ON THE MEASUREMENT OF THE LUNAR DISTURBANCE OF GRAVITY. 115 



I shall now proceed to compute from the formulae (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 mai'k in taking h as the semi-range of 

 equilibrium tide. At the equator the lunar tide has a range of about 

 53 cm., and the solar tide is very nearly half as much. Therefore at 

 spring-tides we may take h = 40 cm. It must be noticed that the high- 

 ness 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 as repre- 

 senting the tides over large areas ; iv, the density of the water is, of 

 course, unity. 



If we suppose it is the Atlantic Ocean and the shores of Europe with 

 Africa, and of North and South America, which are under consideration, it 

 is not unreasonable to take I as 3,900 miles or 6'28 x 10^ cm. Then 

 'lirzjl =2X10-8. 



Taking v jg as3x 10^, that is to say, assuming a rigidity greater than 

 that of glass, we have for the slope in seconds of arc, at a distance z from 

 the sea-shore 



^_f-^, xlog,10x(8-log„.) , _ ^2,) 



= 0"-01008 (8-log,o2) 



} 



From this the following table may be computed by simple multipli- 

 cation : — 



Distance from mean 



water-mark Slope 



1 cm. = 1 cm 0"-0806 



10 cm. = 10 cm -0706 



10^ cm. = 1 meter -0605 



10' cm. = 10 m -0504: 



10' cm. = 100 m -0403 



105 cm. = 1 kilom 0"-0302 



10«cm. = 10 kilom -0202 



2 X 10«cm.= 20 kilom -0170 



5 X 10«cm.= 50 kilom -0131 



10' cm. = 100 kilom "0101 



On considering the formula (22) it appears that z must be a very small 

 fraction of a millimeter before the slope becomes even as great as 1'. 

 This proves that the rounded nick in the surface, which arises from the 

 discontinuity of pressure at our ideal mean water-mark, is excessively 

 small, and the vertical displacement of the surface is sensibly the same, 

 when measured in centimeters, on each side of the nick, in accordance 

 with the first of (21). 



The result (5) of section 1 shows that, Avith rigidity 3 x 10*, the true 

 deflection cf plumb-line dne to attraction of the water is a quai'ter of the 

 slope. Hence an observer in a gravitational observatory at distance z 

 from mean water-mark, would note deflections from the mean position of 

 the vertical 1;^ times as great as these computed above. And as high 



i2 



