200 Mr. Mallet on the Physical Conditions 



The coefScient h is readily obtained, g and I' being both known, ^/l (21' — I) l\ 

 is a mean proportional between 21' — I and I, and is also had when I is given. 

 It may be also shown that 



V=kl,sin,kT, (22) 



T being the time from the commencement of motion, when the extension or 

 compression is I, and the time due to I', the maximum expansion or compres- 

 sion is 



■n being = 3'1416, the ratio of the circumference to the diameter; so that the 

 time of one complete oscillation is double this, or, 



and the number of complete oscillations per second of time 



2T:^i\l'J 2-Kyi\PLj 



in which it may be remarked that inversely N : T^. The mean velocity of 

 oscillation being equal to twice the amplitude 21' divided by the time, or to 

 WN^ is given by the expression 



The effect of the strain P producing acceleration at any moment, T, which eifect 

 is the whole of P at the moments that the oscillation begins and ends, is 



P{l-cosin]cT), (27) 



that is to say, its periodicity varies with I, assuming as is done throughout, that 

 the extensions or compressions are proportionate to the extending or com- 

 pressing forces, and that between o and I' the elasticity of the bar remains 

 perfect. 



124. Case II. — When the straining load possesses an initial velocity. Here 



p 

 the work done by the initial velocity, half the vis viva of — at the moment it 



reaches the bar, together with the work done by P, times the height Z", must be 



