520 Transactions. — Chemistry and Physics. 



the quantities of physical phenomena with experiment upon 

 those phenomena are in some cases not quite so effective as 

 the theory of probability enables them to be made, and that 

 the latter theory has even had a detrimental effect upon the 

 comparison by reason of it having been frequently assumed 

 to have provided a universally satisfactory method — that of 

 least squares — by which we can determine those constants 

 which arise from the unestablished properties of matter, and 

 at the same time more or less tacitly institute a comparison 

 between the theory and the results of experiment in the case 

 of a phenomenon of variation where quantity is both measur- 

 able and supposed determinable by theory, given the proper- 

 ties of matter. The results of the theory of probability will 

 be accepted with regard to the probable value of a single 

 quantity directly measured and its probable error. 



In the present paper the writer proposes to examine the 

 representation of physical phenomena of variation by means 

 of formulae, whether empirical or founded more or less com- 

 pletely upon reason. 



1. The phenomena which will be examined are those 

 where a quantity (Y) varies with a variable (X)— that is to 

 say, takes up magnitudes which, ceteris paribus, depend in 

 some fixed way upon the magnitude of X. If we observe, by 

 experiment, how the variation occurs we obtain knowledge 

 which can be expressed by a graph. We may make the axis 

 of Y the ordinate, that of X the abscissa. We shall consider 

 only such cases where Y has, m fact, although it may not 

 have been observed, one value, and only one, for each value 

 of X, which in general extends from pins to minus infinity.* 



2. The first fact we notice is that in such case we observe 

 values within a limited range. This we may call the " experi- 

 mental range of X." Beyond that range we know nothing, 

 whereas most mathematical expressions will yield values 

 from minus to j^ us infinity. The definite integral is in form 

 a striking exception to this, and from one's experience of text- 

 book formulae it is to be wished that some simple means of 

 indicating experimental range could be brought into general 

 use. This idea of dealing with the experimental range only 

 will be found of fundamental importance in later parts of this 

 paper. 



3. Now, the graph may be of two distinct kinds — (A) that 

 of a curve or curves, or (B) that of a series of datum points. 

 The first kind, that of a curve, contains the same complete 

 statement of values of Y as does the analogous kind of mathe- 

 matical formula, which is defined as holding between limits of 

 range of X. The second kind, that of datum points, contains 



* But see Appendix, III. 



