492 THE PRINCIPLES OF SCIENCE. [chap. 

 ^ . ^ 



form of function has been wrongly chosen. If it appears 

 that the form of function has been correctly ascertained, 

 we may regard the constants as only approximately accurate 

 and may proceed by the Method of Least Squares (p. 393) 

 to determine the most probable values as given by the 

 whole of the experimental results. 



In most cases we shall find ourselves obliged to fall 

 back upon the third mode, that is, anticipation of the 

 form of the law to be expected on the ground of previous 

 knowledge. Theory and analogical reasoning must be our 

 guides. The general nature of the phenomenon will often 

 indicate the kind of law to be looked for. If one form of 

 energy or one kind of substance is being converted into 

 another, we may expect the law of direct simple proportion. 

 In one distinct class of cases the effect already produced 

 influences the amount of the ensuing effect, as for instance 

 in the cooling of a heated body, when the law will be of 

 an exponential form. When the direction of a force in- 

 fluences its action, trigonometrical functions enter. Any 

 influence which spreads freely through tridimensional 

 space will be subject to the law of the inverse square 

 of the distance. From such considerations we may some- 

 times arrive deductively and analogically at the general 

 nature of the mathematical law required. 



The Graphical Method. 



In endeavouring to discover the mathematical law 

 obeyed by experimental results it is often desirable to 

 call in the aid of space-representations. Every equation 

 involving two variable quantities corresponds to some kind 

 of plane curve, and every plane curve may be represented 

 symbolically in an equation containing two unknown 

 quantities. Now in an experimental research we obtain 

 a number of values of the variant corresponding to an 

 equal number of values of the variable ; but all the 

 numbers are affected by more or less error, and the values 

 of the variable will often be irregularly disposed. Even 

 if the numbers were absolutely correct and disposed at 

 regular intervals, there is, as we have seen, no direct mode 

 of discovering the law, but the difficulty of discovery is much 

 increased by the uncertainty and irregularity of the results. 



