391 



" Since v does not vanish when xsaO, - is not obtained by differ- 



dxr 



entiating under the integral sign, but the term a v x=0 must be sup- 



7T 



plied*, so that (observing that v x= o=f(t) by one of the equations 

 of condition) we have 



Hence 



dv d 2 v C 

 i --- 7-5=1 



dt dx 2 J o 



2 ,, .. 1 

 , - -- a, fit) > 



dt if J ^ J J 



and the second member of the equation being the direct develop- 

 ment of the first, which is equal to zero, we must have 



dtff 2 _f/.\ n 



_ +a 3 W __ a /(Y) = o, 



whence 



-?*[* 2 j.,^ a?t j* 

 cr = e I - a/(0e dt, 



J * 



the inferior limit being an arbitrary function of a. But the other 

 equation of condition gives 



therefore 







But 1 e~ tta2 cos ba,da.= jf-J e~4^, 



'o 

 therefore 



r-a2 rf fl/irXi - 



e sin 6 a . ac?a = -< -( - I e 4a 

 rfo 1 2\a/ 

 V. \ / 



whence writing i t', x, for a, i, and substituting, we have 



* f * , -3 _fl 

 *f*Jo 



" Your conclusion as to the American wire follows from the dif- 



* According to the method explained in a paper " On the Critical Values of the 

 Sums of Periodic Series," Camb. Phil. Trans, vol. viii. p. 533. 



