420 Mr. Gr. H. Darwin on Variations in the Vertical 

 constant introduced on integration is zero, we have 



sin^+^sin3^+isin5^ + ...= -Wog(±^)-l]+i^og2. 

 o 0~ (17) 



Then, from (15) and (17), 



ico8 20 + £cos40+...-?jsin0+isin30+...|j 



= _ i ( 1 _^) logr ;, ) _ i(1+ ^) log2 _|.j (i8) 



Integrating (15), and observing that the constant is zero, 

 we have 



psin20+^sin40+...= -i0[log(±0)-l]-i01og2.(19) 



Integrating (17), and putting in the proper constant to 

 make the left side vanish when = 0, we have 



= _±<Plog( + 0) + i0 2 (f+log2). . . (20) 



For purposes of practical calculation, 6 may be taken as so 

 small that the right-hand side of (18) reduces to — £log ( ±26), 

 and the right-hand sides of (19) and (20) to zero. 



Hence, by (8) and (9), we have in the neighbourhood of 

 the coast, 



gwh 2l[l , 1 1 I 1 



= ^x 4x2-1037, I • • (21) 



7TV 



da gwh , ., A , 2irz 



I shall now proceed to compute from the formulas (21) the 

 depression of the surface and the slope, corresponding to such 

 numerical data as seem most appropriate to the terrestrial 

 oceans and continents. 



Considering that the tides are undoubtedly augmented by 

 kinetic action, we shall be within the mark in taking h as 

 the semi-range of equilibrium tide. At the equator the lunar 

 tide has a range of about 53 centim., and the solar tide is very 

 nearly half as much. Therefore at spring-tides we may take 

 /j = 40 centim. It must be noticed that the highness of the 

 tides (say 15 or 20 feet) near the coast is due to the shallow- 

 ing of the water, and it would not be just to take such values 



