Q UA NTITA Tl VE IND UCTION. 1 1 5 



be a closed one, or whether it has infinite branches ; 

 whether such branches are asymptotic, that is, approach 

 indefinitely towards straight lines ; whether it is loga- 

 rithmic in character, oj; trigonometric. This indeed we 

 can only do if we remember the results of previous in- 

 vestigations. The process is still inversely deductive, and/ 

 consists in noting what laws gave particular curves, and 

 then inferring inversely that such curves belong to such 

 laws. If we can in this way discover the class of func- 

 tions to which the required law belongs, our chances of 

 complete success are much increased, because our hap- 

 hazard trials are now reduced within a narrower sphere. 

 But, unless we have almost the whole curve before us, the 

 identification of its character must be a matter of great 

 uncertainty ; and if, as in most physical investigations, 

 we have a mere fragment of the curve, the assistance 

 given would be quite illusory. Curves of almost any 

 character can be made to approximate to each other for a 

 limited extent, so that it is only by a kind of divination 

 that we can fall upon the actual function, unless we have 

 theoretical knowledge of the kind of function applicable 

 to the case. 



When we have once obtained what we believe to be the 

 correct form of function, the remainder of the work is 

 mere mathematical computation to be performed infallibly 

 according to fixed rules*, which include those employed 

 in the determination of empirical formulas (vol. ii. p. no). 

 The function will involve two or three or more unknown 

 constants, the values of which we need to determine by 

 our experimental results. Selecting some of our results 

 widely apart and nearly equidistant, we must form by 

 means of them, as many equations as there are constant 

 quantities to be determined. The solution of these equa- 

 tions will then give us the constants required, and having 



i See Jamin, * Cours de Physique,' vol. ii. p. 50. 

 I 2 



