To produce the revolution of the planetary balls about the sun, a system of vertical concentric tubes is usually employed, which are adjusted very near to each other, but yet so far removed as not to influence each other's motion. These tubes are of different lengths, the innermost being the longest, and to the superior extremity of each a radius vector is attached, and thereby made to revolve once during each revolution of the tube. The lower extremities of the tubes form the arbors or axes of as many toothed-wheels, which are either immediately driven by pinions adjusted to a vertical axle called the 'annual arbor,' or derive their motions indirectly from those pinions by means of an interposed train of wheels. The determination of the relative number of teeth which must be given to the wheels and pinions in order to produce any required motion may be thus explained.
A pinion generally means a wheel consisting of a los.q number of teeth than that which it drives, but in the present article this restric tion is unnecessary. The teeth of a pinion are called leaves. The number of revolutions made by the wheel during one revolution of the pinion by which it is driven, is found by dividing the number of leaves iu the pinion by the number of teeth in the wheel f—thus, if there be 35 leaves and 7 teeth, the wheel will make 35 or 5 revolutions ;luring one revolution of the pinion ; but if there be 7 leaves and 35 teeth, the wheel will make or I of a revolution during one entire 35 E revolution of tho pinion. If to the axle of the wheel be adjitited a second pinion, which drives a second wheel, and if to the axle of this wheel a third pinion be adjusted which drives a third wheel, and so on, then the number of revolutions made by the last wheel during one revolution of the first pinion will be found by multiplying together the number of leaves in the several pinions, and dividing the result by the product of the number of teeth in the several wheels :—thus if there be four pinions, having 7, 8, 9, and 10 leaves respectively, and the same number of wheels, having 20, 21, 22, and 23 teeth respectively, the number of revolutions made by the List wheel during one revolution of the first pinion will be 7 x 8 x 9 x 10 6 or, in other words, 20x 21 x 22 x 23 — 253' the last wheel will revolve six times during 253 revolutions of the first pinion. Conversely the ratio which the product of the number of leaves must bear to the product of the number of teeth, in order to produce any required relative motion between the first pinion and the last wheel, is found by dividing the number of revolu tions made by the wheel by the number of revolutions to be made in the same time by the pinion. The actual number of teeth to be given to the wheels and pinions, as well as the number of wheels and pinions to be employed in any particular case, is matter of convenience, not of necessity: in every instance the employment of a single pinion and a single wheel is theoretically sufficient, but in practice it is generally desirable to avoid the use of wheels or pinions with a very large or very small number of teeth. In the planetarium of the Royal Institution the number of teeth is in no instance under 7, or above 137. In a more complete instrument, constructed by Dr. Pearson in 1813, the limits were 14 and 241. The same gentleman recommends about 10 teeth to the inch, which he considers "sufficiently strong, and not liable to unnecessary shake, when the teeth and spaces are made equal and at a proper depth for action." The lowest number employed by him was 7 to the inch, the highest 13.
Supposing we wish the radius which carries the hill representing the earth to revolve once during each revolution of the annual arbor, it is only necessary that the wheel which is adjusted to the lower extremity of the earth's tube should contain the same number of teeth as the pinion by which it is driven, and which is adjusted to the annual arbor. In this case each revolution of the annual arbor will be the measure of one solar year. If each revolution of the annual arbor be required to
represent any assigned portion of a year, the necessary modification in the relative number of teeth in the earth's wheel and pinion will appear sufficiently obvious from what has preceded; but for the sake of simplicity, we shall assume that the earth's radius vector revolves exactly once during each revolution of the annual arbor, and upon this supposition we have now to fix the relative number of teeth which should be given to the wheels and pinions which regulate the motions of the other planetary balls. It generally happens that the number of revolutions which the radius vector of any one of the planetary balls is required to make during one revolution of the annual arbor is ex pressed in the form of a decimal Suppose, for instance, that the relative motion required were that of the earth and Jupiter. Jupiter revolves in mean solar days; the earth in mean solar days; the number of revolutions made by Jupiter during one revolution of the earth is therefore = .0843045. If this decimal be converted into a continued fraction, the resulting series of fractions, which approximate more and more nearly to '0343045, will be found to be . , 7 29 , 94 sic., any one of which, 11 12 83 344 1,115 according to the degree of accuracy required, may be taken for the ratio which the number of leaves in the pinion must bear to the number of teeth in the wheel, if only a single wheel and pinion bo employed, or the ratio which the product of the number of leaves must bear to the product of the number of teeth, if a train of wheels and pinions be employed. If the first of these fractions, or its equivalent, 7 be taken,,the wheel attached to Jupiter's tube should 77 contain 77 teeth, and the pinion attached to the annual arbor by which it is driven should contain 7 leaves, and Jupiter's radius will then revolve once during 11 revolutions of the annual arbor, that is, in x 11 = 4017.8204 days, which is less than the true period by 3141644 days. In the same manner may be found the time in which Jupiter's radius will revolve when any of the other fractions are taken, as under :— The third of these fractions, or rather its equivalent, was 14 the one employed by Dr. Pearson in the construction of his new planetarium ; so that in that instrument the wheel attached to Jupiter's tube contains 166 teeth, and is driven by a pinion of 11 leaves attached to the annual arbor. In the planetarium of the Royal Institution, a train of wheels and pinions represented by the compound 94 fraction 111 x was employed, which therefore gives a period of 22 40 111 94 365.2564 x x 4330'778 The contrivance by means of which 'ss true elliptic orbit may be produced is extremely simple. For this purpose all that is necessary is that the radius vector which connects the planetary ball with the superior extremity of the tube should consist of two parts or arms, the lengths of which have a determinate ratio dependent solely upon the eccentricity of the orbit, and that while the larger arm revolves about the centre of the ellipse by means of the wheel-work already described, the smaller arm be made to revolve about the extremity of the larger with the same angular velocity but in the opposite direction. This may be effected in two ways. By means of a pulley fired to the planetary tube and connected by an endless silken cord with another pulley free to revolve about a vertical axle situated at the extremity of the larger arm. The effect of this connection will be that the latter pulley will revolve once during each revolution of the larger arm but in the opposite direction. lf, therefore, the smaller arm be attached to this pulley it will revolve in the manner required. The same motion may be produced by employing a double pinion extending the entire length of the larger arm and communicating the rotatory motion given to it at one extremity, to the axle of the smaller arm situated at the other.