82-85] Tidal Figures 83 



Mutiplying equations (218) to (220) by 3, 1, 1 and adding, we find 



which is of course merely the condition that P shall be harmonic. 



The elimination of a/a 2 , /3/6 2 and 7/c 2 from these same three equations 

 gives 



+ _ - + _ 



CL C 



9/3 



Y( Cl c 2 + dCs + 3c 2 c 3 ) 



/ 



9/3 /2$\ 2 



-I [ci (& + c 2 ) + c 2 (3a 2 + c 2 ) + c 3 (3a 2 + 6 2 )] + ( -^ ) = . . .(223), 

 QJ \ a / 



in which we may insert the value of 6 obtained by eliminating p and o> 2 from 

 equations (205) to (207), namely 



o \ / 



r A -J,, + aJc) (224). 



Equation (223) now becomes purely an equation in a, b and c ; it is the 

 equation which determines the first point of bifurcation on any linear series 

 of ellipsoidal configurations. 



84. Let us limit our discussion to ellipsoids such that abc = r 3 . On re- 

 placing c by r */ab, equation (223) becomes an equation in a, b only, and so 

 may be represented by a curve in a diagram such as that in fig. 7 (p. 50). 

 We may examine in particular the points in which this curve will meet the 

 spheroidal series of tidal figures and the Jacobian series of rotational figures, 

 or, more directly, we may search for the first points of bifurcation on these 

 series. 



Tidal Figures 



85. On the tidal series of spheroids, b = c, so that c 2 = C 3 , and for a zonal 

 harmonic distortion we have also @ = 7. Thus equations (219) and (220) 

 become identical, each reducing to 



_3a c 3 c =e ft 



while equation (222) becomes 



_3a == 2/3 

 a 4 c 4 ' 



Eliminating a, ft from these equations, and inserting the value of 6 from 

 equation (81) we find, 



2(9 4 



tf (3c 2 + 2a 2 ) a (3c 2 + 2a 2 ) (c 2 + 2a 2 ) 



62 



