SECT. IL] LINEAR MOVEMENT. 369 



The variable temperature v of a point of an infinite line is 

 expressed by the equation 



TT 



a? denotes the distance between a fixed point 0, and the point m, 

 whose temperature is equal to v after the lapse of a time t. We 

 suppose that the heat cannot be dissipated through the external 

 surface of the infinite bar, and that the initial state of the bar is 

 expressed by the equation v=f(x). The differential equation, 

 which the value of v must satisfy, is 



dt ~ CD dx* 



But to simplify the investigation, we write 

 dv d*v 



which assumes that we employ instead of t another unknown 



i 4 Kt 



equal to ^ . 



If in/ (oj), a function of # and constants, we substitute X+%n*/t 



for a:, and if, after having multiplied by -_ g-* 2 , we integrate with 



VTT 



respect to w between infinite limits, the expression 



1 f+ 

 ^1 d?ie~ na 



satisfies, as we have proved above, the differential equation (b) ; 

 that is to say the expression has the property of giving the same 

 value for the second fluxion with respect to x } and for the first 

 fluxion with respect to t. From this it is evident that a function 

 of three variables f (x, y, z) will enjoy a like property, if we substi 

 tute for x, y, z the quantities 



provided we integrate after having multiplied by 



dn P -n* &L ,-* *3L f - q * 



j= e , ,- e * , ._ e * . 



VTT VTT VTT 



F. H. 24 



