QUANTITATIVE INDUCTION. 123 



culus of differences a correct empirical formula ; for if p 

 be the first term of the series of values, and A^, ?p, 

 A. 3 p, &c., be the first number in each column of dif- 

 ferences, then the with term of the series of values will be 



m i , m-i m-2 > 



p + ra A> + wz- - A"> + m --- A p + &c. 



A closely equivalent but more practicable formula for 

 interpolation by differences, as devised by Lagrange, will 

 be found in Thomson and Tait's ' Elements of Natural 

 Philosophy/ p. 115. 



If no column of differences shows any tendency to 

 become zero throughout, it is an indication that the law 

 is of a more complicated, for instance of an exponential, 

 character, so that it cannot be correctly represented in 

 a formula involving only a few powers of the variable. 

 Dr. J. Hopkinson has lately suggested another method of 

 arithmetical interpolation^, which is intended to avoid 

 much that is arbitrary in the graphical method. His 

 process will yield the same results in all hands, but he 

 remarks that it has no theoretical basis to rest on. 



So far as we can infer the results likely to be obtained 

 by variations beyond the limits of experiment, we must 

 proceed upon the same principles. If possible we must 

 detect the exact laws in action, and then trust to them as 

 a guide when we have no experience. If not, an empirical 

 formula of exactly the same character as those employed 

 in interpolation is our only resource. But the reader must 

 carefully observe that to extend our inference far beyond 

 the limits of experience is exceedingly unsafe. Our know- 

 ledge is at the best only approximate, and takes no account 

 of very small tendencies. Now it may, and in fact usually 

 will, happen, that tendencies small within our limits of 



i ' On the Calculation of Empirical Formulae.' ' The Messenger of 

 Mathematics,' Xew Series, Xo. 17, 1872. 



