with a daily sidereal error of 0".0586, or 21" in the year*. 256. Another mode of indicating sidereal and solar time in the same clock, consists in placing behind the ordinary hour hand a moveable dial concentric with and smaller than the fixed dialt. Both dials must in this case be divided into twenty-four hours. The hand of the clock performs a revolution in twenty-four solar hours, and therefore indicates mean solar time upon the fixed dial as usual, but a slow retrograde motion is given to the moveable dial, so that the same hand shall point upon the latter to the sidereal time, which corresponds to the solar time shewn upon the fixed dial. For this purpose it is evident that during each revolution of the hour hand, the moving dial must retrograde through an angle corresponding to the quantity which sidereal time has gained upon solar time in twenty-four hours; which is 3'.56". 555 = 236".555, and as the entire circumference of the dial contains 86400", we have From this fraction approximate numbers may be obtain ed, by which the proper wheel-work for the motion of the dial can be set out. This is Francœur's result. + This method is due to Mr Margett, the details of his mechanism may be found in Rees' Cyclopædia, art. Dialwork. (4) contains a large prime 487, but is employed by Mr Margett. (B) = 3 x 11 x 19 103 is a better approximation. contains a smaller number, and CHAPTER VIII. ELEMENTARY COMBINATIONS. CLASS B. [DIRECTIONAL RELATION CONSTANT. 257. THE elementary combinations which are the subject of the preceding chapters, include those which are employed in all the largest and most important machines; for the parts of heavy machinery are always made to move with uniform velocity, if possible; and consequently with a constant velocity ratio and directional relation to each other. In the combinations which remain to be considered, either the velocity ratio, or directional relation, or both, vary; but as the arrangement of them is for the most part derived from some one or other of the previous contrivances, it will no longer be necessary to enter so much at large into the explanation of principles and of various forms, as a reference to the preceding chapters will for the most part suffice, at least for the less important machines. For this reason I have not thought it necessary to assign a separate chapter to each division of the classes B and C, as in class A, but shall include these classes each in a single chapter. CLASS B. DIVISION A. COMMUNICATION OF MOTION BY ROLLING CONTACT. 258. It has been already shewn, in Art. 35, that when a pair of curves revolving in the same plane in contact are of such a form as to roll together, the point of contact remains in the line of centers. The two radii of contact coincide therefore with this line, and the tangents of the angles made by the common tangent of the curves at the point of contact with their radii respectively are the same. 259. Ex. 1. In the logarithmic spiral the tangent makes a constant angle with the radius vector. Let two equal logarithmic spirals be placed in reverse positions, and made to turn round their respective poles as centers of motion, and let these centers be fixed at any distance that will permit the curves to be in contact. Then in every position of contact the common tangent will make the same angle with the radius vector of one curve that it makes on the opposite side with the radius vector of the other. The two radii of contact will therefore be in one line, and coincide with the line of centers, and hence, equal logarithmic spirals are rolling curves. Ex. 2. Let a Pm, APM be two similar and equal ellipses of which 8, h; S, H are the foci, and let them be placed in contact at any point P situated at equal distances a P, AP from the extremities of their major axes, and draw tPT the common tangent at P. = Now by the property of the ellipse the tangent makes equal angles with the radii s P, Ph; and because a P = AP, and the ellipses are equal, the tangent makes the same angle with the radii SP PH; whence tPs TPH, and 8 PH is a right line. Also 8 P = SP; .. 8P + PH = SP + PH = AM is a constant distance, whatever be the distance of the point of contact P from the extremity of the axes major. If, therefore, the foci s, H be made centers of motion, and their distance equal to the major axes of the ellipses, the curves will roll together. The logarithmic spiral and ellipse round the focus appear to be the only two rolling curves that admit of simple independent demonstrations of their possessing this property. 260. Suppose fig. 124 to represent any pair of rolling curves, and let r = s P be the distance of their point of contact P from the center of rotation s of the first curve, and 0 = as P the angle made by r with a fixed radius sa, and let r, PH,0,= PHA, be the corresponding quantities in the second curve, and c the distance sH of the centers; then sincer and r, are in the same straight line, = also the lengths of those parts of the curves a P, AP, that have been in contact are equal; ·· √√dr2 + r2d02 = [√ dr2, + r2 ̧d02 and as dr = – dr, ..rde = r,do, = c − r) do ̧. rde Again, is the tangent of the angle the first curve dr r de makes with r, and is the tangent of the angle the dr. second curve makes with r,, and these angles are the same; |