In this table, if the numerators of the different fractions be taken as so many solar years, the respective denomina tors will express the number of tropical revolutions made by the corresponding planets in those years nearly ; and if the last fractions be taken in each line, the coincidence of the two periods will be very exact. In the instance of the earth, the numerators are years, and the denominators days, and, what is extraordinary, the seventh quotient leaves no remainder in the continual division ; consequent 164 ly, the value of is 365d. 5h. 48m. 48s and is 450 269 47 13 359 ble into the compound fraction, or train, 10 X 9 X 5 269 47 or into either of the following equivalents, x x 39 269 10 18 115 10 , or 26 x x , the last of which Dr. W. Pearson 94 has availed himself of in one of his new machines, here after described. From this remarkable coincidence be tween 164359 days and 450 solar years, we arrive at a perfect correction of the Calendar, or can reconcile the solar with the civil years, by simply omitting seven and retaining two centenary leap-years in every 900 years, in stead of omitting 9, as the practice is now established for 365 X 450 + 109 = 164359 ; hence, in every 450 solar years there should be 109 intercalary days, or 118 in 900 such years ; but if we count 24 leap years only in each of nine successive centuries, there will be only 116 intercalary days, instead of 118, and therefore every fourth (or fifth) and ninth century should alternately retain the 29th day of February, in order to make the solar years and civil days accord at the termination of every nine centuries.
Should any, or all of the synodic revolutions be fixed upon for the wheelwork of an orrery, for the sake of con necting the revolving arms without the assistance of tubes, as Dr. W. Pearson has done in his large orrery with equa ted motions, the fractions and trains for such periods may be computed by means of the converging series, as well as any other train. With the inferior planets the numerators will be found the same for both periods, but the denomi nators of the synodic revolutions will be equal to the de nominators of the periodic revolutions, after their numera tors are subtracted therefrom, provided that the earth be one of the two planets from which the synodic period is computed. In this case, a part of the synodic train must be placed on the earth's bar, or arm, which, by its annual motion, while the train is in action, will convert the sy nodic into a periodic revolution. This effect we have ready noticed in the instance of Venus in Ferguson's orrery. If we take which is one of her fractions for 13 producing a periodic revolution, when placed in a station ary position, her corresponding fraction for a synodic revo 13 8 5 8 lution will be provided the 5 revolve in a year, and be placed on the earth's revolving arm in con nexion with the 8, which 8 being central, will, under these circumstances, carry Venus in her periodic time as she has reference to the sun, but in her synodic period as she regards the earth ; hence a periodic becomes a synodic train when unity is ejected. In the computations for the inferior planets, the solar year was the dividend, hut in those for the superior planets, it was the divis. r ; hence the synodic train of Mars, or other superior planet, is derived from its periodic train by considering the denominator as unchangeable, and taking the difference for the nume rator.
When the synodic period, from conjunction to conjunc tion, of any two planets is required, look for one planet at the top, and the other at the side of the table, and the square which is common to both will contain the period sought for. For instance, the time which Mars will re quire in passing from his conjunction with Jupiter to the same relative situation again, will be 816.434 days in mean motion, hut this period will be lengthened or shortened according to the sum or difference of the equations of the two planets at the termination of the mean period. In all cases where synodic revolutions form the data for the computation of wheel-work, the resulting trains must con nect the two arms of the planets in question, and then the slower moving arm will push the quicker round once in the time of its own periodic revolution, without reference to the direct effect produced by the computed train.
In like manner, the synodic revolution naturally arises out of the periodic revolutions of two planetary arms, by reason of the difference of their velocities ; and though such synodic period is not contemplated in computing the periodic revolutions, it is a natural consequence of the relative velocities, provided the trains are in a, permanent situation that produce the respective motions.
When a graduated dial and its index are both revolving in the same direction, but with different velocities, so that either a direct or retrograde quantity is indicated by the difference, in given periods ; as is the case in some orre ries, where the moon's apogee and node have their mo tions pointed out; the period must be first reduced to its lowest terms, in the form of an improper fraction, and then the numerator will be the driven wheel, and the denomi nator must be substracted from or added to the numera tor, to constitute the driver, according as the motion is comparatively progressive or retrograde : for instance, if the moon's progressive motion of the apogee be taken at years, the improper fraction, reduced in the usual way, 7 62 2 will be 6 and --= will be the required wheels ; 7 62 53and when the fraction is in high numbers, for the sake of accuracy in the period, the difference may be applied in a similar manner, and the high fraction thus obtained may afterwards be broken into factors for a train, as has been already explained. As another instance, let us take the moon's node, the period of which is nearly years, or 3 56 56 56 when reduced; hence 3 59 will be the proper wheels; and if 56 revolve in a year, 59 immediately con 56 2 nected with it will revolve in , or years. In this case, also, if higher numbers were taken as the improper frac tion, for the sake of greater precision in the period, after the difference has been applied, the high numbers may be broken into factors for a train retaining the same value as the high simple fraction. Examples where trains have been used, for both the motions of the moon's apogee and node, will occur hereafter, when we come to describe Dr. W. Pearson's Tellurian and Lunarian united in one per fect machine. With respect to the computation of num bers for preserving the parallelism of the earth's axis, any numbers will do for the wheels, provided that the products of the numerators and of the denominators are the same, and provided that the earth is made to rotate backwards once in each year, or in the period during which the an nual bar carries it forwards round the sun.