88 THE TIDAL PROBLEM. 



IV. MOMENT OF INERTIA FOR THE LAPLACIAN 



LAW OF DENSITY. 



Letting / represent the moment of inertia of the sphere m, we have 

 1= I r^cos'^dm= II I o r* cos^ (p dr d(p dd (18) 



Substituting the expression for (t from (11) and integrating, we find, 



/ = ^^[3 (fi'-2) sin /ji-fi (fi'-6) cos fx] (19) 



or, making use'of (12), 



J 2a'm[3 ( /x^— 2) sin/i— /x(//^— 6) cosjm] ^^qn 



3/j.^ sin fi—fi cos fji 



Hence the values of Cj and Cj, occurring in (1), are determined by equations 

 of the form 



^_ 2 [3(//''— 2)sin/t— /f (/x''— 6)cos//] .gn 



3/x^ sin pi— pi cos pi 



When Wi represents the earth and m^ the moon, we find from this equation 

 and the value of /i given in (15) that 



Ci = C2 = 0.33594 (22) 



instead of 0.4, the value for homogeneous spheres. 



