Brown. — Phenomena of Variation. 523 



9. It is intended to confine our attention chiefly to the 

 example, of the last or last but one class, which is called 

 the " power-expansion " or " Taylor's series formula."* It is, 

 however, intended that the objections to the use of an arbi- 

 trarily systematic mode of computation should apply to all 

 classes with respect to systematic instrumental error, and to 

 all but the first with respect to the effect of systematic re- 

 sidual causes which are not allowed for in the function, or 

 of any mistake or incompleteness in the inferring of the 

 function. 



10. Besides the curve and the datum -point graphs, we 

 need to mention an intermediate class— namely, that of 

 experiments which are arranged to give data for many points 

 of X without any attempt to obtain repeated measures at any 

 one point. 



11. We might venture to define the characteristic virtues 

 of the two main types of graph by saying that the curve 

 yields a clear idea of the continuity of a phenomenon with- 

 out allowing any great accuracy to be obtained in the measures 

 of Y, while the datum point allows great accuracy to be 

 attained in the measures of Y, and also permits definiteness to 

 be attained in probable error, but leaves the interpolation to 

 be judged. It maybe put also thus : the curve gives a notion 

 of dY/dX, the datum of Y. It is sometimes possible to form 

 a graph of both kinds of measures— to measure accurately 

 datum points and also to get the slope of the curve near 

 these points. This procedure is analogous to that of con- 

 structing mathematical tables where datum points are often 

 computed exactly and intermediate points found by Taylor's 

 theorem. By such means very full information would be 

 given of the actual phenomena. 



12. A graph of the above-mentioned intermediate class, 

 while it combines the virtues of both main forms, combines 

 also their defects. In contemplating such a graph one 

 would feel more content if a likely value for probable 

 error at a few points of X were provided by the experi- 

 menter. The difficulty with this form of measurement is 

 the very large number of ineasures necessary — theoretically 

 a double infinity. 



13. It is perhaps desirable to point out that in datum 

 measurements we usually cannot get either X or Y exactly 

 the same for each measure, accordingly we have to inter- 

 polate the values of Y to one common or mean value of 



* It is to be observed that, in the case of functions the Taylor's series 

 expansion of which are suffi< iently convergent when applied to the 

 exp* rimpntai range, the result of the application of such a formula is 

 practically identical with the result of the application of an unexpanded 

 function of any class. 



