26 Art. 3.— K. Hirayama: 



in which x and e are written x' and e' where cos d is negative, and 



y_ s'ee'(e^x'~e'^x)+p^ cos d(e^ + e'^) 

 (s'ex—pi cos 6){s'e'x'+2^i ^^^ ^) 



_ s'ee'(x'—x)E+p^ cos 6(e^ + e'-) 

 (s'ex—pi cos 0){s'e'x' -Vp^ cos ^) 



The denominator is positive in both cases. The quantity x —x is 

 positive in Tj^pe 1 and negative in T^^pe 3. E is positive in Type 

 1 and may be positive or negative in Type 3. Hence, for the 

 révolution of Type 1 taking the negative sign for the double sign, 

 the integral becomes negative and consequently C decreases. For 

 the revolution of Type 3, if E is 7iegative or less than a certain posi- 

 tive value, Y is positive and, taking the positive sign for the double 

 sign, the integral becomes positive and therefore C increases. 



32. Since e cannot be zero permanently in the general case 

 of the first order, E decreases without limit so that it will become 

 negative after a certain epoch. Now, when E is negative, the 

 type of the motion is limited to two kinds, namely; the revolution 

 and the libration, of Type 3. Hence we see that the revolutions 

 and the librations, of Type 1 and Type 2, are not permanent forms of 

 the motion ; hut will change ultimately either to the revolution or the 

 libration, of Type 3. Also, since C increases algebraically when 

 negative, it will become positive if it ivas initially negative, and it 

 cannot become negative if it ivas initially positive; but will decrease 

 to some limiting value which is positive and then will increase 

 without limit. 



33. It becomes necessary before proceeding further to find 

 the limiting values of x and e in the libration or the revolution of 

 Type 2, in which "^=0 for d=7:, supposing the efïect of the 

 resistance is null. Let Xi and e^ be the values of x and e at the 

 zero-point of -jr, then 



x,'=C-6p,e, e,'=E + ^x, 



