November 25, 1897] 



NA TURE 



89 



with the use of a totally different method of calculation will, Dr. 

 Brown estimates, cut down the twenty years occupied by 

 Delaunay over the same problem to five. 



One great advantage of an algebraical theory is the facility 

 with which the work can be divided into sections, each consist- 

 ing in the calculation of a group of terms multiplied by the same 

 power of the eccentricities, inclination, and solar parallax, or, in 

 the language of the lunar theory, having the same characteristic. 

 Delaunay's theory unfortunately lost this advantage by its 

 peculiar methods, and substituted another form of subdivision 

 that enormously increased the total labour in the desire to 

 present each step in a moderate compass. Adams appears to 

 have been the first to clearly recognise the advantages of a sub- 

 division such as Dr. Brown has employed, but it was Dr. Hill 

 who actually laid the foundations of the present theory. In the 

 first volume of the American Journal he published his famous 

 " Researches in the Lunar Theory," where, after the lapse of 

 more than a century, he revived Euler's idea of using rectangular 

 coordinates. Confining his investigations to the case when the 

 eccentricities, inclination, and solar parallax are supposed to 

 vanish, he used axes rotating so that the axis of x points con- 

 stantly to the sun, and then replaced x, y by the conjugate 

 complex variables ti, s = x ^y si - \, and the time by another 

 complex quantity f = ^("-'''Xv'-^, and the symbol of differentia- 

 tion by D defined as f —; . He also used the symbol m to denote 



the ratio of the synodic month to the sidereal year, and k to 

 denote the mass of the earth divided by the square of the 

 difference of the mean motions of the moon and the sun. He 

 then arrived at the differential equations, 



(D -1- /«)-« -t- \ m^u + I mh - _ f =o 



{us) 



(D - w)-j + ^ ni^s -r f in'^u 



{us) 



At this point arose a difficulty which, in a closely analogous 

 form, is common to all lunar theories, the presence, that is to 

 say, of the quantity denoting the mass of the earth divided by 

 the cube of the distance. The practical convenience of a theory 

 is perhaps in no way better tested than by examining the manner 

 in which this difficulty is overcome, and it is certainly not too 

 much to say that in this respect Dr. Hill's method has no rival. 

 In a few brief steps he succeeds in eliminating the obnoxious 

 quantity altogether, and he obtains two equations of the second 

 degree in u, s and homogeneous in these variables except for 

 a constant of integration, which may be looked upon as replac- 

 ing k. These equations are easily solved numerically, and 

 denoting the values of the variables with Dr. Hill's modifications 

 of the general problem by the suffix zero, ti^, and its conjugate 

 complex j„ may be henceforth looked upon as known functions 

 of the time. They are, in fact, capable of .expression as infinite 

 series of positive and negative odd powers of ^. The co- 

 efficient of C in «'o 's denoted by a, which is a constant defining 

 the linear dimensions of the orbit. By having recourse to one 

 of the original equations containing k, the value of kja^, which 

 is a mere number, may be found. This completes the in- 

 vestigation of the variation, as this class of inequalities is called. 

 It is at this point that Dr. Brown took up the subject. He 

 replaced Dr. Hill's first pair of equations by the following set 

 of three, the third of which determines z or the moon's co- 

 ordinate perpendicular to the ecliptic, which in the particular 

 case treated by Dr. Hill is zero 



(D + mfu + i mhi + ^ ni^s - 



(D - 7iifs + \ Ill's -F %m^u 

 (D2 - vi'-)z 



In these equations n, represents the part of the disturbing 

 function neglected by Dr. Hill, every term of which is divisible 

 by either the solar eccentricity or parallax. The quantity k can 

 be eliminated from the first two of these equations in a manner 

 arialogous to the methods of Dr. Hill. It can be also 

 elirninated from the third and either of the other two in an 

 obvious manner. The resulting equations need not be written 

 down here ; following Dr. Brown, they will be alluded to as 

 the homogeneous equations. There are thus two distinct sets 



of equations that can be used at any step in the work. In 

 practice one set is used, and a single equation from the other 

 set is used in addition, generally as a mere equation of verifica- 

 tion, but in certain special cases for the actual solution when the 

 equations of the first set are not well adapted for the purpose. 

