24 . -A^rfc- 3. — K. Hirayama 



Differentiating the same equations with respect to t, supposing the 

 effect of the resistance is null, 



2.(^)=6„cos.(^)-6,,sin*(f) 

 \dr)~ds'\dt ) 



Substituting the expressions of -jr, etc. by (4) and taking the 

 differences between the two sets of equations, we obtain 



(5) 



-^= Sae^x + 6ßpie^ cos d 



which are the equations for the variations of the arbitrary con- 

 stants. These equations show that JE always decreases while G 

 increases or decreases depending on x and cos d. Now the signs 

 of X and cos 6 are both negative in the libration of Type 1 and both 

 positive in the libration of Type 3. Hence C always decreases in 

 the libration of Type 1 and increases in that of Type 3. 



dC 

 30. There are three more cases in which the sign of -jr may 



be determined easily, namely; the libration of Type 2 when Cis 

 negative, the revolution of Type 1, and the revolution of Type 3 

 when JS is negative. In the first of these cases the motion being 

 a kind of libration about 0, 6 may increase from to an angle Oq 

 and decrease to— ^o- The limit ^o is not greater than -^ so long 



dC 

 as C is negative and consequently the second term of -rr is always 



positive. Now, we have by the equation (17) of Chapt. II 



no/e*xdt = /—. 3- 



j J sex— Pi cos o 



and 



