528 Transactions. — Chemistry and Physics. 



for the application of Taylor's theorem to the approximative 

 treatment of non-linear functions, and mention two other 

 theorems of the graphical calculus which are of occasional 

 use. In cases where X (or Y) is invariably associated with 

 a constant by addition or by multiplication we get possible 

 graphical operations, for, if the expression is /(X + A) r 

 we may draw a curve to /(X) and then introduce the effect 

 of any value of A by shifting the curve bodily along its X axis; 

 and so also with regard to Y. In the case of multiplication 

 we get a stretch of a drawn standard curve in either one way 

 or in two ways. For, to take the latter case, when / X (A x Y) 

 = / 2 (B x X), having drawn a standard curve to convenient 

 values of A and B, we get the effect of any values of either 

 constant by uniformly stretching the curve in directions 

 parallel to both axes. This can be effected by means of 

 throwing shadows, and appears of value in our subject, since 

 the frequency curve is of this form (with an immaterial rela- 

 tionship between the constants). 



26. Reverting to the question of appealing to the judgment 

 to detect systematic deviation from a formula, we see that we 

 expect the deviation to become evident as a recognisable 

 additive curve — i.e., as if it were representable by a term/(X). 

 Clearly, this is frequently the case even where the deviations 

 may be logically functions of Y, as, indeed, we supposed all 

 errors to be in section 5 ; for, in a graph, if a function of X be 

 represented, the corresponding function of Y is also auto- 

 matically represented by the curve. By such means we can 

 sometimes form an estimate of causes of error or deviation, 

 and sometimes also — as we shall see in the case of the Missis- 

 sippi Problem — be able to form an idea whether it is any use 

 or not to go on complicating the particular formula which we 

 are employing. When our resources are practically exhausted 

 we shall give our formula, together with a statement of its 

 range and the relation between probable (fortuitous) error and 

 observed deviation, exhibiting the latter quantities in a graph 

 of deviations, and leave it for others to judge what degree of 

 likelihood attaches to our formula. Circumstances may lead 

 us to employ least squares, but the value of our experiments 

 cannot be adequately indicated unless we provide at least the 

 equivalent of the details mentioned. 



A Graphic Process for applying Power-expansion or 

 Taylor's Series Formula. 



27. A process will now be described by which it is very 

 easy to graphically apply to date formulae of four or even five 

 terms in ascending powers of the variable. 



28. It follows from Taylor's theorem that, if we use a 



