180 



Dr. A. V. Hill. The Energy 



resistance is unknown, and as the distribution of the return currents is 

 a matter more or less of speculation, it seems advisable to consider only the 

 effect of the currents in the external circuit, where the E.M.F. and the 

 resistance can be accurately observed. The energy of the currents, as 

 calculated below, is less, therefore, than the true amount by the unknown 

 quantity involved in the return currents inside the fibre. 



The most natural assumption as to these return currents is that they are 

 similar to the external currents observed, the E.M.F. being located somehow 

 in the walls of the fibre ; in that case the total amount of energy involved in 

 the electric change is double the quantity calculated below. 



Suppose that the wave of negative potential shown in the lower curve of 

 the figure is represented by the equation 



y = fii-^h), 



the function / being given by the observed form of the electric response. 

 Let E be the resistance per unit length of a tissue, as measured by direct 

 experimental means. The potential at a point x being (t—x/a) at a 

 moment t, the potential at a neighbouring point (x + Bx) at the same 

 moment will be f{t — {x + Sx)/a}, so that the current running in the small 

 element of length 8x between them will be 



p _ f {t—(x + Bx)fa] —f (t—x/a) 

 ^ f(t-x/a) 



where /'(s) is the differential coefficient of f{z) with respect to z. Hence the 

 heat produced in this element of tissue in time t, being 'RSxC'^Bt joules, is 

 equal to 



— ^rr — h ^^^t calories. 

 4-18 Ea2 



Integrating this with respect to x and t, in order to determine the total heat 

 H produced in the whole length I of the tissue by the whole wave, we find 



1 1*^ = i = CO 



dt. 



Now it is obvious that if the wave be propagated unchanged the same 

 amount of heat is liberated at each point of the tissue : hence the value of 



f'^dt cannot depend upon x and must be equal to its value at any 



Ji = 



convenient point on the tissue, e.g., at the electrode used in the experiments. 

 Hence, from a curve relating electrical potential to time at one spot (as 



