the Velocity of Sound. 279 



where 



Ylu)=f^{u)du, Y^{u)=f¥^{u)du, G,[v)=fG{v)dv,8i.c. 



Each of the functions F and G satisfies equation (3.). Sup- 

 pose, therefore, that F=0; then 



^ = G{v) + euG,{v) + '^' G,{v) + ^- G,{v) + &c. 



No inference respecting the pi'opagation of the motion can be 

 drawn from this result unless <p be expressible in exact terms. 

 The nature of the series at once suggests a form of G, which 

 gives to (p an exact expression, and, as we shall see, applies to 

 the present inquiry, viz. the form Ae"'". Since f must not in- 

 crease indefinitely with the time, G is clearly a circular func' 

 tion. Let therefore G(iO = Ae""^~i + B£-'^""'~' ; or, what is 

 equivalent, let G{v) = 7ncos{iiv + c). Then, 



G,(v)= -sin{7iv + c)= 3- \ -' 



„ , . m , , m d'^.cosinv-\-c) 

 G,{v)=--,cos{nv^c)= -,' -^^ 



„ , , m . , , m d^.co&inv + c) 

 G3W= - -,sxn{m + c)=--,' ^ -^ 



&c. = &c. 



Consequently, 



. d.coslnv + c)meu d'^.cos(nv + c)me^u^ „ 

 9=^cosinv + c) 1^ '-^ + ^ •-Ts"-^"' 



= 7« cos 

 = 7« cos 



Let, now, 

 Tlien 



{"(^-S)+^} 



< n{;s—at) {z-\-at) +c > 





''+Z=\/^+ie. 



7t=\/- 

 Hence, finally, 



27r/ / 7>? A 

 ^=mcos — {z—at\/ 1 H ^ +c J. 



