154 Royal Society : — 



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

 of condition) we have 



Hence 



d-v r=° r2 1 



d?=] I ^ «/(0 - a-t<r. j sin ccxdx. 



dv d'v C^ (d-!z „ 2 .,,,1 . , 



e 40, 



therefore 



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

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



+ a-ra' a.f(t) = 0, 



at IT 



whence 



-a^'T' 2 rr,s v-H If 



cr = e I - ex,f{t)e at, 



J '^ 



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



equation of condition gives 



^=e-«'' r I ccf{t)^'Ut =(|)"'4'' e-«^^'/(^y^', 



therefore 



j;= I^J I I f(j')xe~'^ sinaxdocdt'. 



ButTe-""' cosbccdcc=l(^J 



4ai 

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



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

 ferential equation itself which you have obtained. For the equation 



kc — = — ^ shows that two submarine wires will be similar, provided 



the squares of the lengths x, measured to similarly situated points, 

 and therefore of course those of the whole lengths I, vary as the times 

 divided by ck ; or the time of any electrical operation is proportional 

 to kcl-. 



"The equation /-^ — = -r-^ — hv gives h cc l~^ for the additional 



condition of similarity of leakage." 



♦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. 



