238 REPORTS ON THE STATE OF SCIENCE. 



Now, if a change of variables be made according to the following 

 scheme, 



X = ft(x-vt), Y = y, Z = z, T = /3^-^, . . (i) 



c =c+ [vh]/c, h' = h- [ve]/c, 



E x =c' x =e x ?,=/to' r E 3 =/,V H. r =/,,', H/=/3V, R/=ftK' I {U) 

 we find that the equations (a) are exactly equivalent to equations 



Ig— m. -]^-«* • • (A) 



divE=0, divH = 0. 



The immediate consequence of this theorem is that if any solution of 

 equations (a) be found, say 



e = e (as, y, 2, 1), 

 h =7](x, y, z, t), 

 we may thence deduce a solution of equations (A), viz., 



E, =e x = * s [/3(X + vT), y, z, ,3 ( T + ^)] 



E, = j9 (e, - J &,) = /3 [«, { /3(X + »T), y, z, p(* + 3f) } 



_^ : {/3(X + rT),y, 2 , /8(t + 5)}} 



Suppose now that there exists a being capable of following the pro- 

 pagation of electromagnetic disturbances in the aether without himself 

 affecting those disturbances, and not capable of perceiving any other 

 phenomena whatever. Further suppose that all this being requires of his 

 measures of space and time, of electric and magnetic vectors, is that by 

 use of them the laws of his sethereal universe are expressed in the 

 form (a). Then, as a consequence of the invariance of these equations, this 

 tethereal being has no means of discrimination between (xyzteh) and 

 (XYZTEH). As far as he knows, he might sometimes be measuring 

 in one set of variables and at other times in the other set. 



If he adopted (x y zt) as space time co-ordinates then velocity would 



be defined as 



T Ix 



If, on the other hand, he adopted (X Y Z T), then velocity would be 

 defined as 



D 8X 



These lead at once to the transformation of velocities 



W r — V Trr _ W„ W _ W z 



Wx-s- 1 * • . w r = , w » — , w z = 



VIV 



0(1-*-) ' ) ( 1 -?) 



In particular, \i w x = 0, W x = - u ; and if W A - = 0, w x = v. 



If a is the acceleration of a point at rest in the (x y zt) co-ordinates, 



