236-238] 



Rate of Dissipation of Energy 



199 



the </> coordinates which occurs throughout the whole gas, is the same as if 

 the increase, in the case of each single molecule such as we have just been 

 considering, were 



.............................. (484). 



238. This quantity must be evaluated from equation (480). The form 

 of the function U is, however, almost entirely unknown. We are therefore 

 compelled to examine the general properties of the function X + iY defined 

 by equation (480), making as few assumptions about the function U as 

 we can. 



A good deal can be learned from a graphical treatment. In fig. 15, let 

 the thick line be the graph of the function U, regarded as a function of the 

 time, then the dotted curve will be the graph of the function U cospt, and 



FIG. 15. 



the value of X, given by equation (479), is equal to the area of this curve 

 estimated algebraically, divided by ap. The area of this curve can, however, 

 be regarded as made up of the areas A, B ... P, Q, R, S, T ... , and of these 

 the alternate areas, say B ... Q, 8 ... must be estimated negatively. Thus if 

 we denote the numerical magnitudes of the areas by the corresponding letters, 

 we have 



It is at once clear that each term in this sum is very small ; Q is nearly 

 the arithmetic mean between P and R, and so on, and the two isolated 

 terms- at the beginning and end are clearly seen to be small, on reference to 

 the figure. 



It is therefore obvious that X is very small : to see clearly how small, we 

 must have recourse to exact analysis. 



