174 Mr Hargreaves, Examples illustrating : 
defining a correspondence between the components of relative” 
velocity as expressed in the two schemes. 
If we transform in the reverse direction using 
Zz 
Z=y(2—ut), f=y(t yh) 
I pda dy'dz' — p'u,dt'dy'dz' — p'v,'dt'dz'da’' — p'w, dt'da'dy’ 
becomes 
| yp. € + a ) dxdydz— p'u, dtdydz — p'v, dtdzdx 
— yp (w+ w,’)dtdady, 
and so 
p= YP (1 + ye) > Plr=P Ur, PUr—pP Us, 
p(w + w,) = yp’ (w+ w,)...10a), 
and therefore 
Uy Up W + W, 
Ur ao ww ? Uy = ww ’ W + Wy ag ww 
Y @ VP? ) Y (1 1 P?2 ) 1 V2 
or W,= mee (10 b). 
WW, 
We (1 az ) 
The equations (10) are an inversion of (9) through the formula 
, 2 
(1 | (1 = TR | = 
V2 
The geometrical relations of (6) when we form the differentials 
and write 
dz=(w+w,)dt, dx=u,dt,...,dz’ =w, dt’ 
also imply a correspondence between the two expressions of 
relative velocity; which will be found to agree with (9b) or (10b) 
according as we pass from plain to dashed letters or wice versd. 
The agreement is of course a necessary part of the consistency of 
the scheme of transformation. 
Example II. An Electromagnetic Characteristic or Gene- 
rating function. 
The electric and magnetic vectors are given in (1b) which is 
derivable from (7) by the reticular operation. When a single 
source is In question, (7) can be expressed as | rAdu+dd, and d 
