CONTEMPORARY ADVANCES IN PHYSICS 291 



U{x, y, z, t) = s{x, y, z, t - r/c). (3) 



The value of U in any point of the surface 5 at the moment / is the value 

 of 5 which prevailed in that point at the moment when the "wavelet" 

 started forth which was destined to reach the origin at /. It might be 

 said that an observer, stationed in the origin at the moment / and 

 inspecting the surface by means of the wavelets, observes the values of 

 U instead of the contemporary values of 5. Thus a star-gazer viewing 

 the sky perceives, not the stars as they now are or as at some one 

 past moment they all were, but each star separately as it was at some 

 past epoch peculiar to itself; and the apparent arrangement of the 

 heavenly bodies is one which in fact has never existed. 



We shall be concerned not only with the value of 5, but with the 

 values of the space-derivatives ds/dx, ds/dy, ds/dz, which prevail at 

 each point of the surface at the moment when the wavelet starts forth ; 

 for all of these will influence the value of 5 at the origin when the wave- 

 let arrives there. These may be written as derivatives of U; but one 

 must be careful here, for U is a function of x, y, z not only explicitly, 

 but also implicitly through r; and there is a distinction to be made 

 between total and partial derivatives, a distinction having physical 

 importance. 



To grasp this, denote by (x, y, z) the coordinates of some particular 

 point on S, and by {x 4- dx, y, z) those of a nearby point, and by r and 

 r -\- dr their respective distances from the origin, and by U and 

 U -\- dll the values of If in these points at the instant t. Now, U 

 and U -\- dU are values of 5 which existed at different instants of time, 

 as may be seen by writing down the expressions : 



U = s{x, y,z,t - r/c) ; U -\- dU = s{x -\- dx, y, z, t - r -\- dr/c). (4) 



Therefore, if I form the total derivative dU/dx in the classical way, I 

 am not obtaining the value of ds/dx which prevailed in (x, y, z) at 

 {t — r/c). To obtain this value, I must begin by subtracting the value 

 of 5 prevailing in (:r, y, z) at (/ — r/c) from the value of 5 prevailing in 

 {x -{- dx, y, z) at the same moment; that is, I must form the differ- 

 ence between {U -\- dU) and U, meaning by the former symbol : 



U -{- dU = s(x + dx, y, z, t — r/c). (5) 



I must then divide this difference by dx, and pass to the limit. But 

 this is the classical way of forming the partial derivative of U with 

 respect to x. Therefore the values of the derivatives ds/dx, ds/dy, 

 ds/dz prevailing at the moment of departure of the wavelet which is 

 destined to reach the origin at t, are the partial derivatives dU/dx, 



