WILSON AND LEWIS. — RELATIVITY. 



401 



IG". In rotation an area becomes an equal area.^^ 

 10. We are now prepared to discuss in some detail the general 

 characteristics of our rotation. 

 Consider (Figure 8) a series of rota- 

 tions about 0, whereby the point P 

 assumes the positions P', P", .... 

 Let the parallelograms on OP, OP', 

 OP",.... as diagonals and with 

 sides along the fixed lines be con- 

 structed. Then by 16° the areas 

 of these parallelograms are equal, 

 and in terms of the intervals on 

 the fixed lines 



OAXOB = OA' X OB' 



= OA" X OB". Figure 8. 



The point P thus traces a curve which in ordinary geometry would be 



with the condition 



0(a)0(t) = </,(«?>) 



necessitated by 13°. This is a functional equation of which the onlj' (con- 

 tinuous) sokitioa is 0(«) = a''. Hence rotation must be of the form 



u' = an, v' — a^v. 



The unit parallelogram on the axes of u and v is hereby transformed into a 

 parallelogram on these same axes with intervals a and «'' along u and r. By 

 VI the area of the new parallelogram is therefore a*""^!. If this is to be unity, 

 r = — 1. The ti'ansformation equations for rotation are therefore 



u = au, 



v/a, 



where a is necessarilj^ positive because points do not change from one side of 

 the axes to another. 



The intrinsic significance of these equations should not be overlooked. A 

 rotation may be represented as a multiplication of all intervals along one of 

 the fixed lines by a constant factor and a division of all intervals along the 

 other fixed line by the same factor. Or, increasing the unit intei-val along 

 one fixed line and decreasing it in the same ratio along the other is equivalent 

 to a rotation. (This process effected along any other axes than the fixed lines 

 would leave the area unchanged, but would not be a rotation). As the unit 

 interval along one fixed line cannot be compared either by translation or by 

 rotation with the unit along the other, and as one of these units is arbitrary, 

 we have additional evidence that there is no natural zero of angle. 



13 Such a postulate is unnecessary in Euclidean geometry owing to the 



Ceriodic nature of the Euclidean rotation. Postulate 16° could be replaced 

 y one involving only the notion of symmetry between rotations in opposite 

 directions. . 



