AND THIN ELASTIC RODS. 365 



Let y-2 (i'nSin — 1 . 



Then ^ .^'-^'^^ Sl 

 df a" df df '^ M'' 



(Stopping at the term P3, which will render the results sufficiently approximate.) 



Hence substituting for Q, and writing M for ma, and putting 



M + iM' 

 2M' " *' 



ePP, «FP3 6V* 



e 



d<^ df 2M' 



d»P. d^P, 

 + e 





df df 



whence P, and P3 can be found when e is given. 



Let ^ be the statical central deflection due to the mass M', then 

 If we assume e - 3, which makes M = 8M', we shall find 



P, + 240^3 = - 1-94S [1 - cos V ^ X 30-4« j nearly, 



P. - ^ = -9^ [1 - cos V^ -Si^) nearly; 



the greatest depression is 1 'SS nearly. 



If we assume e = 1, or the mass of the girder indefinitely small, in proportion to that 

 of the load, the greatest depression = 2^ within two places of decimals 



= 2o'(^-^ia, 

 \ir* 81 ttV 



a very close approximation to its true value 2^ in that case. 



To determine the motion of an elastic rod, fixed at one end and free at the other. This 

 problem is much more difficult than that of the vibration of a rod, of which the two ends 

 are at rest and the intermediate parts only in 3 state of vibration. I tried it by many 

 methods, but returned to the one I first thought of, as after all the easiest and best for 

 numerical calculation. I may observe by way of preliminary, that Fourier's series do not 

 apply to this case, on account of the values of the derived functions of /(«), the equation of 



