34 Art. 3.— K. Hirayama : 



39. The equation (8) becomes, when ê is very small and 

 negligible, 



dH s'x 



dt 4^^ cos d 



[-2;9s+|-(2« + s'/3>^j 



— dTT 



Accordingly, if x be negative, the sign of —57- is determined 



by the sign of cos^. Applying this result to the limiting cases of 

 the revolution and the libration, of Type 1, we see at once that the 

 quantity II at the point very near to the axis of x, increases in 

 the revolution and decreases in the libration. Now, the quantity 

 11 is very small and negative in both cases. Accordingly it 

 approaches in the revolution and recedes from in the libration. 

 Hence we conclude that the revolution of Type 1 in its limiting case 

 changes to the libration of Type 1. 



The same result may be obtained in the limiting cases of the 

 revolution and the libration, of Type 2, when C is positive and less 

 than C2. Thus we see that the libration of Type 2 changes to the 

 revolution of Type 2, in this case. 



In the limiting cases of the revolution and the libration, of 

 Type 3, the sign of the quantity in the brackets is not definite. 

 It becomes positive when ß is very small compared with a. It this 

 case ^increases in the revolution and decreases in the libration, 

 and, since H is very small and positive, it approaches in the 

 libration and recedes in the revolution. Hence, if ß be very small 

 compared with a, the libration of Type 3 changes to the revolution of 

 Type 3. 



40. Let us next consider the extreme case of the libration of 

 Type 1 in which -jr is very small for ^=;r. We have by (9) 



Now s'e,Zi+pi is negative, 3s'ei^ + Xi is positive by (7), and therefore 

 N is negative. The quantity 6*— e^ is negative for all values of 



e. The lower limit of e is /J-^ — Ci hy (ß), and 



