472 THE PRINCIPLES OF SCIENCE. [CHAP 



of which it is a function ; but the subject is 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 quantities, the 

 values of which depend upon the successive powers of the 

 variable quantity. If y be a function of x then we may 

 say that 



y = A + Ex + Ca? + T)x* + Ex* 



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 coefficients A, B, C, &c., may be 

 zero or negative ; but whatever they be they are fixed. 

 The quantity x on the other hand may be made what we 

 like, being variable. Suppose, in the first place, that x and 

 V are both lengths. Let us assume that T^,^^ 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* = yp-.^, and 

 if C be less than unity, the term C y? will be inappreci- 

 able, being less than we can measure. Unless any of the 

 quantities D, E, &c., should happen to be very great, it 

 is evident that all the succeeding terms will also be in- 

 appreciable, because the powers of x become rapidly 

 smaller in geometrical ratio. Thus when x is made small 

 enough the quantity y seems to obey the equation 



y = A + Ba;. 



If x should be still less, if it should become as small, 

 for instance, as T.^nnunny of an inch, and B should not 

 be very great, then y would appear to be the fixed 

 quantity A, and would not seem to vary with x at all. 

 On the other hand, were x to grow greater, say equal to 

 T 1 inch, and C not be very small, the term C x 2 would 

 become appreciable, and the law would now be more 

 tH)mplicated. 



We can invert the mode of viewing this question, and 

 suppose that while the quantity y undergoes variations 

 depending on many powers of x, our power of detect- 

 ing the changes of value is more or less acute. While 

 our powers of observation remain very rude we may be 

 unable to detect any change in the quantity at all, that 

 is to say, B x may always be too small to come within 



