) Nov. 2, 1882 } 
NATURE 
3 
of mathematics to calculate what these heigh ~ - ould be, if the 
earth’s mass were absolutely unyielding. But the tides of long 
period are nearly free from the dynamical influences which 
render those of short period so intractable to calculation, and 
must in fact nearly follow the laws of the ‘‘ equilibrium theory.” 
In 1867 it was not, however, even definitely known whether 
or not the tides of long period were of sensible height at any 
station. Although there has been a continual advance in the 
knowledge of tidal phenomena since that time, it is only within 
the last year that there is a sufficient accumulation of tidal obser- 
vations, properly reduced by harmonic analysis, to make it 
possible to carry out Sir William Thomson’s suggestion. The 
great advances in knowledve that have been recently made are 
principally due to the adoption of systematic tidal observation at 
a great number of stations by the India) Government. The 
results of these observations are now being issued yearly by the 
Secretary of State for India in the form of tide-tables for the 
principal Indian ports. I have had the pleasure of carrying out 
the examination of the tidal records, and a detailed account of 
the work will appear at § 848 of the new edition of Thomson 
and Tait’s ‘* Natural Philosophy,” now in the press. 
The tides chosen for discussion were the lunar fortnightly 
declinational tide, and the lunar montbly elliptic tide. These 
tides must be free from the meteorological disturbances which 
make the heights of all the solar tides quite beyond prediction. 
The fortnightly and monthly tides consist in an alternate increase 
and diminution of the ellipticity of the ellivtic spheroid of which 
the sea level (after elimination of the tidal oscillations of short 
period) formsa part. There are two parallels of iatitude respec- 
tively north and south of the equator which are nodal lines, along 
which thewater neither rises nor falls. When, in the northern 
hemisphere, the water is highest to the north of the nodal line of 
eyanescent tide, it is lowest to the south of it, and wice versd ; 
and the like is true of the southern hemisphere. If the ocean 
covered the whole earth, the nodal lines would be in latitudes 
35 16’ N. and S. (at which latitudes }—sin? /a¢. vanishes) ; but 
when the existence of land is taken into consideration, the nodal 
latitudes are shifted. Now according to Sir William Thomson’s 
amended equilibrium theory of the tides, the shifting of the 
nodal latitudes depends on a certain definite integral, whose 
limits are determined by the distribution of land on the earth’s 
surface. 
For the purpose of examining the tidal records, it was there- 
fore first necessary to evaluate this integral. Approximation is 
of course unavoidable, and for that end the irregular contours 
of the continerts were replaced by meridians and parallels of 
latitude, and the integral evaluated by quadrature. This pro- 
cedure will give results quite accurate enouzh for practical 
purposes. It appeared as the result of the quadrature that, if 
we assume the existence of a large Antarctic continent, the lati- 
tude of evanescent tide is 34° 40’, and if there is no such 
continent it is 34°57’. Hence the displacement of the nodal 
latitudes due to the existence of land is very small. 
This point having been settled, the mathematical expressions 
for the fortnightly and monthly tides are completely determinate, 
according to the equilibrium theory, with no yielding of the 
earth’s mass, 
If there is yielding of the earth, either with perfect or imper- 
fect elasticity, and with frictional resistance to the motion of the 
water, the height of tide and the time of high water must depart 
from the laws assigned by the equilibrium theory. This conclu- 
sion may also be stated in another way, which is more conve- 
nient for practical purposes; for we may say that at any station 
there must actually be a tide with a height equal to some fraction 
of the full equilibrium height, and with high water exactly at the 
theoretical time, and a second tide, of exactly the same nature, 
with a height equal to some other fraction of the equilibrium 
height, but differing in the time of high water by a quarter- 
period from the the retical time, viz. ahout three-and-a-half- 
days for the fortnightly, and a week for the monthly tide. These 
two tides may, according to geometrical a: alogy, be called per- 
pendicular component tides. According to the theory of the 
composition of harmonic motions, the two components may be 
compounded into a single tide, with time of hivh water occurring 
within a half-period of the theoretical time ; and this is the way 
in which the results of elastic yielding and frictional resistance 
were first stated above. ‘Thus the actual tide at any station 
involves two unknown fractions, x and y, being the factors by 
which two components, each of the full theoretical height, are 
to be multiplied in order to give the two components in proper 
amount to represent the reality. 
