ENERGETICS 35 



the logarithm is due to the fact that the differential coefficient of the logarithm of x to base 



e is - , i. e. , 

 x 



d log a; 1 j , i dx 



' - = - and a log x = , 

 dx x x 



and therefore, conversely, the integral of is log, x, and that of is log, v, or, when 

 integrated between the limits of v z and v lt is 



log, i> 2 - lg v i or log,- 2 . 

 "i 



Details of the way in which, by a simple application of the binomial theorem, the 

 differential coefficient of a logarithm is obtained may be found in the books mentioned 

 (Nernst-Schonflies, pp. 82-85, or Mellor, p. 51). We may note that the quantity e, chosen 

 as the base of natural logarithms, is one of the most important in mathematics. As the 

 sum of the infinite series : 



its value can be obtained to as many places of decimals as required. 



The differential coefficient of log x is the ratio of the amount by which log x increases when 

 x increases by an infinitesimal fraction of its value, say it becomes x + h, to the increase 



h itself. That is, we want the value of ? ^ _LL_8j? when h becomes so small as to 



ft 



approximate to zero. When the expression is expanded by the binomial theorem, we 



finally arrive at another expression in which - appears multiplied by log e, i.e., 



x 



_ . 



dx x 



There are many reasons for taking e, as the base of a system of logarithms in dealing with 

 mathematical formulae, and when this is done, log e to the base e becomes unity. Our equation 

 is then simply : 



d log, x _ 1 

 dx x 



This digression into the region of pure mathematics is merely for the purpose of explaining 

 the appearance of a logarithm in the expression for the work done in compressing a gas. 



Attention may be called to the frequent occurrence of processes whose 

 magnitude at any given moment depends on how much of the process has been 

 already completed, or, when an equilibrium is being approached, on the nearness 

 to the end the process is. In the case before us, the work needed to cause the 

 same actual diminution in volume of a gas increases the more the gas has been 

 already compressed. Perhaps the simplest case is that of the absorption of light 

 by a coloured liquid. Suppose that we allow 100 units of light of a certain wave 

 length to enter the liquid and that, after it has passed through one centimetre, 

 it has lost 0-1 of its original intensity and has become 90 units, or 100 x -9 ; after 

 the next centimetre, this 90 units will have lost O'l of 90 and become 81, or 

 100 x 0'9 x 0'9, i.e., 100 x 0'9 2 , and so on. Hence, after passing n centimetres, its 

 value will be 100 x 0'9". Note that three layers do not absorb three times as 

 much as one layer, but less, so that the value of the light transmitted is not 70 

 but 72'9. The application of this law (that of Lambert) will be found in the 

 spectro-photometer, which has played so large a part in the investigation of 

 haemoglobin. 



In such kinds of processes, then, we have to deal, not with simple linear 

 relationships, but with exponential or logarithmic ones. 



Other aspects of the question may be found in Newton's " Law of Cooling," one 

 of the earliest cases 'to which the infinitesimal calculus was applied. Here the 

 rate of cooling depends on the difference of temperature between the hot body and 

 its surroundings, so that it steadily diminishes as the temperature difference 

 becomes less ; in theory, equality of temperature is attained only after an infinite 

 time, asymptotically, as it is called, after the straight lines to which such a curve 

 as the hyperbola continually approaches without actually reaching ; this is due to 

 the fact that each succeeding portion of the curve moves towards the asymptote 

 a little less than the previous portion did. In such cases as loss of heat, or the rate 



