It is true that our expressions, composed of periodic terms, are nothing more than translations into analytical language of the epicycles of the ancients;* but they are evolved directly from the fundamental laws of force and motion, and as many new terms as we please may be obtained by carrying on the same process; whereas the epicycles of Hipparchus and his followers were the result of numerous and laborious observations and comparisons of observations; each epicycle being introduced to correct its predecessor when this one was found inadequate to give the position of the body at all times: just as with us, the terms of the second order correct the rough results given by those of the first; the terms of the third order correct those of the second, and so on. But it is impossible to conceive that observation alone could have detected all those minute irregularities which theory makes known to us in the terms of the third and higher orders, even supposing our instruments far more perfect than they are; and it will always be a subject of admiration and surprise, that Tycho, Kepler, and their predecessors should have been able to feel their way so far among the Lunar inequalities, with the means of observation they possessed. LONGITUDE OF THE MOON. 64. We shall firstly discuss the expression for the moon's longitude, as found Art. (51). 0=pt+2e sin (cpt− a) + že3 sin2 (cpt — a) + me sin {(2-2m−c) pt−2B+a} — 3me' sin (mpt + B −5) -1ksin2 (gpty). * See Whewell's History of the Inductive Sciences. The mean value of 0 is pt; and in order to judge of the effect of any of the small terms, we may consider them one at a time as a correction on this mean value pt, or we may select a combination of two or more to form this correction. We shall have instances of combinations in explaining the elliptic inequality and the evection, Arts. (66) and (70); but in the remaining inequalities each term of the expression will form a correction to be considered by itself. 65. Neglecting all the periodical terms, we have which indicates uniform angular velocity; and as, to the same order, the value of u is constant, the two together indicate that the moon moves uniformly in a circle, the 2π period of a revolution being which is, therefore, the p expression for a mean sidereal month, or about 27 days.* The value of p is, according to Art. (50), given by and as m is due to the disturbing action of the sun, we see that the mean angular velocity is less, and therefore the mean periodic time greater than if there were no disturbance. Elliptic inequality or Equation of the Centre. 66. We shall next consider the effect of the first three terms together: the effect of the second alone, as a correction of pt, will be discussed in the Historical Chapter, Art. (109). *The accurate value was 27d. 7h. 43m. 11.261s. in the year 1801. See Art. (99). 0=pt+2e sin (cpt − a) + že2 sin 2 (cpt — a), which may be written 0 =pt+2e sin [pt−{a+(1−c) pt}]+že2 sin 2[pt−{a+(1−c) pt}]. But the connexion between the longitude and the time in an ellipse described about a centre of force in the focus, is, Art. (13), to the second order of small quantities: 0 =nt + 2e sin (nt — a') + že2 sin2(nt — a′), where n is the mean motion, e the eccentricity, and a the longitude of the apse.* Hence, the terms we are now considering indicate motion in an ellipse; the mean motion being p the eccentricity e, and the longitude of the apse a+(1−c) pt; that is, the apse has a progressive motion in longitude, uniform, and equal to (1-c) p. 67. The two terms 2e sin (cpt- a) + že sin 2 (cpt- a) constitute the elliptic inequality, and their effect may be further illustrated by means of a diagram. Let the full line AMB represent the moon's orbit about the earth E, when the time t commences, that is, when the moon's mean place is in the prime radius Er from which the longitudes are reckoned. The angle YEA, the longi E B tude of the apse, is then a. At the time t, when the moon's mean longitude is YEM=pt, the apse line will have moved in the same direction through the angle AEA' = (1 — c) YEM, * The epoch & which appears in the expression of Art. (13) is here omitted; a proper assumption for the origin of t, as explained in Art. (34), enabling us to avoid the ε. 7 and the orbit will have taken the position indicated by the = 2e sin A'EM + že2 sin 2A'EM = 2e sin A'EM (1 + že cos A'EM ) ; which, since e is about, is positive from perigee to apogee, 68. The angular velocity of the apse is (1-c) p, or, if for c we put the value found in Art. (48), the velocity will be m'p. Hence, while the moon describes 360°, the apse should describe m2. 360° 1° nearly, m being about 1. = But Hipparchus had found, and all modern observations confirm his result, that the motion of the apse is about 3° in each revolution of the moon. See Art. (112). This difference arises from our value of c not being represented with sufficient accuracy by 1-4m3. Newton himself was aware of this apparent discrepancy between his theory and observation, and we are led, by his own expressions (Scholium to Prop. 35, lib. III. in the first edition of the Principia), to conclude that he had got over the difficulty. This is rendered highly probable when we consider that he had solved a somewhat similar problem in the case of the node; but he has nowhere given a statement of his method: and Clairaut, to whom we are indebted for the solution, was on the point of publishing a new hypothesis of the laws of attraction, in order to account for it, when it occurred to him to carry the approximations to the third order, and he found the next term in the value of c nearly as considerable as the one already obtained. See Appendix, Art. (94). c=1-3m2 - 225 m3 ; ..1-c=3m2 + 225m3 = 3m2 (1 + 75m); .. (1−c) 360° = (1+754) (value found previously) 10 = 23° nearly, thus reconciling theory and observation, and removing what had proved a great stumbling-block in the way of all astronomers.* When the value of c is carried to higher orders of approximation, the most perfect agreement is obtained. The motion of the apse line is considered by Newton in his Principia, lib. 1., Prop. 66, Cor. 7. Evection. 69. The next term + me sin ((2-2m-c) pt-2B+a)} in the value of ✪ has been named the Evection. We shall consider its effect in two different ways. Firstly, by itself, as forming a correction on pt. 0=pt + 15 me sin {(2 – 2m — c) pt — 2B+a}. then pt mpt+B a' = (1 − c) pt + a = mean longitude of apse 0=pt+ 15me sin[2 { pt − (mpt + B)} — { pt − (1 − c) pt + a}] =pt+ 15me sin {2 ( ) − ☺) − ( ) — a')}. The effect of this term will therefore be as follows: is In syzygies 0 =pt- 15me sin( ) — a'); *See Dr. Whewell's Bridgewater Treatise. F |