If the equilibrium theory is fulfilled without sensible elastic 
yielding of the earth, the first component has its full value, or 
x is equal to one, and the second component vanishes, or y is 
zero. If fluid friction exercises a sensible influence, y will have 
a sensible value ; and if the solid earth yields tidally, x will be 
less than unity. The amount of elastic yielding, and hence the 
average modulus of elasticity of the whole earth may be com- 
puted from the value of x. After rejecting the observations 
made at certain stations for sufficient reasons, I obtained from 
the Tidal Reports of the British Association and from the Indian 
Tide Tables, the results of thirty-three years of observation, 
made at fourteen different ports in England, France, and India. 
These results, when properly reduced, gave thirty-three equa- 
tions for the « and thirty-three for the y of the fortnightly tide, 
and similarly thirty-three for the x and thirty-three for the y of 
the monthly tide ; in all 132 equations for four unknowns. 
The x and y of the two classes of tide were in the first instance 
regarded as distinct, but the manner in which they arise shows 
that it is legitimate to regard them as identical, and thus we 
have sixty-six equations for x and sixty-six for . 
The equations were then reduced by the methods of least 
squares, with the following results :—- 
For the fortnightly tide— 
% =—6176) )056, 07, = 020055. 
And for the monthly tide— 
x = ‘680 + 258, y = ‘ogo + ‘218. 
The numbers given with alternative signs are the probable 
errors. 
The very close agreement between the x and y for the two 
tides is probably somewhat due to chance. 
The smallness of the two ’s is satisfactory; for, as above 
stated, if the equilibrium theory were true, they should vanish, 
Moreover, the signs are in agreement with what they should be, 
if friction is a sensible cawe of tidal retardation, But consi- 
dering the magnitude of the probable errors, it is of course more 
likely that the non-evanescence of the y’s is due to errors of 
observation or to the method of reduction, 
I haye already submitted to the British Association at this 
meeting a paper on a misprint, discovered by Prof. Adams, in 
the tidal report for 1872. This report forms the basis of the 
method of harmonic analysis which has been employed in the 
reduction of the tidal ohservations, and it appears that the 
erroneous formula has been systematically used. The large 
probable error in the value of the monthly tide may most 
probably be reduced by a correct treatment of the original tidal 
records, 
It has been already remarked that it is legitimate to combine 
all the observations together, for both sorts of tide, and thus to 
obtain a single x and y from sixty-six years of observation. 
Carrying out this idea, I find: 
x ='676 + ‘076, y="029+'065. 
These results really seem to present evidence of a tidal yield- 
ing of the earth’s mass, and the value of the x is such as to show 
that the effective rigidity of the whole earth is about equal to 
that of steel. 
But this result is open to some doubt for the following 
reason : — 
Taking only the Indian results (forty-eight years in all), 
which are much more consistent than the English ones, I find 
aw = -931 £'056, y = "155 + “068. 
We thus see that the more consistent observations seem to 
bring out the tides more nearly to their theoretical equilibrium 
values with no elastic yielding of the solid. 
Tt is to be observed however that the Indian results being 
confined within a narrow range of latitude give (especially when 
we consider the absence of minute accuracy in my evaluation of 
the definite integral) a less searching test for the elastic yielding 
than a combination of results from al! latitudes. : . 
On the whole we may fairly conclude that, whilst there is 
some evidence of a tidal yielding of the earth’s mass, that yielding 
is certainly small, and that the effective rigidity is at least as 
great as that of steel. 
SCIENTIFIC SERIALS 
The Yournal of Physiology, vol. iii. Nos. 5 and 6, August, 
1882, with Supplement number. No. 11 contains :—Optical 
illusions of motion, by H, P. Bowditch and G, S. Hall,—On 
