106 Mr Bryan, On the beats in the [Nov. 24, 



Substituting these expressions for w, Bw and integrating by parts 

 the terms containing dSv/dd, we find the equation of motion 



.. d 2 v . dv , „ x d 2 v 



v -de^^de +{( °-^aW 



T ( d 2 v #v\ #_(#_. \* _ n 

 + ^ V d& + m <ra* dF \dd 2 + i ) v - {} - 

 To find the frequencies, assume that 



v = A cos (s6 + pt) (8). 



Then the last equation gives (substituting for T its value by (1)) 



(1 + s 2 )p 2 - k<ops = (o> 2 - fi) s 2 (s 2 - 3) + -^ s 2 (s 2 - l) 2 (9), 



therefore 



/ 2sa> \ 2 4a>V , , ,s 2 ( s 2 -S) , /3 s 2 (s 2 - l) 2 



{P-s^Ti) s= (?Ti) 5 + (<B -">-?+!- + ^-F+I--< 10 >- 



> 4a)V , 2 x s 2 (s 2 -3), /3 s 2 (s 2 -l) 2 /n1N 



Let w 2 = -2 — rr-, + (w - h) g , -, + -^4 — n; — =-^...(11), 



(s + 1) v s+1 <ra s +1 v J 



then the two values of p are p v p 2 , where 



2s&) 2,9&) 



and the corresponding motions of the small corrugations relatively 

 to the mass are determined by 



V = il COS JS0 + -j— T * + **.*[ ' 



and v=Acos\s0 + s ■ -. t — 'srj 



respectively, together with the relation w = — dv/dd. 



To find the actual motion in space, substitute <j> — oot for 6 in 



the two last equations ; the positions of the corrugations will now 



be referred to the fixed initial line. We find for the two types of 



oscillation, respectively 



{ , s 2 -l j 



v = A cos l scp — £-— y stot + i&jy , 



and v = Acos\s<fi ^— r scot — vrM. 



If the amplitude (A) is the same in both, we see, by addition, that 



their resultant is given by 



/ s 2 — 1 \ 

 v = 2A cos'utJcoss (<£ — a-— r tot) (12). 



