Liquid Spheroid and the Genesis of the Moon. 263 



Equations (28) and (29) now give the two values of n^ 



VIZ. : 



V=2..p[3|^, -7W) (^ + «/^+/')] = (^ 



(37) 



and 



-/ = 47r7P3^,[3^-^-:^^ (38) 



8. The condition that the expression for n^ in (37) should 

 be positive is Riemann's condition of stability. Using the 

 condition (31) of steady motion, we find that this requires 



/(3 + 7/2) >(3+ 8/H/^) tan-y. 



27r . /Sa 



The period — given by (38) reduces to ir A/ — in case 

 2 y 



the spheroid becomes a sphere of the same mean density as 

 the earth, and there is no rotation. This is the period given 

 by Sir William Thomson *. 



The period of rotation given by Professor Darwin in his 

 first paper f as most probable for the earth-moon system when 

 the two bodies formed a single mass is 5 hrs. 36 mins. : in a 

 later paper he finds that this is doubtful, but that the period 

 was most likely between two and four hours |. 



I append two Tables, of which I. gives the shorter period 

 27r/ni and the density for different values of/^, the period of 

 rotation being 5 hrs. 36 mins. ; and II. gives the longer 

 period 27r/»2, and the density for different values of /^ and 

 different values of the period of rotation from three to six 

 hours. 



Table I. 



p. 



e . 



P- 



T,. 



018 



0-02077 



5-77 



2957 sec. 



0-19 



002177 



5-50 



3054 sec. 



0-20 



002295 



5-22 



3177 sec. 



0-30 



0-03170 



3-78 



3842 sec. 



0-40 



03969 



301 



4454 sec. 



This Table gives the value of the shorter period T^ and the 

 density when the length of the day is 5 hrs. 36 mins. 



* Phil. Trans. 1863. t Ibid. 1879. % Ibid. 1880. 



