372 Sir W. Thomson on Problems relating 



discovered by Fourier : and the effect of simultaneously pla- 

 cing other quantities of heat, positive or negative, at other 

 points will, as he has shown, be determined by finding the 

 effect of each source separately by proper application of the 

 same formula, and adding the results in accordance with the 

 principle of the superposition of thermal conductions stated 

 above. Hence the effect of simultaneously placing equal po- 

 sitive and negative quantities, +Q and — Q, at two points, 

 (a, /3, 7), (a', /3', 7'), will for any subsequent time t be ex- 

 pressed by the formula 



O * f _ (q?-a)2-Hff-/3)2 + (g-Y) 2 _ (tt-a')2-Ky-/3') 2 +(z-y') 2 ^ - 



— — 1~'*{ e at — e at ' I. 



Ss/kV I J 



If in this expression we take a=Ja, a ; =— \a, /3 = 0, B' = 0, 

 y = 0, 7 /= =0, and suppose a to be infinitely small, we 

 find what it becomes by differentiating the first term with 

 reference to a, writing a instead of dot, and taking a = 0, /3 = 0, 

 7 = 0. The result constitutes the solution of the proposed 

 problem ; and thus, if v denote the required temperature at 

 time t and point (a, y, z) of the solid, we find 



Qa 



_ «a+ y 2 +g s 



= ^jr-g . xt~*e at 



A more convenient formula* to express the solution will be 



* In this formula k denotes what 1 have called the thermal diffusivity 

 of the substance — that is to say, its thermal conductivity divided by the 

 thermal capacity of unit bulk of the substance. Diffusivity is essentially 

 reckoned in units of area per unit of time j or, as Maxwell puts it, its di- 



— I. Its value (141 square British feet per annum for the 



trap-rock of Oalton Hill, used further on in the text) was taken from my 

 paper on the " Reduction of Observations of Underground Temperature," 

 published in the Transactions of the Royal Society of Edinburgh for 

 April 1860, where it was found by the application of Fourier's original 

 formula to a harmonic reduction of Forbes's observations of underground 

 temperature. Reducing this number to square centimetres per second, 

 and expressing similarly the results of my own reduction of Forbes's ob- 

 servations for two other localities in the neighbourhood of Edinburgh, 

 and of Professor Everett's reductions of the Greenwich Underground 

 Observations, we have the following Table of diffusivities : — 



Diffusi Tities. 



Trap-rock of Calton Hill -00786 of a square centim. per second. 



Sand of experimental garden .... -00872 „ „ „ 



Sandstone of Craigleith Quarry . . -02311 „ „ „ 



Gravel of Greenwich Observatory "I §m 9 AO 



Hill / U1 ^ 4y " » » 



These numbers were first published by Everett, in his ' Illustrations 

 of the Centimetre -Gramme-Second (C. G. S) System of Units,' published 

 by the Physical Society of London (1875), a most opportune and useful 

 publication. 



