AND THIN ELASTIC RODS. " 367 



Assuming for the vibration of the rod 



y - 2 {(A sin/',* + 5„ cosp^s + C„e^-' + D^e-^-') cos plbt\, 



where J^, &c. are the »*'' particular values of the general ^,, &c., we find the above ex- 

 pression yet further reduced, and , ■ ' 



J„= i)„{l+e-^.»(-i)"}. 



5„=-Z)„{l+e-''.»(-ir'}. 



These expressions are only useful in computing the first one or at most two values of C„, 

 jff„, and A^, as after that e'^'" may fairly be assumed = 0. 



Perhaps it might be possible to develope directly any /(«) in a series of terms similar 

 to those we have found as particular solutions of the equation, but in any case it would be 

 labour thrown away, as it is not necessary to consider more than the first two values of pa, 

 and, by equating the differential coefficients in the statical curve, 



y = ; as - , where s = 0, 



we shall obtain an approximation quite close enough. Of course the most simple and natural 

 method of solving any partial differential equation between two variables *, t, would be to 

 obtain a particular solution 



where p„ is one of an infinite number of roots of the equation -^ (A) = 0. 



Then, if we could develope any other given function of s, say \|/(s) in a series of the 

 form 2. {^„(/p„s)^, we could always determine 2)„ directly, supposing we knew the value 

 of y when t = 0, But such expansions are not always possible, or at all events practicable. 



2 ■ ' 



Roots of cos pa = - 



By trial and error, we find 



25 

 PiO = 1.875 = — nearly (measure of 107". 27')> 

 8 



6'"' = 6.521, e-^>''=.153, .. 



cospia=--.3, sin p^a = .954,^ ^ 



p^a ■= 4.69, cos Ugft == , 



55 



Vol. X. Part II. 47 



