114 



REPOET 1882. 



and a quarter of an inch to unity for the ordinates ; it is of course an 

 enormous exaggeration of the flexure actually 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 curvature near the coasts. 

 It is this curvature, scarcely perceptible in the figure, which is of most 

 interest for practical application. 



The series (8) and (9) are not convenient for practical calculation 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 + i cos 29 + ^ cos 39 + . . . = - log^ (± 2 sin 10) . . . (1.3) 

 cos - i cos 20 + ^ cos 30 + . . . = loge (2 cos 10) .... (14) 



Where the upper sign in (13) is to be taken for positive values of and 

 the lower for negative. 



For the small values of 0, for which alone we are at present con- 

 cerned, the series (13) becomes — logg (-+; 0) and the lower loge 2. 



Taking half the diiference and half the sum of the two series we have 



1 cos 20 + 1 cos 40 + = - 1 log (± 0) - 1 log 2 ... . (15) 



COS0 + 1 cos 30 + i cos 50 + = - i log (± 0) + + log 2 . . . . (16) 



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

 introduced on integration is zero, we have 



sin + isin 30+1^ sin 50 = - 10 [log (± «)-!]+ 10 log 2 . (17) 



Then from (15) and (17) 



1 cos 20 + 1 cos 40 + 



-H 



sin + ^ sin 30 + 



»)-i(i.|l),„g._| 



•} 



(18) 



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



. = - i^ [log (± 0) - 1] - i01og2 (19) 



Isin 20 + isin 49 + 



Integrating (17) and putting in the proper constant to make the left 

 side vanish when = 0, we have 



^^+^+i + 



]_3 ^ 33 ^ 53 ^ 



■ - (p -^ + Is «os 39 + . . . ) 



= _ i 92 log (± 0) + 1 02 (I + log 2) (20) 



For purposes of practical calculation may be taken as so small that 

 the right-hand side of (18) reduces to — ^ log (± 29), and the right- 

 hand sides of (19) and (20) to zero. 



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



nivJi 21 



— xr2 



givh 



■L+L + L + 



13 33 ^ 53 



= ^i^ x^x 2-1037 



TTV TT' 



da qivli 1 1 rv 1 



27rz 



(21) 