 Dr. Brown's procedure is as follows : let 



U = «(, -f- «M + U^ Z = z'^ + Zx 



Where u^, z^ denote the terms already calculated, «x. 2x the 

 new terms of characteristic \ to be calculated in the next step 

 of the process of solution. Either Ux or zx is always zero ac- 

 cording as \ contains an odd or even power of the inclination. 

 These values are then substituted in either set of differential 

 equations, and the terms of order A picked out. It can be 

 readily seen that the right-hand side of the equations contain 

 only known terms, and the unknown new terms occur in the 

 first degree and multiplied by functions of u^ s^ only. If the 

 first set of equations be used, the terms containing kjr^ must be 

 expanded by Taylor's Theorem into series proceeding according 

 to powers of n^ + ux, Su, -f- Sx, z^ -f Zx with coefficients con- 

 taining k, Uq, Sq only. These coefficients are easily deducible 

 from Dr. Hill's value of «(,, the method of special values being 

 in general used. One remark, however, requires to be made. 

 Every time a set of terms is calculated whose arguments are the 

 same as the terms in Uq, there arises the opportunity of modify- 

 ing the meaning of the linear constant a. It is otherwise 

 evident that any solution remains a solution when a is replaced 

 by a new constant a' defined by the relation a/ a' - i =an arbitrary 

 series of powers and products of the squares of the eccentricities, 

 inclination, and solar parallax. The value of k is of course 

 simultaneously modified also. Consequently we should be 

 liable to have the values of such quantities as k/r^^ varying from 

 time to time as the approximation proceeds. This would be 

 obviously inconvenient, and Dr. Brown has used the power of 

 modification at his disposal so that k/a^ remains invariable 

 throughout the solution, and therefore, since in Dr. Hill's 

 papers it is a function of m only, it always remains so. 



In the first set of equations therefore the unknown terms 

 enter in the form 



C-HD + m)''ux + MC-^ux + NCsx 

 and 



D-'zx - 2Uzx 

 where 



M = hm^ + hA, 

 Po' 



■ N = WC-^ + 



Po" 



NO. 1465, VOL. 57] 



the same form at every approximation. (A misprint in the 

 algebraical value of N, on p. 63, should be noticed ; the factor 

 (~^ being there omitted. This is merely a printer's error, for 

 the arithmetical value on p. 90 is correctly given. Indeed, 

 were it otherwise, the discordance of the results from those of 

 other theories would long ago have been noticed.) 



When the new terms to be calculated have the same argu- 

 ments as Ue or zjt, the principal elliptic or inclinational terms, a 

 new term in the motion of the perigee or node (of order \/e or 

 \/k) has to be calculated. The unknown term Cxjt appears 

 multiplied by 2(D + ;n)ue in the first equation, and gx^t appears 

 multiplied by 2Dr^ in the second equation. These coefficients 

 ' are independent of \ : since, however, A must be of at least the 

 i third order for the point to arise, it does not properly enter into 

 I the part of the work already published. 



j A series with indeterminate coefficients is then assumed for 

 j Ux or Zx : and a number of simultaneous equations formed, 

 j from which the coefficients are found. The labour of forming 

 I the known terms in these equations increases rapidly with the 

 I characteristic ; but the operations required are mere multiplica- 

 tion of series, and can to a great extent be left to a computer. 

 The results are readily checked by computing independently the 

 I value when C= i> 



The unknown terms enter at each step into the equations under 

 the same algebraical form, or rather under one of two forms, ac- 

 cording as the new terms belong to u or z. These forms unfortun- 

 ately involve the symbol of differentiation D, so that the different 

 sets of simultaneous equations have different arithmetical co- 

 efficients ; but whenever more terms with an old set of arguments 

 are being calculated, the arithmetical coefficients are the same 

 as before, and it is only the right-hand sides which are different. 

 This greatly facilitates the labour of solution ; but the advan- 



