that on which the moving pressure P, upon the first drum is applied. This disposition of the moving and working pressures being supposed, and this case being inves tigated by the same steps as the preceding, we shall arrive at precisely the same expressions (equations 223 and 224) for the relation of the moving and the working pressures, and for the modulus.

In estimating the value of the initial tension T (equation 225) it will, however, be found, that the angle e, subtended at the centre C, of the second drum by the arc KML, which is embraced by the band, is no longer in this case represented by -a, but by +a,. This will be evident if we consider that the four angles of the quadrilateral figure C,KIL being equal to four right angles, and its angles at K and L being right angles, the remaining angles KIL and KCL are equal to two right angles, so that KC,L=~—«, ; · but the angle subtended by KML equals 2-KC,L; it equals therefore,. If this value be substituted for in equation (225) it becomes

Now the fraction in the denominator of this expression being essentially greater in value than that in the denominator of the preceding (equation 225), it follows that the initial tension T, which must be given to the band in order that it may transmit the work from the one drum to the other under a given resistance P, is less when the two parts of the band cross than when they do not, and, therefore, that the modulus (equation 224) is less; so that the band is worked with the greatest economy of power (other things being the same) when the two parts of it which intervene between the drums are made to cross one another. Indeed, it is evident, that since in this case the arc embraced by the band on each drum subtends a greater angle than in the other case, a less tension of the band in this case than in the other is required (Art. 185.) to prevent it from slipping under a given resistance, so that the friction upon the axis of the drums which results from the tension of the band is less in this case than the other, and therefore the work expended on that friction less in the saine proportion.

THE TEETH OF WHEELS.

197. Let A, B represent two circles in contact at D, and moveable about fixed centres at C, and C,. It is evident that if by reason of the friction of these two circles upon one another at D any motion of rotation given to A be communicated to B, the angles PC,D and QC,D described in the same time by these two circles, will be such as will make the arcs PD and QD which they subtend at the circumferences of the circles equal to one another. Let the angle PC,D* be represented by &,, and the angle QC,D by 4,; also let the radii C,D and C,D of the circles be represented by r, and r,. Now, arc PD=r11, arc QD=,,; and since PD-QD, therefore r,,r,,;

Or rather the arc which this angle subtends to radius unity.

The angles described, in the same time, by two circles which revolve in contact are therefore inversely proportional to the radii of the circles, so that their angular velocities (Art. 74.) bear a constant proportion to one another; and if one revolves uniformly, then the other revolves uniformly; if the angular revolution of the one varies in any proportion, then that of the other varies in like proportion.

When the resistance opposed to the rotation of the driven circle or wheel B is considerable, it is no longer possible to give motion to that circle by the friction on its circumference of the driving circle. It becomes therefore necessary in the great majority of cases to cause the rotation of the driven wheel by some other means than the friction of the circumference of the driving wheel.

One expedient is the band already described, by means of which the weels may be made to drive one another at any distances of their centres, and under a far greater resistance than they could by their mutual contact. When, however, the pressure is considerable, and the wheels may be brought into actual contact, the common and the more certain

method is to transfer the motion from one to the other by means of projections on the one wheel called TEETH, which engage in similar projections on the other.

In the construction of these teeth the problem to be solved is, to give such shapes to their surfaces of mu tual contact, as that the wheels shall be made to turn by the intervention of their teeth precisely as they would by the friction of their circumferences.

m

198. That it is possible to construct teeth which shall answer this condition may thus be shown. Let mn and m'n' be two curves, the one described on the plane of the circle A, and the other on the plane of the circle B; and let them be such that as the circle A revolves, carrying round with it the circle B, by their mutual contact at D, these two curves mn and m'n' may continually touch

B

one another, altering of course, as they will do continually, their relative positions and their point of contact T.

It is evident that the two circles would be made to revolve by the contact of teeth whose edges were of the forms of these two curves mn and m'n' precisely as they would by their friction upon the circumferences of one another at the point D; for in the former case a certain series of points of contact of the circles (infinitely near to one another) at D, brings about another given series of points of contact (infinitely near to one another) of the curves mn and m'n' at T; and in the latter case the same series of points in the curves mn and m'n' brought into contact necessarily produces the contact of the same series of points in the two circumferences of the two circles at D.

To construct teeth whose surfaces of contact shall possess the properties here assigned to the curves mn and m'n' is the problem to be solved. Of the solution of this problem. the following is the fundamental principle:

199. In order that two circles A and B may be made to revolve by the contact of the surfaces mn and m'n' of their teeth, precisely as they would by the friction of their cir

cumferences, it is necessary, and it is sufficient, that a line drawn from the point of contact T of the teeth to the point of contact D of the circumferences should, in every position of the point T, be perpendi cular to the surfaces in contact there, i. e., a normal to both the curves mn and m'n'.

To prove this principle, we must first establish the following LEMMA:-If two circles M and N be made to revolve

about the fixed centres E and F by their mutual contact at L, and if the planes of these circles be conceived to be carried round with them in this revolution, and a point P on the plane of M to trace out a curve PQ on the plane of N whilst thus revolving, then is this curved line PQ precisely the same as would have been described on the plane of N by the same point P, if the latter plane, instead of revolving, had remained at rest, and the centre E of the circle M having been released