58o 



NATURE 



\Sepi. 26, 187J5 



base-line c d is then measured with the greatest possible accuracy, 

 and the angle formed at c and D in the directions e and / are 

 obtained. Thus the two triangles c D ^ and c D/are completely 

 given ; for in each is a side (c d) and the two adjacent angles 

 known. But thus also are their heights eh and fh given, and 

 these added give the side ^/"in the great triangles. From e andy 

 the two triangles formed towards A and B are measured, and this 

 gives completely the two triangles Kfe and "Bfe, as also their 

 heights A /6 and B /', which added give the distance sought.^ It 

 will be seen at once that this method offers great advantages, 

 especially if it be possible to obtain with the greatest accuracy 

 the small base-line. This latter condition Bessel fulfilled at first 

 to an astonishing degree, as he, by the introduction of a base- 

 apparatus, attained the greatest accuracy. Bessel and Baeyer 

 accomplished a degree-measurement between Memel and Tranz 

 in 1831-36. They obtained for the mean latitude of the measured 

 arc (54° 58' 25""5) a degree-length of 57142 toises. An opera- 

 tion was carried out by Maclear between 1836 and 1848 at the 

 Cape of Good Hope, by which for south latitude 35° 43' 20" a 

 degree-length of 56933 toises was obtained. 



{To be continued.') 



ON THE PRECESSION OF A VISCOUS 

 SPHEROID^ 



T HAVE been engaged for some time past in the investigation 

 ■^ of the precession of a viscous spheroid, with the intention 

 of seeing whether it would throw any light on the history of the 

 earth in the remote past. As some very curious results have 

 appeared in the course of the work, I propose to give an account 

 of part of them to tha British Association. 



The subject is, however, so complex and long, that no attempt 

 will be made even to sketch the analytical methods employed. 



In a paper of mine read before the Royal Society in May last, 

 a theory was given of the bodily tides of viscous and imperfectly 

 elastic spheroids ; and this paper formed the foundation of the 

 present investigation. 



For convenience of diction I shall speak of the tidally dis- 

 turbed body as the earth, and of the disturbing bodies as the 

 moon and sun ; moreover, in all the numerical applications, the 

 necessary data were taken from these three bodies. 



The effect of the internal friction called viscosity, is that the 

 bodily tides in the earth lag, and are less in height, than they 

 would be if the earth were formed of a perfect fluid. 



An analytical investigation proved that the action of the sun 

 and moon on the tides in the earth is such that the obliquity 

 to the ecliptic, and the lengths of the day and month all become 

 variable ; the alteration in the length of the year remains, how- 

 ever, quite imperceptible. 



But I will now explain, from general considerations, how the 

 lagging of the tides produces the effects above referred to. 



Let the figure represent the earth as seen from above the 

 south pole, so that s is the pole, and the outer circle the equator. 

 The rotation of the earth will then be in the direction of the 

 curved arrow close to s. Within the larger circle is a smaller 

 concentric one, one-half of which is drawn with a full line, and 

 the other half with a dotted line. The full line semicircle is part 

 of a small circle in S. latitude and the dotted one part of another 

 small circle in the same latitude, but to the north of the equator. 

 Generally, dotted lines indicate parts which are behind the plane 

 of the paper. 



It will make the explanation somewhat simpler, if we sup- 

 pose the tides to be raised by a moon and antimoon diametri- 

 cally opposite to one another ; this, as is well known, is a justi- 

 fiable modification of the true state of the case. 



Then let M and m' be the projections of the moon and anti- 

 moon on to the terrestrial sphere. 



If the substance of the earth were a perfect fluid, or were 

 perfectly elastic, the apices of the tidal spheroid would be at M 

 and m'. If, however, there be internal friction, the tides will 

 lag, and we may suppose the apices of the spheroid to be at T 

 and t'. In order to make the subject more intelligible, the tidal 

 protuberances are then supposed to be replaced by two equal 

 heavy particles T and t', which are instantaneously rigidly con- 



* It is here assumed that /e is at tight angles to r D, and A B at right 

 angles to /e. There is no necessity for this condition, and it could never 

 actually occur. 



2 A paper read at the Dublin Meeting of the British Association, by 

 G. H. Darwin, M.A , Fellow of Trinity College, Cambridge. 



nected with the earth. This same idea was, I believe, made use 

 of by Delaunay, in considering the ocean tidal friction. 



Then the attraction of the moon on T is greater than on x' ; 

 and that of the antimoon on x' greater than on x. Hence, 

 besides equal and opposite forces acting at the earth's centre, 

 directly towards M and m', there are small forces (varying as 

 the square of the tide generating force) acting in the directions 

 X M and x' m'. 



We will consider the effect on the obliquity first. These two 

 forces, TM, x'm', clearly cause a couple about" the axisLL' in 

 the equator, which lies in the same meridian as the moon. The 

 couple is indicated by the curved arrows at l and l'. Now, if 

 the effects of this couple be compounded with the existing rota- 

 tion of the earth, according to tha principle of the gyroscope, 

 it is clear that the south pole s tends to approach M and the 

 north pole to approach m'. Hence supposing the moon to move 

 in the ecliptic, the inclination of the earth's axis to the ecliptic 

 diminishes ; in other words, the obliquity of the ecliptic 

 increases. 



Next with regard to tidal friction ; the forces x M and x'm' 

 produce a couple about the earth's axis, s, which tends to 

 retard the earth's rotation. 



Lastly, since action and reaction are equal and opposite, and 

 since the moon and antimoon produce the forces T M, x' m' ' 

 on the earth, therefore the earth must cause forces on the moon 

 and antimoon in the directions M X and m' x'. These forces are in 

 the same direction as the moon's orbital motion ; hence the 

 moon's linear velocity is augmented. The consequence of this 

 is that her distance from the earth is increased, and with that 

 increase comes an increase of periodic time round the earth. 



The consequences of the lagging of the earth-tides, therefore, 

 are an increase of the obliquity to the ecliptic, a retardation of 

 the earth's rotation, and a retardation of the moon's mean 

 motion. 



In this general explanation it is assumed that the lagging tides 

 are exactly the same as though the earth were perfectly fluid, 

 and as though the t\A&-raising moon were more advanced 

 in her orbit than the true moon, whilst the moon which attracts 

 the tidal protuberances was the true moon. That is to say, it is 

 assumed that the tides raised are exactly the same as though the 

 earth were a perfect fluid, save that the time of high tide is late, 

 and that the tides are reduced in height. 



Now although this serves in a general way to explain the 

 phenomena which result from the supposition of the earth's 

 viscosity, yet it is by no means an accurate representation of the 

 state of the case. 



In fact the internal friction sifts out the whole tide-wave mto 

 its harmonic constituents, and allows the different constituents 

 to be very differently affected as regards height and phase. 



Thus the lagging tide-wave is not exactly such as the general 

 explanation supposes, and the nearer does the spheroid approach 

 to absolute rigidity the greater does the discrepancy become. 



The general explanation is a very fair representation for 

 moderate viscosities, but for large ones it is so far from correct 

 that the tendency for the obliquity to vary may become nil, and 

 for yet larger ones the obliquity may tend to decrease. 



