36 PRINCIPLES OF GENERAL PHYSIOLOGY 



of a chemical reaction, one may look upon the driving force as becoming less and 

 less. The velocity of chemical reactions will be dealt with in Chapter X. 



The increase of money lent at compound interest follows a similar law ; for 

 this reason, the general law in which a function varies at a rate proportional to 

 itself, an exponential function, was called by Kelvin, " the compound interest law." 

 On this point, pp. 56-64 of Mellor's book (1909) will repay perusal. 



The name "function" has just been used without explanation and it may be useful here to 

 refer to some terms often met with in descriptions of phenomena from the mathematical 

 standpoint. The volume of a given mass of a particular gas is different, according to the 

 pressure to which it is exposed ; but it is always the same, other conditions being unchanged, 

 when the same pn-ssim- is applied. The volume of a gas is said to be a "function" of the 

 pi insure. A function, then, is a quantity which changes according to some definite law when 

 another quantity, of which it is said to be a function, changes. This is expressed in symbols : 

 r =f (p) t in the case of Boyle's law ; or, generally, y f (x), 



which means that, to every value of .r, there is a determinate value of y. x and y are CM! led 

 " ftriable*." Any quantity which remains unchanged during a particular mathematical 

 operation is called a "constant." When the value of one variable depends on that of the 

 other, as in the example given, the first is called the " dependent mr table," the second, the 

 " independent mriable." Which of the two is chosen as the independent variable is a matter 

 of convenience. In cases involving time as one variable, it is usually taken as the independent 

 variable, since its changes are the most uniform. When the values of y are simple arith- 

 metical multiples or fractions of those of x, so that the graph is a straight line, y is said to 

 be a " linear function" of x. When y varies as a power of x, it is said to be an "exponential 

 Junction," and so on. 



Speaking generally, the object of scientific research is to find out how one 

 thing depends on another, in fact, what " function " the one is of the other. 



To return to our main theme, we find that the work done in compressing a gas 

 isothermally from the volume v. 2 to t\ is : 



RT log , a. 



Further, since, by Boyle's law, pressures are inversely as volumes, we have : 



V '2 = Pl 

 V l P* 



and writing c t and c., for osmotic or molar concentrations of any two solutions ,-is 

 being proportional to p l and/).,, we have a formula which gives the work done in 

 concentrating a solution from the value c } to c.,, as in the case of the kidney when 

 secreting urine of an osmotic pressure different from that of the blood, as will be 

 seen later. 



Or, again, if c t and c. 2 represent the concentration of an ion in two solutions in 

 contact with electrodes of the same substance, we have the electromotive force of 

 the battery, due regard being taken as to the units in which R is expressed. We 

 shall see later how this fact is made use of to determine the real acidity of a 

 solution, and how it is related to the electrical -changes taking place in acti\<- 

 organs. 



For further details as to this important law, the reader is referred to the work of Nernst 

 (1911, pp. 51 and ~v2), and the essay of Benjamin Moore (1906, pp. 21, etc ). 



The practical bearing of the logarithmic form of the equation may be seen in 

 the case of a concentration battery in hydrogen ions, as used for determining the 

 true acidity or alkalinity of a complex fluid like blood, for example. If the 

 relative concentration of the hydrogen ions in the two solutions compared is, in 

 one case, as 2 to 1, and, in another case, as 10 to 1, the electromotive force in the 

 second case will not be five times that in the first, but in the ratio of log 10 to log i', 

 that is, as 1 to 0-301, or about 3'3 times. Thus the actual E.M.F. of a battery, 

 composed of a standard calomel electrode combined with a hydrogen electrode in 

 one-tenth noimal hydrochloric acid, is 0-394 volt, while if one hundredth normal acid 

 is taken, the value is 0-452 volt. It will be noted that the logarithmic form of the 

 equation lessens the delicacy of the method. 



