Brown. — Phenomena of Variation. 529 



formula of n terms to approximate to a given curve, we obtain 

 exactly the same choice of approximations whatever the scale 

 of the variable may be or wherever its origin may be. We 

 may therefore elect to make the experimental range unity in 

 a new variable, and make the beginning of the range the 

 origin. Thus, if the experimental variable X ranges from p to 



p + q, we take as a temporary variable x = : — — . It may 



be noted that in the case of a continuous function the corre- 

 sponding Taylor's series becomes — 



* = * + [*(£),]••+[£(£■),]■*+* 



29. If this expression were very convergent our analysis 

 would lead us to the values of the bracketed quantities. 

 Since, however, curves in general cannot be said to be repre- 

 sentable by continuous functions, and particularly convergent 

 ones, we cannot expect to make this conception of curves 

 being built up of the effects of initial rates of change our 

 basis of operations. We may with great convenience utilise 

 the average rates of change for the whole range. Something 

 of the sort is done in using an interpolation-table method, 

 such as that given in " Thomson and Tait " (1890, i., 

 p. 454). 



30. The graphic process consists in taking for the first 

 term the initial value of Y as given by the graph ; for the 

 second the average rate of change for the whole experi- 

 mental range ; for the third the average curvature for the 

 whole scale expressed in terms of a parabola ; for the fourth 

 the difference in curvature of the first and second halves of 

 the range expressed as a cubic standard formula ; and so 

 on. Up to the fourth term at least there is no difficulty 

 whatever in keeping the effect of each of these three opera- 

 tions in the mind, and in forming one's conclusions whether 

 a certain formula is as good as can be possibly got. The 

 standard formulae which the writer has used for this pur- 

 pose are for the parabola (x — x*), and for the cubic 

 (x — Sx 2 + 2x 3 ). This is as far as we shall go for the pre- 

 sent, but a table is given of some standard functions which 

 might be used up to x 8 (or formulae of nine terms) if one were 

 clever enough to perform the work with all of them at 

 once, or under special circumstances. These formulae will 

 be reverted to again. 



31. The practical work is now very simple. We draw the 

 graph of the experiments in terms of the temporary variable 

 (which we should have mentioned is better not arranged to 

 have its scale exactly equal to that of the experimental 

 variable, but as nearly as is practicable, keeping p and q 



34 



