due to Elasticity of the Earth's Surface. 419 



ing attention to the manner in which the signs of the series 

 occur, obtain the values of a corresponding to 0, +\ it, +\ir, 

 ±f'7r, ±fT, ±£w> ±6 7r > ±f 7r - The resulting values, 

 together with the slopes as obtained above, are amply suffi- 

 ient for drawing a figure, as shown annexed. 



The straight line is a section of the undisturbed level, the 

 shaded part being land, and the dotted sea. The curve shows 

 the distortion, when warped by high and low tide as indicated. 



The scale of the figure is a quarter of an inch to \tt for the 

 abscissas, and a quarter of an inch to unity for the ordinates ; 

 it is of course an enormous exaggeration of the flexure actu- 

 ally possibly due to tides. 



It is interesting to note that the land-regions remain very 

 nearly flat, rotating about the nodal line, but with slight cur- 

 vature near the coasts. It is this curvature, scarcely percep- 

 tible in the figure, which is of most interest for practical 

 application. 



The series (8) and (9) are not convenient for practical cal- 

 culation in the neighbourhood of the coast, and they must be 

 reduced to other forms. It is easy, by writing the cosines in 

 their exponential form, to show that 



cos 6 + i cos 20 + 1 cos 30 + 



log e (±2sin|0), 



cos 0—\ cos 20 + \ cos 30-. . .= log e (2 cos ±0), 



(13) 

 (14) 



where the upper sign in (13) is to be taken for positive values 

 of 6 and the lower for negative. 



For the small values of with which alone we are at present 

 concerned, the series (13) becomes — log e (±0) and the lower 

 log e 2. 



Taking half the difference and half the sum of the two 

 series, we have 



icos2(9 + icos40+ =-ilog(±0)~|lpg2, . (15) 



cos0 + icos30 + icos5<9+ = -ilog( + 0)+-|log2. . (16) 



Integrating (16) with regard to 0, and observing that the 



