xxii.] QUANTITATIVE INDUCTION. 491 



process by drawing them out in the form of a curve. We 

 can in this way ascertain with some probability whether 

 the curve is likely to return into itself, or whether it has 

 infinite branches ; whether such branches are asymptotic, 

 that is, approach infinitely towards straight lines ; whether 

 it is logarithmic in character, or trigonometric. This 

 indeed we can only do if we remember the results of pre- 

 vious investigations. The process is still inversely deduc- 

 tive, and consists in noting what laws give particular curves, 

 and then inferring inversely that such curves belong to 

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

 functions to which the required law belongs, our chances 

 of success are much increased, because our haphazard 

 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 divina- 

 tion that we 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, 1 which include those employed 

 in the determination of empirical formulae (p. 487). 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 form by means 

 of them as many equations as there are constant quantities 

 to be determined. The solution of these equations will 

 then give us the constants required, and having 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 acknowledge that our 

 1 Jamin, Cours de Physique, vol. ii. p. 50, 



