Astronomy and High-speed Inertia, 85 



which contains a slightly modified constant, and also a 

 periodic term. It will suffice to take the case o£ orbits 

 sufficiently circular to enable us to treat v as constant, so as 

 to put a mean value in the small correction term instead of 

 introducing av 2 = /bi(2au — 1). In that case also we shall 

 have no trouble about the precise meaning of 6. In so far 

 as the angle between tangent and radius vector is not a right 

 angle, i. e. in so far as the velocity 6 and the position 6 are 

 not the same, the effect for small excentricities is to modify 

 the solitary k in the denominator of equation (5) below into 

 k(l — 2e/'d); but, as this is just the k which can be neglected, 

 the slight complication will not be here attended to. 



Before solving (2), we may note that this equation has the 

 form of the ordinary resonance equation, 



x + tcx -f n 2 x = E cos pt, 



with x equal to n minus the constant terms above, but with 

 the damping coefficient tc zero and with the frequencies n 

 and p equal. So it is an equation which is liable to give 

 infinite values; and even in practice it gives large accumu- 

 lative amplitude when the damping is small. It is the 

 equation on which all tuning or syntony in Wireless Tele- 

 graphy is based. The ordinary particular integral for this 

 case, 



E 



X— a „ COS Vt, 



rr—p* 



gives an oscillation of infinite amplitude, the main infinite 

 part of which, however, may be got rid of by combining it 

 with the supplementary part of the complete solution with 

 arbitrary constants, 



A cos nt + B sin nt. 



For putting p=n-\-z, and proceeding to the limit when z 

 is zero, we get 



_ E cos (n + z)t _ E cos nt ~Ezt sin nt 

 x ~ n 2_( n " fr )2 -" - 2 nz 2n~z 



The first term is infinite, but when combined with the 

 arbitrary term A cos nt it disappears, and the solution 

 is left, 



# = ^E/n . t sin nt, 



which exhibits an amplitude steadily increasing with time. 



