Brown. — Phenomena of Variation. 



537 



a high degree to a simpler expression with very little error, 

 but only for the range between the limits of the variable 

 (original or transformed) and 1. 



Some Symmetrical Standard Functions. 

 (Abscissa, x, or z = — (1 — 2x).) 



Postscript. — Since writing the above I have had occasion 

 to employ formulae of more than four terms, and certain 

 practical points have come to light. Suppose the graph con- 

 sists of a " smooth curve," or, if it is of the datum-point 

 variety, a smooth curve can be satisfactorily drawn through 

 the data, then the analysis may proceed as follows : The con- 

 stant, linear, and parabolic (standard) terms are obtained as 

 before, and we draw in the base-line corresponding to this 

 formula of three terms. Thus we have reduced the deviations 

 to zero at x = 0, \, and 1. We then scale off the devia- 

 tions from this formula at x = \ and f, and then, using the 

 standard cubic and a quartic the formula of which may be 

 x(l — x)(l — 2x)' 2 , and which is obviously zero at the same 

 three points as the cubic, we compute the amounts of these 

 functions required to reduce the deviations to zero at the 

 quarter points of x. 



