114 REPORT—1863. 
time; otherwise we should have Q=yCt, where y would be a second use- 
less and absurd coefficient. From equations (1) and (2) it follows that only 
two of the electrical units could be arbitrarily chosen, even if the natural 
relation between electrical and mechanical measurements were disregarded. 
Thus if the electromotive force of a Daniell’s cell were taken as the unit of 
electromotive force, and the resistance of a metre of mercury of one milli- 
metre section at 0° were taken as the unit of resistance, it would follow from 
equations (1) and (2) that the unit of current must be that which would be 
produced by the Daniell’s cell in a cireuit of the above resistance, and the 
unit of quantity would be the quantity conveyed by that current in a second 
of time. Such a system would be coherent; and if all mechanical, chemical, 
and thermal effects produced by electricity could be neglected, such a system 
might perbaps be called absolute. But all our knowledge of electricity is 
derived from the mechanical, chemical, and thermal effects which it pro- 
duces, and these effects cannot be ignored in a true absolute system. Che- 
mical and thermal effects are, however, now all measured by reference to the 
mechanical unit of work; and therefore, in forming a coherent electrical 
system, the chemical and thermal effects may be neglected, and it is only 
necessary to attend to the connexion between electrical magnitudes and the 
mechanical units. What, then, are the mechanical effects observed in con- 
nexion with electricity? First, it has been proved that whenever a current 
flows through any circuit it performs work, or produces heat or chemical 
action equivalent to work. This work or its equivalent was experimentally 
proved by Dr. Joule to be directly proportional to the square of the current, 
to the time during which it acts, and to the resistance of the circuit ; and it 
depends on these magnitudes only. In mathematical language this is ex- 
pressed by the equation W=C" Re, 2 2)... fs ns ode Gl slae de « (3), 
where W=the work equivalent to all the effects produced in the circuit, and 
the other letters retain their previous signification. This is the third funda- 
mental equation affecting the four electrical quantities, and represents the 
most important connexion between them and the mechanical units. From 
equation (3) it follows (unless another absurd coefficient be introduced) that 
the unit current flowing for a unit of time through a circuit of unit re- 
sistance will perform a unit of work or its equivalent. If every relation 
existing between electrical and mechanical measurements were expressed by 
the three fundamental equations now given, they would still leave the series 
of units undefined, and one unit might be arbitrarily chosen from Which the 
three other units would be deduced by the three equations; but these three 
equations by no means exhaust the natural relations between mechanical and 
electrical measurements. For instance, it is observed that two equal and 
similar quantities of electricity collected in two points repel one another with 
a force (F) directly proportional to the quantity Q, and inversely to the 
square of the distance (¢) between the points. This gives the equation 
F Sa a a 2 rains) 
from which it would follow that the unit quantity should be that which at 
a unit distance repels a similar and equal quantity with unit force. The 
four equations now given are sufficient to measure all electrical phenomena 
by reference to time, mass, and space only, or, in other words, to determine 
the four electrical units by reference to mechanical units. Equation (4) at 
once determines the unit of quantity, which, by equation (2), determines the 
unit current ; the unit of resistance is then determined by equation (3), and the 
unit electromotive force by equation (1). Here,then, is one absolute or coherent 
x 
