220 OUTLINES OF NATURAL PHILOSOPHY. 



the latter case, we must assume the simplest func- 

 tion that can represent the observations, and this 

 naturally consists of a series of terms, proceeding 

 according to the powers of one of the variable 

 quantities, with co-efficients which areconstant, but 

 unknown quantities, to be found from the observa- 

 tions. This is called the method of Interpolation, 

 because it inserts a term in the midst of a number 

 of others. 



232. If oc and y are two variable quantities, of 

 which several values have been determined 

 from observation ; if y be assumed equal to a 

 series of the powers of x, beginning from 0, and 

 going on to as many terms as there are obser- 

 vations, viz. y = A + B x + C a? -f D #*, &c. ; 

 then, if for y and x, be put their corresponding 

 values, as determined by observation, as many 

 equations will arise as there are unknown co-ef- 

 ficients, A, B, C and D to be found, from which 

 they will become known. 



The most useful interpolations are, when the time is 

 one of the unknown quantities, and when the in- 

 tervals between the observations are equal, as is 

 supposed in what follows : 



rr 



233. Let a, a, a', a", a", be any number of 

 quantities determined from observations made 



; Dflfi c ; 15 , 



dl 



