Skating on Thin Ice. 19 



and if d is the deflexion at the free end, 

 d Bj 

 h 



^- = ^-rj 2 +V3— V* = (th gZ + tan ql) sh ^Z sin ql 



-t>3 JD3 



+ sh(//cos (/Z — chgZsin ql 



sh g-Z sin ql _ sh 2^— sin 2^Z 

 ~~ cos ql ch g>Z 2 ch ^Z cos ^Z ' 



P^ sh2?Z-sin2?/ 

 />6 * ch 2 ql + cos 2 <^Z 



(17) 



(18) 



This reduces to the preceding value of d in (4) § 8 when 

 ql is small, and the buoyancy is neglected. 



Velocity of Sound in Air. 



15. The question of sound velocity may receive recon- 

 sideration here. 



For the velocity W of longitudinal vibration through ice 

 or a solid substance, we have taken 



W ! = <j»f = <^, (1) 



P 



so that the velocity is that acquired in falling freely under 

 gravity through half the elastic length. 



But in air the cubical elasticity, p~r, which on Boyle's 



isothermal law is equal to the pressure, becomes y times the 

 pressure in the adiabatic compression and expansion of rapid 

 vibration ; so that the velocity of sound in air is taken as 

 the velocity acquired in falling through y times half the 

 height of the homogeneous atmosphere ; and this y is the 

 ratio of the specific heat (S.H.) at constant pressure and 

 constant volume, taken on the average at 7=1*4. 



Suppose, however, a shower of rain is falling : how does 

 this affect the value of 7, and the velocity of sound ? 



The thermodynamical influence is the same as that in- 

 vestigated by Sir Andrew Noble in his " Research on the 

 Pressure and Work of fired Gunpowder," Phil. Trans. 

 1875-94. 



There he found by his experiments on the pressure in 

 a closed vessel, that the law connecting pressure with volume 

 and density was something intermediate to the isothermal 

 law, with 7=1, and the adiabatic law, 7=1*4; and he 

 accounted tor the difference by the influence of the heat 

 stored up in the solid particles of powder in the gas. 



C2 



