368 Prof. Challis's Determinationqf the Velocity of Sound 





^■=z-\-at, v=z—aty Gi(v)==/G(v)(afv, 



G2(v)=/Gi(v)rfv, &c., 

 it was found that 



^=G(v) + .^Gi(v)+ '^G,(v)+ £|-^G3(v) + &c. 



No inference respecting the propagation of the motion can be 

 drawn from this result, unless <^ be expressed in exact terms. 

 The form of the series shows that this can be done only by 

 satisfying the following equations : 



1 r* f \ _i_ d.Giy) 



&c.= &c. 



Now since Gi(v)= /'G(v)<?v, it follows that 



^i^)=G(v) 



dv ^ 



or 



d^.GAv) _ „ „ , . 

 -^— +^'-Gi(v)=0. 



The upper sign would give to the function G a logarithmic 

 form, which is clearly excluded by the nature of vibratory 

 motion. Taking, therefore, the lower sign, the integral of 

 the last equation gives 



G,(v)= sin(Mv4-c). 



Consequently 



G(v)=wcos (wv + c). 



This form of G satisfies all the other equations. Hence 



