for Addition of Velocities. 



773 



seen from I he ground, no matter at what rate, under light- 

 velocity, the platform is moving. Assuming physical con- 

 tinuity up to light-velocity, we reach a conclusion contra- 

 dictory to the last. 



Mathematically the paradox lias its origin in the fact that 

 the expression {u-\ v)j(\-\-vv) is indeterminate for the values 

 n— — r — 1, and approaches different limits in the two ways 

 in which we have reached these values of the variables. 



Physically it means, I suppose, that the measured speed 

 of 300,000 kilometres per sec. cannot be attained by matter, 

 and that, at speeds very close to this, there would be an 

 extraordinary instability in the measured value of the re- 

 sultant of opposite velocities of that order of magnitude. 



Fiff. 2 shows the variation of the magnitude of the resultant 



when in- are at right angles to each other, in accordance with 

 the formula (u 2 + v 2 — u 2 v 2 )i. The curves are again hyper- 

 bolas with centres at the origin and asymptotes 



On the old theory they would be hyperbolas, passing through 

 the same points on the ?/-axis with asymptotes y= +a\ 



In the general case, when uv are inclined at an angle y 

 the magnitude of the resultant is 



(u 2 4 r' 2 -f 2uvcoay—u 2 v 2 sm i y)y(l + uvcosy). 



