﻿498 Prof. E. V. Huntington on a Neiv 



arrive simultaneously is said to lie on the perpendicular 

 through (m, 0). To each of the points that lie on this per- 

 pendicular an ordinate, y, is now assigned in an obvious way, 

 by means of light-signals started from the point (m, 0). 

 In particular, the points (m, 1), (in, 2), ... ; (m, — 1), 

 (in, — 2), . .., are determined. 



7. In describing* this rather novel process of laying out a coordinate 

 system by light-signals, we have tacitly assumed that each stage of the 

 process is possible — in other words, that the coordinate system thus 

 obtained will be permanent, so that the coordinates assigned to any given 

 station will always be the same whenever the process is repeated. This 

 assumption is obviously j ustified in the case of a system at rest in the 

 eether; but a moment's reflexion will show that this will not in general 

 be true in the case of a system which is moving with respect to the 

 sether. 



For example, consider a circular platform rotating with constant 

 velocity about its centre O. Here all the clocks could readily be sjm- 

 chronized with a central clock at 0, according to the rule in § 4 ; but 

 two clocks A and B on the circumference would then not be synchronous 

 with each other according to the same rule *. 



In one special case, however, the process is legitimate, namely, in case 

 of a system moving through the ceiher with uniform velocity in a straight 

 line, and to the proof of this we shall turn our attention in Part II. 

 For the present, we consider only the case of a system at rest. 



Definition of distance. 



8. A pair of coordinates having been assigned by the 

 method of light-signals to every station in the plane, the 

 " distance " between two points (jci, yi) and (a: 2) yi) is then 

 defined as the quantity 



^(x 2 -x i y^(y 2 -y l Y i 



and the observers on the platform are now in a position to 

 develop the whole theory of coordinate geometry for the plane. 

 For example, all points whose coordinates were found to 

 satisfy the equation x 2 + y 2 = r 2 would be said to lie on a 

 circle of radius r, &c. 



Definition of observed velocity. 



9. The observers are now able to discuss the velocity of an 

 object, say an aeroplane, that flies in a straight line across 

 their platform. To determine the velocity, two observers 



* The relation "synchronous with," as here defined, is therefore, 

 although symmetrical, not necessarily transitive ; it resembles rather 

 the relation " friend of." A and B may both be friends of C, and yet 

 not be friends of each other. 



