44 ^it- 3.— K. Hirayama : 



The equations for the variations of the constants become 

 ' = 'àae'x + ^ßipß'^'' cos d' ^^- (^ + V)e' 



dC 



dt 



E always decreases, as in the previous cases. We can prove also 

 that C increases if negative, that it decreases in the revolution of Type 

 1 and that it iiicreases in the revolution of Type 3 if E be legs than a 

 certain 'positive value. 



53. We have to consider the orders of the smaU quantities in 



this case. The eccentricity e may be supposed to be a quantity of 



the order of 10'\ and the constant j9„ being a quantity multiphed 



by ?/i'= 1/1047, may therefore be supposed to be the third order 



of e. Hence, assuming that the orders of the three terms in the 



fii'st equation of (12) are the same, x becomes a quantity of the 



'i + 3 

 order — :^. Consequently if i>o, the order of x will be higher 



than tliat of e' by a unit order at least, so that e may be supposed 

 to be a constant in the first approximation. In the second member 

 of the third equation of (12), the order of p,é~\ being /+1, is 

 higher than that of x by a unit order at least, if i>?). Hence we 

 may write 



-— = SX — Q COS u 



7lo dt -^ 



where y is a constant. 



54. The e(iuation corresponding to (H) becomes in this case 

 -|-(^ê-^-')= [^è^ + ^(r-r)+,9(.-2)^4-?^(r-ê^')] -^'^'e 



in which 



^r_ s' Ss'e'—i(i — 2)x -,( 



^ 7— ^^> — -•> T Zi7~^ 



I S X — l'Pfi' - COS , 



