326 BEPOKT— 1880. 



ordinates can be obtained by calculation in any way and to any extent 

 that may be demanded. Very different considerations present themselves 

 in dealing with physical data, even when the given abscissjs and ordinates 

 are measui'ed with absolute precision. Here a further assumption of 

 simplicity is required, namely, that there is continuity (in the sense in 

 which that term is used in Cauchy's proof of Taylor's theorem) of the 

 same order as the number of differences used. This does not necessarily 

 exist. In a pressure diagram, for instance, the pressure may, as matter 

 of fact, have varied discontinuously between the selected ordinates, while 

 the assumption of process is, that it has varied continuously. 



It is worth while here to mention another problem in which arbitrary 

 processes are often used as if they were definite, and that is, the averaging 

 of discontinuous or irregular phenomena. To fix the ideas, consider the 

 population of a small island represented by the ordinate of a line, the 

 abscissa being proportional to the time. The population line will not be 

 a curve, but a series of steps, falling one unit at every death, and rising 

 at every birth — horizontal between. If the population be large, this will 

 hardly be distinguishable from a curve — probably a continuous one. But 

 if it is endeavoured actually to reduce it to an equivalent continuous curve, 

 this cannot be done with indefinite approximation, unless some further 

 assumption be introduced ; for any fair curve lying between one drawn 

 through the top edges of the steps and another through the bottom edges 

 will answer our indefinite question. The general answer is then uncertain 

 to the extent of this difference, at least. Each condition that it is subjected 

 to — as, for instance, equality of areas, of moments, of moments of inertia 

 or the like — merely introduces an equation which must be satisfied by the 

 coefficients of any algebraical formula which may be selected to represent 

 the data. 



Some degree of indeterminateness in amount must always remain, 

 unless the conditions are exhausted by assumptions suflicient to render 

 the curve determinate, or by evidence extraneous to the actual data of 

 observation. 



This is obvious enough when stated, being merely an extension, from 

 two to more quantities, as well as in genere, of the remark, that a mean is 

 indeterminate until we know what mean is meant. Yet it is no uncommon 

 thing to see observations discussed in disregard of this, and treated, not 

 as affording means of determining the parameters of a law assumed, or 

 derived aliunde, but as affording the means of determining the law itself 

 with completeness. 



The effect of discontinuity increases as we differentiate, and decreases 

 as we integrate. Thus a corner in a curve alters the tangent, or first 

 differential coefficient, discontinuously, and makes the second infinite, 

 while the ordinate only changes its rate of variation, and the change is 

 still less observable on the integral or area. The amount of discontinuity, 

 as well as its order, is to be considered. For many pur^DOses a small 

 discontinuity of low order is of about the same importance as a considerable 

 discontinuity of a higher order of diffei'entiation. 



The ordinary assumption, both in interpolation and in quadratures is, 

 that one quantity may be finitely expressed as a rational integral function 

 of the other, with a sufficient approximation, and that the constants of 

 this expression may be determined also with sufficient approximation, 

 from the known values of the quantity assumed to be so expressed as a 



