92 THE PRINCIPLES OF SCIENCE. 



of any other gas 11 . This law is probably true only by 

 approximation, but it is obvious that it would be true of 

 the perfect gas with infinitely distant particles x . 



Mathematical Principles of Approximation. 



The whole subject of the approximate character of 

 physical science will be rendered more plain if we con 

 sider it from a general mathematical point of view. 

 Throughout quantitative investigations we deal with the 

 relation of one quantity to certain other quantities, of 

 which it is a function ; but the subject is quite sufficiently 

 complicated if we view one quantity as a function of 

 one other. Now. as a general rule, a function can be 

 developed or expressed as the sum of certain other quanti 

 ties, the values of which depend upon the successive 

 powers of the variable quantity. Thus, if y be the one 

 quantity which is regarded as a function of x, then we 

 may say that 



7/ = A + Brr + C;r 2 + Dx 3 + Ex 4 + .... 

 In this equation, A, B, C, D, &c., are fixed quantities, of 

 different values in different cases. The terms may be 

 infinite in number or after a time may cease to have any 

 value. Any of the co-efficients A, B, C, &c., may be 

 zero or negative ; but whatever they may be they are 

 fixed. The quantity x on the other hand may be made 

 what we like, being variable at our will. Suppose, in the 

 first place, that x and y are both measurable lengths. Let 

 us assume that 10 1 UOO part of an inch is the least that we 

 can take note of. Then when x is one hundredth of an 

 inch, we have x 2 = 10 , T 000 , and if C be less than unity, 

 the term C x 1 will be inappreciable, being less than we 



u Joule and Thomson, Philosophical Transactions, 1854, vol. cxliv. 



P- 337- 



x The properties of a perfect gas have been described by Rankine, 



Transactions of the Royal Society of Edinburgh, vol. xxv. p. 561. 



