U Prof. F. Slate 



on a 



It is a striking feature about the forms referring to the 

 frame (U) that y(ii) recurs as a sort of " weighting factor " 

 which also designates the particular frame that is associated 

 with (F). Comparing the last forms in equations (21, 26), 

 we may say that y(u) disturbs the symmetry or completes it; 

 we may seek symmetry in the first member or in the second; 

 and one additional step reaches either end. Consideration 

 of the work-equations helps to analyse these connexions 

 further. Direct integration of either extreme member in 

 equation (18 J, referring to equations (5, 21), gives 



\ T (v -it)dt= \ T a 'v e '<it = c*-\(y(v e ')-l). . (31) 



Ju Jo 7{ U ) 



From equation (22) the work obtained is, remembering 

 equations (24, 26), 



°To'(v '+u)dt=\ C T a v c dt = chn (y(v c )-ry(u)). . (32) 



vQ Ju 



Complete the set ; adding in reliance upon equation (26) 



T o 'v 'dt = c*m y(u)( 7 (v o ')-l). . . . (33) 





Equations (6. 32) express work done in (F) ; equations (31, 33) 

 work done in (U). The scheme of substituted velocities and 

 new limits is symmetrical in all four. But transition from 

 (F) to (U) divides by y(tt) in one case and multiplies by it 

 in the other. The deciding fact is patent : Equation (6) 

 presents the original (or physical) quantity as belonging to 

 (F), and makes allowance for its "appearance v as judged 

 in (U); equation (33) takes its start in the physics of (U) 

 and calculates what appears in (F). Substituting for equa- 

 tion (26) the form 



T "=(i-^ 2 At„ (34) 



which would embody the reversed reduction with the same 

 (T a ), gives immediate verification. Finally it is plain that 

 the various reduction factors, in combination with original 

 totals and with fractions of them, constitute a mode of 

 projection *. 



The above group of relations brings out the idea that they 

 approach a treatment of motion from rest in (U) as though 



* This furnishes a strong hint about their affiliations with imaginary 

 angles, and the natural adaptation of hyperbolic trigonometry to relativity. 



