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

 The slope of the surface is T-or-r^; thus 

 ^ = ^{i C os20 + icos40 + J r cos60+...} 



dz 



77V 



^ - (sin 0+ 4^30+ ^sin50 + ...l, 



(9) 



The formulae (8) and (9) are the required expressions for 

 the vertical depression of the surface and for the slope. 



It is interesting to determine the form of surface denoted 

 by these equations. Let us suppose, then, that the units are 

 so chosen that gwld/iPv may be equal to one. Then (8) 

 and (9) become 



a=^ 2 sm20+^ 2 sin4<9 + ...+ |jpCos0+ ^ 3 cos30 + ... j (10 



^=icos20 + icos40 + ...-?;{psm0+ i s in30 + ...}. HJ 



When is zero or ±tt, da/ dd becomes infinite, which de- 

 notes that the tangent to the warped horizontal surface is 

 vertical at these points. The vertically 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 case in 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 millimetre. Thus this depar- 

 ture from reality is of no importance whatever. 



When 0=0 or ±ir, 



'=l{p + ^ + ^.-.} = lxi-<m=^o,(i2) 



being + when 0=0, and — when 0= ±ir. 



When 0= ±^tt, a vanishes; and therefore midway in the 

 ocean and on the land there are nodal lines, which always 

 remain in the undisturbed surface, when the tide rises and 

 falls. At these nodal lines, defined by 0= ±iir, 



n , 2/ 1 11 I 



do 



d0' 



= -•3466 + -6168= --9634 and +-27 



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

 By assuming successively as \ir, \tt, ^tt, and summing 

 arithmetically the strange series which arise, we can, on pay- 



