TRANSACTIONS OF SECTION A. 473 



logical disturbances which make the heights of all the solar tides quite beyond pre- 

 diction. The fortnightly and monthly tides consist in an alternate increase and 

 diminution of the ellipticity of the elliptic spheroid of which the sea-level (after 

 elimination of the tidal oscillations of short period) forms a part. There are two 

 parallels of latitude, respectively north and south of the equator, which are nodal 

 lines, along which the water neither rises nor falls. "When, in the northern hemi- 

 sphere, the water is highest to the north of the nodal line of evanescent tide, it is 

 lowest to the south of it, and vice versa ; and tlie like is true of the southern 

 hemisphere. If the ocean covered the whole eartli the nodal lines would be in 

 latitudes 35° 16' N. and S. (at which latitudes ^ — sin" lat. 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 therefore first necessary to 

 evaluate this integral. Approximation is of course unavoidable, and for that end 

 the irregular contours of the continents were replaced by meridians and parallels of 

 latitude, and the integral evaluated by quadrature. This procedure will give 

 results quite accurate enough for practical purposes. It appeared as the result of 

 the quadrature that, if we assume the existence of a large antarctic continent, the 

 latitude 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 imperfect 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 conclusion may also be stated in another way, which is more convenient for 

 practical purposes ; for we may say that at any station thei'e 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 fi-action of the equilibrium height, 

 but differing in the time of high-water by a quarter-period from the theoretical 

 time— viz. about three-and-a-half days for the fortnightly and a week for the 

 monthly tide. These two tides may, according to geometrical analogy, be called 

 perpendicular component tides. According to the theory of the composition of 

 harmonic motions, the two components may be compoimded into a single tide, with 

 time of high-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, 

 .r 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 fidfilled without sensible elastic yielding of the 

 earth, the first component has its full value, or .r 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, .v will be less than 

 unity. The amount of elastic yielding, and hence the average modulus of elasticity 

 of the whole earth may be computed from the value of a: After rejecting the ob- 

 servations made at certain stations for suflicient reasons, the author 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 diflferent ports in 

 England, France, and India. 



These results, when properly reduced, gave thirty-three equations for the .r and 

 thirty-three for the y of the fortnightly tide, and similarly thirty-three for the .r 

 and thirty-three for the y of the monthly tide; in all 132 equations for four 

 unknowns. 



