which the numerator is a prime. But this fraction may be by the above method resolved into two: And if these fractions be employed for the trains μ and v, the axes Aa, Bb will revolve with the required ratio, And the periods are inversely as the synchronal rotations. If, therefore, a period of twelve hours be given by a clock to the axis Bb, Aa will receive a period accurately equal to a lunation. The mechanism may be thus represented in the notation already explained. If the fraction be resolved into a difference instead of a sum, mechanism, by making the trains μ and v of different signs, that is, by making their extreme wheels revolve different ways. 414. Ex. 2.-Mean time is to sidereal time nearly as 8424: 8401. and we obtain the following train, which differs from fig. 294 only in fixing the wheels E and K upon a single axis, which also carries a wheel of 39, geering with a wheel of 31 upon Aa, as appears in the following notation. Ꮓ 415. Second case. The fraction in the first case has been supposed to have a decomposable denominator. Let now both denominator and numerator be prime. Form two fractions A and in which A is an arbitrary quantity and commodiously decomposable into factors, and proceed to obtain from each of these fractions the sums or differences of two decomposable fractions as before, which may be employed in wheel-work as follows. Let an axis Aa, fig. 294, be connected to one axis Bb, by two trains and an epicyclic train, as in the figure, and also to another axis Ce by a precisely similar arrangement. Then if the synchronal rotations of the axes Aa, Bb, Cc be A, a and a,,, v the trains which connect Aa with Bb, and μ, v, the trains that connect Aa with Cc, we shall have will be the ratio of the synchronal rotations of Bb and Cc. Suppose for example that it be required to make one axis perform 17321 turns, while another makes 11743; both being prime numbers, the fraction is irreducible, and indecomposable into factors. 17321 Assume a divisor 5040-7 x 8 x 9 x 10, and form separately two trains whose velocities are represented by 87 and as in the first method. (Art. 412.) 63 40' 74 whence the trains If we represent the wheels which in the left-hand train correspond to F, G and H, by f, g and h, we have the following notation of the resulting machine. EXAMPLES OF THE THIRD USE OF EPICYCLIC TRAINS. 416. The third employment of epicyclic trains, is to produce a very slow motion. In the formula" Art. 401, all the = μεν p E 1 trains are at present taken positive. Let be made negative, and let μ and have different signs, in which, by properly assuming the numbers of the trains, a may be made very small with respect to p, and therefore the arm to revolve very slowly. This leads to such an arrangement as that of fig. 289 (Art. 400), and in this expression the two terms of the numerator having no common divisor, may be so assumed as to differ by unity, by which an enormous ratio may be produced. For example, put a, c, e, g each equal 83, and we get α Р = = 106 × 83 (832+84 x 65) 108646502 If in this machine we suppress the wheels h and e by making a turn both b and g, and d turn both ƒ and c, we have* 417. If on the contrary we wish to make the shaft, whose revolutions are p, revolve slowly with respect to the arm; then the α numerator of the fraction must be a sum, and the denominator Р a difference; therefore & must in the expression = a με ν Р -3 be positive, and nearly equal to unity, and μ and must have different signs. Fig. 295 is a combination that will answer the present purpose: mp is a fixed axis upon which turns a long tube, to the lower end D n of which is fixed a wheel D, and to the upper a wheel E; a shorter tube turns upon this, which carries at its extremities the wheels A and H. A wheel C is enG gaged both with D and A, and a trainbearing arm mn, which revolves freely upon mp, carries upon a stud at n the united wheels F and G. The epicyclic train therefore is formed of the wheels EFG and H, and is plainly positive, the extreme wheels EH revolving in the same direction. Let H be the first wheel; .. E= HF GE' and v=C with different signs, since A and D revolve different ways; D * Putting a=20, b=100, c=101, g=99, and f=100. This latter combination is given with these numbers by White (Century of Inventions). put A=10, C=100, D=10, E=61, F=49, G=41, H=51, and a we shall obtain =25000, that is, 25000 rotations of the train bearing arm mn will produce one of the wheel C. 418. Generally, however, the first wheel of the epicyclic train n is fixed, in which case the formula becomes 1-ε. If be a positive and very near unity, this will be very small, or n small with respect to a, that is, the motion of the last wheel of the train slow with respect to that of the arm. In the simple forms of epicyclic trains, figs. 285, 286, and 287, the two latter are excluded, because is negative, but the former with the train is usually selected, A being a fixed wheel, and A b. -E D AE bD is made as small as possible; which is effected by but as these large numbers are inconvenient for the wheels that are carried upon the arm, 419. This combination is used for registering machinery for the same purpose as the contrivances in Arts. 395 and 396; and since the concentric wheels A and D (fig. 285) are very nearly of the same size, the pinions b and E carried by the arm may be made of the same number of teeth, or in other words, a thick pinion |