116. Mr. A. Stephenson on the Forcing of 



2. The equation of motion under such action is 



.i/ + /A 2 (l-f 2«i cos ??£ + 2a 2 cos 2nt + . . .) x=a sin (gt + e). 



The properties o£ the forced motion follow readily from 

 consideration of the free motion, the solution for which we 

 shall therefore briefly recapitulate*. For the free motion, 

 i. e. a = 0, we have 



CO 



%= %A r sm{(c — rii)t + e'} 

 where ~°° 



A r {^ 2 -(c-r;0 2 }+^ 2 {ai(A r _i + A r+1 )+a 2 (A r ._ 2 + A r+2 )4-...} = O. 

 On eliminating the A's, we obtain the infinite determinant 



= 0, 



a 2 cl x [ — 1] «! a 2 



u 2 « : [lj «! « 2 



where [r] denotes/^ 2 — (e — rn) 2 . This equation determines o, 

 and the roots are all included in the form +c —rn. If c is 

 real the motion is made up of simple elements of constant 

 amplitude, but if c is complex the amplitudes continually 

 increase, as one part of the solution contains a factor e K \ 

 where \ is a positive quantity. The forced motion due to 

 a sin (qt-\- e) is given by 



00 



x— X A r sin {{q — rn)t + 6}j 

 where _0 ° 



Aj^ 2 -(g~m) 2 }+^^ r a 1 (A r _ 1 + A r+1 )+a 2 (A r _2 + A r+2 ) + ...} = 0, 



except in the case r=i), when 



A o(^ 2 -5 2 )+^!«i(A-i + A 1 ) + a 2 (A_ 2 + A 2 ) + ...}=a. (0/ 



These equations determine a convergent series whatever 

 the value of a. The conditional equation (r) f is the same as 

 the corresponding equation (r) in the free motion with q for c, 

 except that in (0) / the right-hand side is a instead of zero. 

 If, therefore, c is real, the A's become large without limit as 

 q is taken nearer and nearer to c ; and in the limit when 

 q = c the solution passes over into the form 



x = tX'Br cos {(q — rn)t + e\ +2C r sin {(q — rn)t-\- e}. 



* The method is due to G. W. Hill, " On the Motion of the Lunar 

 perigee," Collected Works, vol. i. The properties of the infinite 

 determinant and the numerical evaluation of c are dealt with therein. 



