HARMONIC ANALYSIS AND PREDICTION OF TIDES. 23 



Equation (37) must be true for all points in the surface of equi- 

 librium, so if a point be taken in the surface where the tide-producing 

 potential is zero — that is, where cos 6= Vl/3 — and let a represent the 

 value of r at this point we have 



lE 



V=^ = the constant (38) 



a 



Substituting this in (37) 



r^^--j^ — (3 cos^ 6—1)= — (39) 



from which, by transposing and dividing, 



Ma 

 Ed- 



i^{3 cos^ e-i)=5(r-a) (40) 



Let 



r = a + u ■ (41) 



so that u will represent the equilibrium height of the tide due to the 

 principal lunar force, referred to an undisturbed spherical surface of 

 radius a. 



Substituting (41) in (40), we obtain 



i-^rr, 3 cos^ ^-1) = 7 — ^ — r^ = — 3 - ) +6( - -etc. (42) 

 Ed^ [a + uy a \aj \aj 



u . 

 The fraction - is approximately the ratio of the semirange of tide 



to the mean radius of the earth, and if we assume a range of 40 feet, 

 the numerical value of this fraction would be about 0.000001. It is 

 evident, therefore, that we may neglect the powers above the first, 

 and write 



|=i^'(3 cos^ d-1) (43) 



or 



u = i^{Scos' d-l)a (44) 



as the equilibrium height of the tide due to the principal lunar force. 

 In the preceding formulas a was taken as the radius of the earth 

 along which the tide-producing potential of the force under con- 

 sideration was zero. Let us now see whether this is the mean radius 

 of the earth; that is to say, the radius of a perfect sphere having 

 the same volume as the earth. It is evident that this volume must 

 remain constant without regard to any deformation to which the 

 surface may be subjected. Referring to Figure 7, consider the 

 volume of the earth to be divided into infinitesimal solids by a series 

 of planes with their common intersection in the line OC, the angle 

 between two consecutive planes being designated hj d <p; a series 

 of right conical surfaces with their common apex at and common 

 axis in line OC, the angle between the generating line and the axis 

 being designated by 6] and a series of spherical surfaces with their 



