Brown. — Phenomena of Variation. 531 



the " Notes on the Graphs," the numerical values of these 

 functions (even better formulae may be obtained for this par- 

 ticular purpose) become very small, within the range, in com- 

 parison with the numerical value of the coefficient of x n (that 

 is, the highest power of x in the standard formula). This 

 means that in the expansion as given in section 32. and when 

 it is converging, we may for values of x from to 1, by throw- 

 ing the series into standard form, eliminate one or more of the 

 higher-power terms, and so obtain an expression which is 

 practically as accurate as the simple Taylor series, and is less 

 in degree. The extent to which this may be expected to go is 

 to be seen in the decreasing numerical or percentage values 

 of the ordinates as the degree becomes large — with the octic 

 it is already 1*3 parts in 10,000. Of course, in doing this 

 we sacrifice all pretence to accuracy outside our defined 

 range. 



35. To sum up, we may emphasize the importance of the 

 idea of the experimental range, as we have seen this leads to 

 a great accession of power in the case of what we have ven- 

 tured (not without precedent, of course, but, still, with some 

 misgiving) to call "Taylor's series formulae." An analogous 

 idea is familiar enough in the " period " of the " Fourier series 

 formulae." Secondly, we venture to think that too much 

 stress cannot be laid on the necessity for the statement of 

 probable error in the individual data. This matter is strongly 

 stated in the extract from Sir G. B. Airy's works given in 

 section 38. Even the warnings of so great an authority as 

 the late Astronomer Boyal seem to have been greatly disre- 

 garded. 



36. Thirdly, however plausible or apparently authoritative 

 the theory of a physical phenomenon of variation may be, 

 the experimental data upon it should be so prepared that the 

 precise support given to the theory by the observations should 

 be made evident, as can often be done by a graph either of 

 the observations themselves or of the deviations from the 

 aforesaid plausible theory, the graph exhibiting probable error 

 in the way mentioned in section 5. 



37. Finally, the writer wishes to disclaim any novelty in 

 the foregoing, with one exception, and to apologize for lack 

 of references, which are, indeed, very incomplete in Welling- 

 ton. His object has been to collect a number of what he 

 believes to be true although, no doubt, trite remarks, with 

 the object of collecting an argument which he has been un- 

 able to find in any of the works to which he has access, and 

 which is necessary for the development of another paper, to 

 which reference has been made. The portion for which it is 

 thought some novelty may be claimed is that of the treat- 

 ment of the Taylor's series formulae and similar linear 



