366 Mr ROHRS, on THE MOTION OF BEAMS 



the curve being unknown at the limits « = o, s = a;. The plan I have adopted is as follows. 

 I take a particular solution of the differential equation 



dt" " ds* ' 



where mb'^ = 6', involving exponentials and sines of a with the argument p, and then obtain 



an equation with an infinite number of real roots to determine p, from the conditions of the 



cPy d?y dy 



problem, viz. that , , --— must = 0, a = a, and y and — must be = when « = 0. We' 

 d^ dr ds 



have in this way an infinite number of solutions y = D„f(s, t), where Z)„ is an arbitrary 



constant ; consequently, putting t = 0, we can take any number m of such solutions, and 



determine the m constants by the condition that y = S {Z>„/(s, 0)^ may coincide in m points 



with the bent spring at rest ; it will be found that, taking only the first three values of Z)„ , 



we shall have a very close approximation. Let then 



y = cos p^b't (^1 sinps + B^ cosps + C^^e^' + D^e~P") 



+ sin p^b't (^2 sin pa + /3a cos pa + Ci e^' + D^ e"^). 



This, it will be observed, satisfies the equation 



d'y ,d*y , 



-r-r = —¥ — , where 6' = mh '. 

 df ds* 



Also the factor of sin p''b't will be zero, according to the condition that the rod starts from a 

 position of rest. 



Hence, by the conditions 



we have 



d'y d^y :, dy , 



_, __ = 0, a^a, and-andy-0. « - 0, 



^1 + Ci + D, = 0, 

 ^1 + Ci - Z>, = 0, 



- Bi cos pa — Ai sin pa + C^e^'* + Z),e~^ = 0, 



^1 sin pa - Ai cos pa + C, e^" - A e'P" = ; 



— 2 

 whence cos pa = j 



e^" + e~^ 



pas 



/COS pa + sm pa + e~'^\ 

 ' \ e^" + cos pa - sin pa / ' 



/gpa _ 2 sin pa — e-*"'\ 



Ai= Di \ : ; 



V.e'"' + COS pa — sin pa J 



feP" + e~P" + 2 cospa\ 

 B\ = — D. I : I . 



* 'VeP'' + cos*''-smpa/ 



