162 PROFESSOR W. THOMSON ON THE 
diagram) being previously filled up with insulating matter, such as that used in 
the composition of the bars. Let the complex solid cube thus formed be coated 
round its sides with infinitely thin connected sheets of the standard metal, so 
thin that the resistance to the conduction of electricity along them is infinitely 
great, compared to the resistance to conduction experienced by a current tra- 
versing the interior of the cube by the zigzag linear conductors imbedded in it. 
(For instance, we may suppose the resistance of four parallel sides of the cube to 
be as great as, or greater than, the resistance of each one of the zigzag linear con- 
ductors.) Let an infinite number of such cubes be built together, with their struc- 
tural directions preserved parallel, so as to form a solid, which, taken on a large 
scale, shall be homogeneous. A rectangular parallelepiped, ac, of such a solid, 
with its sides parallel to the sides of the elementary cubes, will present exactly 
the thermo-electric phenomena expressed above by the equations (31) and (32) 
provided the thermo-electric powers ,, @,, @,’,@,", @,, Wy, Wy", W,”, and 
@,, Wy, D,;", @;”, of the metals used in the three systems, fulfil the following 
conditions :— 
1(@,+ @, + @,’ + w,”) = 8, 
+ (@, re @,’) AL e, t (@, <6 i w,’”) aed 6”, 
1(@, + @, + @,’ +@,")=, a . 
1@,—m)=$ He ow) =9" a 
4(@,+ @, + 0,'+ @;") =, 
4(@@, —@,) =, + (@;" — @,”) = V. 
160. To prove this, let us first consider the condition of a bar of any of the 
three systems, taken alone, and put in the same thermal circumstances as those in 
which each bar of the same system exists in the compound mass. If, for instance, 
we take a bar of the first system, we must suppose the temperature to vary at the 
rate w per unit of space along its length; at the rate v across it, perpendicularly to 
two of its sides; and at the rate w across it, perpendicular to its other two sides. 
If 7 be its length, and ¢ the breadth of each side, its ends will differ in temperature 
by wl; corresponding points in one pair of its sides by ve, and corresponding 
points in the other pair of sides, by we. Now, it is easily proved that the longi- 
tudinal electro-motive force (that is, according to the definition, the electro-motive 
force between conductors of the standard metal) would, with no difference of 
temperatures between its sides, and the actual difference w~/ between its ends, be 
equal to 4 (aw, + w,) w/, if only the first of the zigzag conductors existed imbedded 
in the bar, or equal to $ (w," + w,”) w/, if only the second ; and, since the two have 
equal resistances to conduction, and are connected by a little square disc of the 
standard metal, it follows that the longitudinal electro-motive force of the actual 
bar, with only the longitudinal variation of temperature, is 
4 (@, + BW, + W,’ + W,”) ul. 
