ON THE MEASUREMENT OF THE LUNAR DISTURBANCE OF GRAVITT. 113 



It is interesting to determine the form of surface denoted by these 

 equations. Let us suppose then that the units are so chosen that 

 gwhljn'^u may be equal to one. Then (8) becomes 



« = |sin2fl+lsin49+. . 



dn 



r cos 20 + ^ COS 49 + . . . 



TT ll2 



gCosff+KsCOsSe + 



}(10) 



sin0 + isin39+ . . j. .(11) 



When 9 is zero or + tt, dujdB becomes infinite, which denotes that 

 the tangent to the warped horizontal surface is vertical at these points. 

 The verticality of these tangents will have no place in reality, because 

 actual shores shelve, and there is not a vertical wall of water when the 

 tide rises, as is supposed to be the ca,sein the ideal problem. We shall, how- 

 ever, see that in practical numerical application, the strip of sea-shore along 

 which the solution shows a slope of more than 1" is only a small fraction 

 of a millimeter. Thus this departure from reality is of no importance 

 whatever. 



When 9 = or -I- 7r, 



being + when 9 = 0, and — when 9 := itn-. 



670: 



When B =■ ±^Tr, a vanishes, and therefore midway in the ocean 

 and on the land there are nodal lines, which always remain in the undis- 

 turbed surface, when the tide rises and falls. At these nodal lines, de- 

 fined by 9 = ± ^TT, 



dn 



= -il0ge2 + 



2/1-1 



TT l 1« 3^ 



+ ^- 



} 



= - -3406 T -6108 = - -9634 and + -2702 



Thus the slope is greater at mid-ocean than at mid-land. By assuming 

 9 successively as c '''' 4 ""> 3 ""> ^^^^ summing arithmetically the strange 

 series which arise, we can, on paying attention to the manner in which 

 the signs of the series occur, obtain the values of a corresponding to 

 0. ± i^, ± i^, ± I'T, ± Itt, ± At, -!- f TT, ± f TT. The resulting values, to- 

 gether with the slopes as obtained above, are amply sufficient for drawing 



a figure, as in the annexed diagram. 



w^jm^y- 



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

 being la,nd, 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 ^t for the abscissas, 



1882. I 



