practical Instrument of Physical Research. 489 



Then shall 



n P G - B / aa \ a /ao-\ a/aon 



{ b ~*-.(»s)~*(*sJ"n*sJl 



_rrS r _ 5 / Bffiru 9 / 3(nru a / d(nr) \ 



We may omit the first term on each side since it is easy to 



see that 'dGfdv='dT/'d(j). 



Expressed in quaternion language this may be put thus : — 

 Let v be any scalar function of p a vector, and G any 



scalar function of p, v, and yvi Let p be a function of p'. 



When expressed in terms of p', let v be denoted by v f ; and 



when expressed in terms of p\ v', and yV [y' standing 



towards p' as y towards /o], let G be denoted by G'. Lastly, 



let 



_ Sdpjdpbdpc 



where <^/o tt , dp b , dp c are three arbitrary increments of p and 

 rfp a ', dpi,', dp c \ the consequent increments in p'. Then shall 



m- 1 S V VvG = Sv'Vv'(^- 1 G'). 



where v, v' stand for yu, yV, and y„, V«' stand towards u, i/ 

 as y, V' towards p, p f . 

 We have 



say. Also 



Sefyy y = Sc/p'y V = Sd/o^'y V? 

 where as usual %' stands for the conjugate of ^. Hence 



or generally 



Again, 



Hence 



v=xV, 



Vi = %Vi- 



y v 'G'=xy»G. 



Hence 



SA'xx-V.. G/ + S Xl '- I xViV„'G' 

 = S V 'Vv'G' + Sxi'-'xVi'Vv'G'. 



