280 Mr. J. H. Jeans on the Vibrations 



energies of these vibrations will be expressible in the forms 



2T = ^,^rH«2^>+ (1) 



2V = ai7)iW + «2P2W+ .... (2) 



where p^^ j)2i • • • ^^^ the frequencies of vibration. 



We call one molecule A, and suppose that a second mole- 

 cule B approaches A, has its velocity altered in direction by 

 the forces exerted upon it by A, and finally recedes from A. 

 Let the instant at which the force between these molecules 

 begins to be appreciable be taken to be ^ = 0, and the instant 

 at which they are finally clear of each other's sphere of 

 action be ^ = t. At any instant during the encounter let the 

 forces exerted by B upon A be derivable from a force- 

 function 



Vi8(/)i + V23^2 + (3) 



Then the equations of motion are of the form (dropping 

 suffixes) 



a^ + a/(^ = Y (4) 



§ 3. Before ^ = the molecule would be describing a free 

 vibration, say 



(^ = C cos pi + D sin |7f (5) 



The impulse Ndt acting from i — O to t^dt sets up an 

 additional free vibration of initial displacement zero and 

 velocity Ydt/a. ; the displacement of this additional vibration 

 at any subsequent time is therefore Yc/i sin /^i/a;?. By com- 

 pounding all these vibrations with the original vibration (5) , 

 we obtain for the displacement at any instant subsequent to 

 t = T, the well-known solution 



1 r^-'^ 



<j) = Ccos2Jt-\-'Dsmpt+— \t^t'^inp{t-t')di^. . (G) 



This may be written 



^=(C-Y) cos;9i + (D + X)sin|7r, ... (7) 



where X and Y are obtained by equating real and imaginary 

 parts in 



X + iY= — Ty^H/^ (8) 



§ 4. The square of the amphtude of vibration (5) is C^ + D^; 

 that of (7) is (C-X)2 + (D -)- Y)^, so that the increase caused 

 by collision is 



X^^ + Y2-2(CX-DY) (9) 



