WILSON AND LEWIS. — RELATIVITY. 



463 



duces no change in w, and in like manner a displacement dr in the 

 plane perpendicular to that of w and 1 does not affect W. Hence we 

 mav write 



/v 1 1 1 



(73) 



vr+dw 



(^r^dl 



\ (8) 



Figure 23. 



Figure 24. 



To compute O^ = — O (I'W), we may write 



0(l-w)= (C>l)-w+ «>w).l. 



Here <0>w is already known. To find <C>1 observe that dl = dT'<C}l 

 is equal to r/r when r/r is along 1 (Figure 24). Further if r/r is elsewhere 

 in the hypercone, for instance, in the plane perpendicular to that of 1 

 and w then also c/l = dr. But when f/r = wds is along w the differ- 

 ential dl vanishes. Hence we may write 



Ol= I-/ w= 1+ hw. 

 ^ I'W it 



where I is the idemfactor. Thus we have 



0(l-w) = (l + ^^lwVw-^lc.l, 



(74) 



1 + 1-c 



R 



or, performing the multiplication hy w, 



OR= -O(i-w) = -w + 



From this it follows at once that 



Op= -^,(0^)w + ]^Ow 



1 /, , 1 + 1-c, 



1. 



(75) 



(76) 



