714 
MR. A. McAULAr ON THE MATHEMATICAL 
the work done on the boundary by the viscosity stress, (2) the work done on the 
boundary by the frictional electric forces, (3) a flux — 0 & Vx of energy at every point 
of space. This may be put in, perhaps, the more familiar form:—the increment of 
heat in the region consists of three parts, (1) the work done against the frictional 
forces throughout the region, (2) the work done by the frictional forces (viscosity and 
electric) on the boundary, and (3) the surface integral taken inwards at the boundary 
of a flux — 0qVx. Stated in this way we see that equation (38) is equivalent to 
saying that the conduction of heat is due to a flux of heat — 0qVx at every point of 
space, and that the frictional forces are sources of heat. # These are statements (2) 
and (3) of § 37 (except that here we have — 0qVx, and there we have a more definite 
form for flux of heat due to conduction.) 
To prove eq. (38), note that the expression on the left 
= jjj’{(0/ + SEC + Sec) cfe - SF p'cW] + {^{(0/, + SE,C + Se,c )ds - SF S p'ds'} 
= J|[ {O + f - 6>SV 0 Vx + SC {cVx + a + VY) + Sc c Vafj ch - S P '<P\V 1 cW ] 
+ j] {Sc& {0 @ Vx + VaH/47r - YC) + SpV dt}. + 
Now put Jjh = jj — JJ S [equation (37)], and transform the integral |J by means of 
equation (4) § 5 above into a volume integral. In doing this note that by reversing 
the process of § 39 we get 
- fff S fV,V', cW + ff Sp'4>' dt = fff S'Ffcdatfds. 
* It is possible at this stage that two objections maybe taken to this reasoning. First it may be said 
that there ought to be no terms in the surface integral leading to the result that the frictional electric 
forces do work on the boundary. That this is not a sound objection will come out more clearly below, 
when the effect of TC will be found to in no way alter the ordinary views of the transference of electric 
energy through the field, and the effect of VaH will be only to modify them in a way which would 
naturally be anticipated from the new hypothesis that Hhas some influence on the frictional forces of the 
field. Secondly, it may be said that besides the three terms mentioned in the text as contributing to rate 
of increase of heat, there should be a fourth due to such causes as the Thomson and Peltier effects. 
This statement is, however, undoubtedly wrong, as will appear more clearly when we come to the con¬ 
sideration of these effects. The explanation is that these effects are explained by terms in /. Hence, 
in equation (38) they are included on the left. If this is not considered convincing, let me call attention 
to equation (25), § 49 below, which asserts that the rate of increase of intrinsic energy (including that of 
the Thomson effect, &c.),in any space = rate of doing work throughout the region of the external forces 
which are not due to friction + the rate of heat supply from external sources situated in the region + such 
a surface integral as now is under consideration (i.e., confined to the true boundary). 
f We have here for the sake of the next transformation added the term JJ,;SdSaH/4w, since from 
the equations [VIL'H] a + { = 0, [VUi'a] a + j = 0, it follows that [SUi'aH] a+ j = 0. 
