AND THIN ELASTIC RODS. 363 



Ist. We observe from (2) that if v = 0, and vt = -, the displacement at the centre by 

 our series is 



as it ought to be, and the series is so convergent that taking only the first term we get a 

 good approximation. 



2ndly. The statical value of S„ is not much increased ; as an extreme case, let 



^ = 1, a «= 420, V = 60, 



3^ = 1, "- = i, 

 a 7 



ZQ 



\ 5 X 32 49/ 



or the denominator is diminished in the ratio of 



12 



225 



Now as the smallest value of ttV is 10 nearly, this is about 201 : 200 nearly. 



But J„ = --~ V-^-B» 



Aim, V —Bi=- •OjBi nearly. 



5 



If o = 460, ^' = - , u = 44, 

 » 6 



Ai will = - -04251 nearly ; this is the same result almost exactly as was obtained by Professor 

 Stokes in this example (the Britannia Bridge). 



W 



Srdly. We observe from (2) that if — be very small, B„ varies little from its statical 



Q 



value provided v be not very great ; but if ( - J be very considerable, then the dynamical 

 part of the denominator in B„ may rise to importance. 



To prove the relations assumed between ^, 5, and 6*, we proceed thus. 



Let Q be the weight at the middle of tlie girder, R, R' the reactions at the ends of the 



girder. Then 5 = ± 6' -— at those points positive at one end and negative at the other. 



