Prof. Thomson on the Electric Telegraph. 153 



to be placed at these points. For the effects of Oj and O' will 

 balance one another as far as regards the potential at O. 

 *' So will those of Og and O'j. 

 O3 and O'g. 

 &c. &c. 

 And again, O and O' would alone keep the potential at E, zero. 

 So would Oj and 0\. 

 O., and O'a- 

 &c. &c. 



r r- 1 



0' A 0; 



G 



0' 



J. 



Hence if we denote 2lkc by a, for brevity, the general solution is 





}• 



+ (z—a)e Mt-0)^(2—'2a)e ^(i-e; +. 



where F(d) is an arbitrary function such that F(<) expresses the 

 potential sustained at O by the battery. 



" The corresponding solution of the equation 

 , dv d^v , 



dt ~~ dj^ 



2t* Jo (f-0)f -- r J 



by which the effect of imperfect insulation may be taken into 

 account." 



Extract of Letter from Prof. Stokes to Prof. W. Thomson (dated 

 Nov. 1854). 

 " In working out for myself various forms of the solution of the 

 dv d"v , , 

 equation -ff^'T'"- ^^^^er the conditions v=0 when ^=0 from x=0 to 



j7=Q0 ; v=f(t), when x=0 from t=0 to <=oo , Hound that the so- 

 lution with a single integral only (and there must necessarilybe this 

 one) was got out most easily thus : — 



" Let V be expanded in a definite integral of the form 



V=l CT(<, I 



which we know is possible. 



" Since r does not vanish when ^=0, — ^ is not obtained by differ- 



dir 



Q 



entiatmg under the integral sign, but the term - a rj,=omust be sup- 



, a) sin axdx. 



Phil. Mag. S. 4. Vol. 11. No. 70. Feb. 1856. 



M 



