NA TURE 



[November 23. 1905 



Travelling this evening between Plymouth and Exeter, 

 1 pulled the screens over the light in my compartment to 

 ■enjoy the moonlight, and was rewarded by seeing a fine 

 display of aurora borealis, which was, I hope, witnessed 

 by some other of your readers. 



Between 9 p.m. and 9.30 p.m., when near Totnes, there 

 w.i- a bright flattened arch near the northern horizon, with 

 white streamers rising from it at intervals, and very bright 

 patches of rose-red, extending from north-east to north- 

 west, and passing nearly overhead. At 9.15 one of these 

 patches, on the right ol the Great Bear, was a veritable 



pillar of flame," and was more remarkable because of 

 its '■ ntrasl to the moonlight, which was very brilliant. 



I think I am right in saving that a similar display 

 has nut lx<pn seen in the south of England for twenty-five 

 or thirty years, and the last " rose-red " display that 1 

 can remember was in 1870. R. Langton Cole. 



Ni vember 1 ,. 



.1 LUNAR THEORY FROM OBSERVATION. 



OX June 3, visitation day at the Roval Observatory, 

 ( ireenwich, the editor, who is a member of the 

 board of visitors, asked me to write an account of my 

 researches on the moon for Nature. I delayed doing 

 this tor a few months in order to render my account 

 more complete. 



The moon's longitude contains about 150, and the 

 latitude about 100, inequalities over o"-i. The argu- 

 ments of tie se inequalities, and the mean longitude 

 i>: the moon, require a knowledge of three angles 

 connected with the moon, viz. the moon's mean 

 longitude, the mean longitude of perigee, and the 

 mean longitude of the node. The other angles in- 

 volved in the arguments define the position of the sun, 

 planets, the solar perigee, &c, and their values are 

 to lie determined from other observations than those 

 of the moon. 



The problem that I have had in view, therefore, is 

 to determine the values of three angles as functions 

 of the time, and to give a list ol some 250 inequalities 

 in all a- accurately as possible. 



Before the time of Newton, this was clearly the 



only way the problem of the m i's motion could he 



attacked, only the limit worked to was then more 

 nearly 500" than o"-i. Since the time of Newton, the 

 method has been almost entirely abandoned. Many 

 mathematicians have attempted to calculate how the 

 moon ought to move ; the comparison between its 

 observed and theoretical course has been rough in the 

 extreme. No attempt has been made to verify from 

 observation the coefficients of those inequalities for 

 which a theoretical value had been calculated; observ- 

 ation lias merely been required to furnish values for 

 tlu.se constants which are theoretically arbitrary, and, 

 as 1 shall show, the determination of these constants 

 has often been rendered less accurate than was 

 necessary by the tacit assumption that all theoretical 

 terms had been accurately computed. 



My point of view, as I have said, is that which was 

 necessarily the only one before the time of Newton. 

 I. et us consider the application of this most ancient 

 ol all methods to the time when no observations were 

 possible except a record of ei lipses. 



The two principal inequalities of the moon's longi- 

 tude arc 



22640" sin ^ + 4586" sin (2D-g), 

 where g is the mean anomaly and D the mean 

 elongation of the moon. Whenever the moon is 

 either new or full, 2D = o; at such times, therefore, the 

 two in, ■qualities are indistinguishable from a single 

 inequality 



22640"— 4586"= 18054" sin §-. 

 ■lie "evection," as the smaller inequality is called, 

 could evidently not have been discovered until the 

 NO. 1882, VOL. 73] 



moon was observed near its quarters; moreover, a 

 correct value of the eccentricity of the moon's orbit 

 could never have then been obtained. On the other 

 hand, so long as the sole object of astronomers was 

 to obtain places of the new and full moons it did 

 not matter whether the two inequalities were separ- 

 ated or not. Roughly speaking, material of a limited 

 class is always good enough for generalisations con- 

 fined to the same class ; it is unsafe to extend the 

 generalisation to a wider class, as in this instance it 

 would be wrong to predict for the quarters of the 

 moon from the formula 18054" sin g. 



When we have an extended series of observations 

 and wish to determine whether a term x sin at 

 runs through the errors, and, if so, to determine x, the 

 theory of least squares directs us to multiply each 

 error by sin at and add. But before equating 



a:2 sin- at = ii e sin at 



we must pause and consider whether there may not be 

 some other error y sin 8t running through the observ- 

 ations such that 



VS sin at sin Bt is not zero. 



Now an interfering term of this sort may arise in 

 two ways:— (1) B may differ so little from a that 

 throughout the whole series id' observations the differ- 

 ence between at and Bt does not take indiscriminately 

 all values from o° to 360 ; 12) the difference between 

 at and Bt may be exactly equal to the mean elongation 

 of the moon, in which case, since the observations 

 are not uniformly distributed round the month, the 

 two inequalities are liable to be confounded, just as 

 the elliptic inequality and the eviction were con- 

 founded in the early days of astronomy. Interference 

 of the first kind can be eliminated by sufficiently 

 extending tin- series of observations, but no amount 

 of observations will obtain a correct result in the 

 second case if the mathematical point is overlooked. 



As a result of attending- carefully to these consider- 

 ations, I have succeeded in obtaining practically the 

 same value of the eccentricity of the moon's orbit 

 from two different series of observations compared 

 with two different systems of tabular places. Hansen 

 and Airy have given values of the same quantity 

 differing by more than one second of arc. For the 

 same reason, the value of the parallactic inequality 

 of the moon obtained by me corresponds closely with 

 the value of the solar parallax obtained in other ways. 

 The consideration neglected bv Airy in this case was 

 the possibility of error in the tabular semi-diameter. 



I have determined from the observations the 

 coefficient of every term the coefficient of which was 

 known to exceed o".i. This constitutes, as I have 

 said, the solution of the problem of the moon, as it 

 presented itself before the time of Newton. It forms, 

 too, the proper basis for comparing- observation with 

 theory. Previously the only thing known about the 

 vast majority of terms was that whereas the apparent 

 errors of Airv's tabular places frequentlv exceeded 

 20", those of Hansen's seldom differed from the mean 

 of neighbouring- observations bv so much as 5", a 

 quantity that might be attributed to errors of observ- 

 ation entirely. When, however, Newcomb in 1876 

 came to re-determine the value of the moon's eccen- 

 tricity (in his immediate object he was not particularly 

 successful owing to the neglect of the considerations 

 1 have just set down), he brought to lig-ht a term the 

 coefficient of which is one second, and the argument 

 of which was at the time unknown. The discovery 

 of this term shows how unsafe it is to test the tables 

 by the mere inspection of the series of errors of in- 

 dividual observations. However, in all my far more 



