116 THE PRINCIPLES OF SCIENCE. 



now the actual function we can try whether it gives with 

 sufficient accuracy the remainder of our experimental 

 results. If not, we must either make a new selection of 

 results to give a new set of equations, and thus obtain 

 a new set of values for the constants, or we must acknow 

 ledge that our 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 approxi 

 mately accurate and may proceed by the Method of Least 

 Squares (vol. i. p. 458) 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 pro 

 portion. 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 

 in which a force acts influences its action, trigonometrical 

 functions must of course enter. Any force or 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 sometimes arrive deduc 

 tively and analogically at the general nature of the mathe 

 matical law required. 



The Graphical Method. 



In endeavouring to discover the mathematical law 

 obeyed by experimental results it is often necessary, 



