WILSON AND LEWIS. 



RELATIVITY. 



411 



value of the area is therefore one-half the nuinerieal \'ahie of dr, that 

 is, one-half the infinitesimal interval of are. From our definition of 

 unit angle (§ 12), it is evident that an angle is eciual to the arc sub- 

 tended upon a unit pseudo-circle centered at the vertex of the angle. 

 This might, in fact, have been made the definition of the measure of 

 angle. It is evident from these considerations that a rotation turns 

 all non-singular lines through the same angle. 



Angles may he classified according to the classes of their sides. If the 

 two sides are (7)-lines, the angle will be designated as of class (77); 

 if they are (5)-lines, the angle is of class (55). Consideration of angles 

 (75), which have one side a (7)-line and the 

 other a (5)-line, and which cannot be gener- 

 ated by rotation, need not detain us here. (See 

 Appendix.) 



If any line (Figure 13) through the center 

 be taken from which to measure angle, posi- 

 tion upon the unit pseudo-circle may be 

 expressed parametricaliy in terms of the 

 angle as follows. Let the given line be a 

 line of class (7) (the pseudo-circle then being 

 of class (5)), and construct the perpendicular 

 line of class (5). These two lines may be 



taken respectively as axes of .ri and .1*4 with the unit vectors ki and 

 k4 along them. The equation of the unit pseudo-circle is then 



Figure 13. 



r-r = (.riki + .r4k4)-(.riki -|- .r4k4) = .ti 



X,' = 1. 



The differential of angle or arc is in this case 



dd = ds= V_f/r.f/r= V (kid.vi-^-k^dxi) . {kidxi+kidxi) = ^dx^^-dx/ 

 Whence, by differentiation of .Ti" — .T4- — 1, 



J^ J ^'^ J Vl + .T42 J V.ri2_i' 

 and Xi = cosh 6, x^ = sinh 6, 



where 6 is the angle between the Xi-axis and the radius ^•ector, and 

 therefore of the class (77). If the given line had been of class (5) 

 (the pseudo-circle of class (7)), and if the angle had been of class 

 (55) measured from the .r4-axis to the radius vector, the results 

 would have been 



