WILSON AND LEWIS. — • UKLATIVITV. 443 



Of these the last is a 3-vector function, whicii clearly vanishes identi- 

 cally. The first three are 1-vcctor functions, and are connected by 

 the relation 



w^f = v(v-f)- v-vf, 



as may be seen by expansion or by the application of (34). 



Kinematics and Dynamics in a Plane. 



35. The three dimensional non-Euclidean geometry which we 

 have developed is adapted to the discussion of the kinematics and 

 dynamics of a particle constrained to move in a plane. The two 

 dimensions of space and the one of time constitute the three dimen- 

 sions of our manifold. xVny (7)-plane in this manifold may be called 

 space, and extension along the complementary (5)-line may be called 

 time. As in the simpler case, any (5)-line represents the locus in 

 time and space of an unaccelerated particle, and any (5)-curve the 

 locus of an accelerated particle. If we choose any two perpendicular 

 axes X\, x-i of space, and the perpendicular time axis X4, then if the locus 

 of an\' particle is inclined at the non-Euclidean angle </> to the chosen 

 time axis, the particle is said to be in motion with the velocity V 

 of which the magnitude is v = tanh 4>- 



For the locus of a particle let 



= /v.- 



x^ — dx^ — dx-^ 



be the arc measured along the (5)-curve, and let r be the radius vector 

 from any origin to a point of the curve. Then the derivative of r by 

 s is the unit tangent w to the curve. We have 



, dxi . , dx-i , , d^i 



w = ki , + ko , + k4 -^ 



as as ds 



If the velocity v is V = ki - + kj , ' 



a.r4 dxi 



dx4 1 



then since -r~ = cosh</) — 



ds ' Vi _ ^2 



we write ^^ 



1 f dxx . . dx'> . .\ V+k4 ..„,. 



w = , -^ ki + k. , + k4 = 7=: • (42) 



Vl _ 2,2 V dx4 dx^ J Vl — 1,2 



31 By a transformation to a new set of axes we may derive at once the general 

 form of Einstein's equation for the addition of velocities. 



