July, 1901.] 



KNOWLEDGE. 



155 



natuie of the subject, but it would hardly place him 

 in a position to cai'ry on the caknilations for himself 

 to the accuracy required for forming tables. Adams to 

 a great extent achieved success in a middle course. He 

 divided and got the mastery of hife subject. He left 

 on one side those higher approximations that involve 

 merely labour in computing, and illustrate no principle. 

 but he pushed the lower approximations to a high 

 degree of accuracy, and not merely showed how to 

 obtain, but actually did obtain numerical values for the 

 principal parts of the motions of the node and apse 

 and the co-efficients of the principal inequalities. The 

 great value of his lectm-es may be illustrated by the fact 

 that when ill-health compelled Adams to cease lecturing 

 shortly before his death, notes of his lectures were 

 borrowed and copied by the younger generation. Pro- 

 fessor Sampson was fortunate to be among the last that 

 attended the course, the present writer was among the 

 first to make a copy. 



The lectures were last delivered twelve years ago, 

 and were constantly revised during the whole period in 

 which they were delivered. And yet they are already. 

 and have for some time been, to a great extent out of 

 date. The last two chapters of Adams' lectures deal 

 with Dr. Hill's methods. These methods have since been 

 developed by Professor Brown, and made the basis of 

 lectures in Cambridge by Professor G. H. Darwin. The 

 concluding portion of this notice will be to attempt 

 to explain the points of superioi'ity of Hill's method 

 over all that preceded it. 



Any co-ordinate of the moon, that is to say, its longi- 

 tude or its latitude, or its radius vector, or the pro- 

 jection of its radius vector in any direction, may be 

 developed in a series of periodic terms, whose arguments 

 are the sums of any integral positive or negative mul- 

 tiples of four fundamental angles that increase uni- 

 formly with the time and at rates that are incom- 

 mensurable. The immense complexity of the problem 

 may be conceived by imagining four piles of coins of 

 incommensurable value. One pile, we may suppose, 

 consists of sovereigns, and, in addition, '' negative 

 sovereig^is, that is to say acknowledgments of debt to 

 the extent of a sovereign ; the other piles are to consist 

 of francs, mai-ks. and krones with corresponding 

 " negative " coins, or acknowledgments of debt. Then 

 for every sum of money that can be made up from these 

 coins, there is a periodic term in the lunar co-ordinate, 

 and if not more than seven to twelve coins are used to 

 make up the sum, then the co-efiicient of the corre- 

 sponding term in the lunar theory is important enough 

 to be calculated. In addition a very great number of 

 these co-efficients are large and correspondingly difficult 

 to calculate, for the same reason that a very slight 

 muscular effort, constantly iepeat«d at regular intervals, 

 will set a swing into violent oscillation, or that a ship 

 that remains moderately steady in some seas will roll 

 violently in others when the interval between one wave 

 and the next happens to be of a certain length. 



Now if this illustration has given any idea of the 

 gigantic nature of the task, it will be clear that it 

 will be a great convenience if the theory is such that 

 the terms can be calculated in groups of a small number 

 at a time, first one gi-oup and then another group, just 

 as Nature builds up a living organism by " upigcncsis," 

 or the adding of one cell to another cell. When tiiis 

 can be done, another stage can be added at any time, 

 should it be considered convenient to do so. When it is 

 not the case, not only is the labour much increased, 

 but it is nearly impossible to extend the work to 



approximations higher than those at which the original 

 computer left it. This criterion is a condemnation of 

 Dclaunay's theory, which, while it is in many ways the 

 most elegant from the mathematician's point of view, 

 has probably proved the most laborious of all in its 

 computations, and has nevertheless not been pushed 

 to such accuracy as might have been desired. But in 

 this respect both the theory adopted by Adams and 

 that propounded by Hill leave nothing to be desired. 



Both Adams and Brown (who is working out Hill's 

 theory) divide the terms into groups. Now there is 

 only one way of dividing into groups, and the first group 

 of terms constitutes what is called the variation, which 

 is an inequality in the moon's motion that goes through 

 its period in half a synodic month. Now Adams and all 

 the theorists before Hill start by pointing out, that 

 owing to its immense distance the sun's influence is verjr 

 slight, and if it could be neglected altogether the 

 moon s orbit would be an ellipse. Then it is found 

 that although the sun's influence is slight, yet in some 

 respects it is cumulative and not periodic, and hence 

 we have a rotating ellipse introduced to our notice 

 which, doubtless, is a very convenient geometrical re- 

 presentation of the moon's path, but it is hardly a good 

 " intermediate orbit " or first approximation for a dyna- 

 mical calculation, since it does not represent an orbit 

 that would result from an approximation to the actual 

 forces that govern the path of the moon. Now Hill 

 takes the first terms that are calculated — the variation 

 — and interprets them dynamically. These terms and 

 these terms only represent a possible path for the moon 

 that could coi-respond to the actual case of Nature, with 

 one modification ; the sun must be supposed to be at 

 infinite distance, while retaining influence enough upon 

 the earth-moon system to maintain the length of the 

 year at its present value. The next group of terms to 

 be selected for calculation (for although the grouping 

 is determinate, there is a little choice as to the order 

 in which the groups are to be taken) may modify the 

 orbit (by adding what are known as the parallactic 

 inequality, because it only exists in consequence of the 

 sun's parallax not being zero, and the annual equation 

 due to the ellipticity of the earth's orbit round the sun) 

 into an orbit that the moon might pursue, only it 

 happens not to do so. And then the rest of the work 

 consists in this only : the moon is considered to oscillate 

 in two ways about the above-mentioned orbit that it 

 might pursue: in the first place it does not remain in 

 the plane of the ecliptic but oscillates from side to side 

 of it ; in the second place it keeps crossing and recros- 

 sing the orbit it might pursue. The size of these oscil- 

 lations must be dctWmined by observation, for. as has 

 been pointed out, they need not exist at all, and their 

 size is subject to no" conditions, and might have been 

 anything including their actual values or zero. More- 

 over, the periods of these oscillations must be determined 

 by calculation, and the lunar theory is then complete. 



Calculation shows that the periods of these oscillations 

 are neither of them exactly a month; and, therefore, 

 to confine our attention to the oscillation across the 

 plane of the ecliptic, the node or point of crossing is 

 not the same in each revolution, or, in other words, the 

 node revolves. This is not contrary to expectation ; 

 the surprising thing would have been if the period of 

 oscillation had turned out to bo exactly a month ; and 

 it leads to no inconvenience, such as in the older 

 theories, when it compelled the introduction of rotating 

 ellipses with no dynamical interpretation. 



