Prof. Thomson on the Electric Telegraph. 147 



It is now communicated to the Royal Society, although only in an 

 incomplete form, as it may serve to indicate some important practi- 

 cal applications of the theory, especially in estimating the dimen- 

 sions of telegraph wires and cables required for long distances ; and 

 the author reserves a more complete development and illustration of 

 the mathematical parts of the investigation for a paper on the conduc- 

 tion of Electricity and Heat through solids, which he intends to lay 

 before the Royal Society on another occasion. 



Extract from a letter to Prof. Stokes, dated Largs, Oct. 28, 1854. 



" Let c be the electro- statical capacity per unit of length of the 

 wire ; that is, let c be such that c.'v is the quantity of electricity 

 required to charge a length / of the wire up to potential v. In a 

 note communicated as an addition to a paj)er in the last June Num- 

 ber of the Philosophical Magazine, and I believe at present in the 

 Editors' hands for publication, I proved that the value of c is 



—j, if I denote the specific inductive capacity of the gutta 



2 log - 

 ^ R 



percha, and R, R' the radii of its inner and outer cylindrical surfaces. 



" Let k denote the galvanic resistance of the wire in absolute elec- 

 tro-statical measure (see a paper ' On the application of the Principle 

 of Mechanical Effect to the Measurement of Electromotive Forces 

 and Galvanic Resistances,' Phil. Mag. Dec. 18.51). 



" Let y denote the strength at the time t, of the current (also in 

 electro-statical measure) at a point P of the wire at a distance x from 

 one end which may be called O. Let v denote the potential at the 

 same point P, at the time t. 



" The potential at the outside of the gutta percha may be taken as 

 at each instant rigorously zero (the resistance of the water, if the 

 ■wire be extended as in a submarine telegraph, being certainly inca- 

 pable of preventing the inductive action from being completed in- 

 stantaneously round each point of the wire. If the wire be closely 

 coiled, the resistance of the water may jiossibly produce sensible 

 effects). 



" Hence, at the time t, the quantity of electricity on a length dx of 

 the wire at P will be vcdx. 



" The quantity that leaves it in the time dt will be 



dt -^ dx. 

 ax 

 " Hence we must have 



-cdx^dt=dt^dx d). 



dt dx ^ ^ 



"But the electromotive force.in electro-static units,at the point P, is 



dv 



dx' 



and therefore at eacli instant 



*>=-£ (^)- 



