636 



REPORT — 1900. 



and if we assume 



and therefore 



du' dv , dw n 

 ax dy dz 



dF dG dB. r^ , „ 



,- + -- + - = everywhere, 

 dx dy dz 



yre hare 



Hence is deduced 



dy dS n-r A 



d>j dz 



.^ K d^F 

 477 dt"- 



C3) 



(4) 



•where V= /x>is thevelocityof propagationof adlstm-bance. Also ^(Fu + Gi'+ Il^t') 



is the energy per unit of Tolume at x y z (o) 



3. The theory in this form is open to objection. If u', the current at x' y' z', 



JT^ "1 



changes with the time, we have a corresponding change of F given by --,= -_ 



But owing to (4) no physical quantity at .r y s can be immediately affected by the 

 change at x' y' z' . The change can have no effect whatever at x y z till after the 



expiration of the time ;^. If therefore F be any physical quantity, we must have 



dF 



— , = 0, which is inconsistent with our definition. If F be not a physical quantity, 

 Fu cannot denote energy, which is inconsistent with (5). 



4. The fact that Vis very great, and ^- very small, does not meet the difficulty, 



because ^ -^- is not generally small. 



5. It is proposed to substitute for u', the current at x'y' z'at the given instant, 

 t/t' the current which did exist at x' y' z' ''.seconds ago, so that our definition will 



be F( = -^dr. F, is used by way of distinction from F. In this form of F the 



objections above taken cease to have effect. 



As u' and all its derived coefficients according to the time are supposed finite, 



we may write tt'( = it'— ~ - 'i + |- L.' '1, &c., or symbolically, for convenience 

 only, 



V- df 



rd , 

 Vi/1 w 



F, = j*l 



T d f 



(6) 



6. It is shown \)i^t, prima fade, F , Gi, H(, so defined, satisfy the differential 

 equations (1) as well as do the ordinary F G H. So as regards the differential 

 equations the proposed substitution makes no difference in form to the theory. 



7. An objection is considered that F and Ft cannot both satisfy Poisson's 

 equations (1) because if they did we should have v'^F = v'^F,, and this cannot, it is 

 said, be true because 



F = F + 



J Ydt -J V^rff^ a.3J Y'dt^ 



