42 Art. 3.— K. Hirayama : 



wliere y/=co-^' is the minimum value of cos^' and e is tlie corres- 

 ponding value of e. Now 



^o/ ip'—'^')ài-=-^ I (x—x)dt 



_ 8 ('^ ( X — X __ X' — X \ja, 



~ Ss' J ^ Vs'ic — 4^2 cos<?' s'x'—^p^co^d'j 



dd' 

 X corresponding to the portion in Avhich —rr is negative, and 



y_ {8'x — A:p^Q^o^d'\x — x') _ —ip.2(cosd'—cosd'){x--x') 



~ (s'x — 4^2 COS d'){s'x' — 4j?2 COS d') ~ {s'x — 4p^ cos d'){s'x' — 4^2 cos 6') 



The denominator is negative. The differences x—x and cos^'— 

 COSÖ' are positive and therefore Z is always positive. Hence 







Now 



I e{ê--e^)ài-j {e-e){é--é')dt-V-è (é'-e')dt 

 and therefore 



I e{é'-e')dt^O 



Hence 



r^dt<o 



J ,, dt 



always. Accordingly the llbrations ultimately change to revolutions. 

 The lihration of Type 2 changes to the revolution of Type 2 or to 

 that of Type 1, and the lihration of Type o to tlie revolution of 

 Types. 



