of Variable Quantities. 3 



In those cases in which an algebraic law can be assigned, 

 to which the ordinates ought to conform, the best method of 

 treating the problem is to determine the constants involved in 

 the function by the method of least squares ; but it might 

 often be not worth while to cany out this process, as, for ex- 

 ample, where the deviations of the various observations from 

 the law are large, and where it would accordingly be pedantic 

 to assign values to the constants with precision. 



Where the law is unknown and the observations are equi- 

 distant, a method of treatment might, perhaps, be devised by 

 ih^ assumption of some form of function containing fewer 

 constants than the number of given points, and consisting of 

 a number of simple harmonic terms, none of which go through 

 a large fraction of their period in passing from one ordinate to 

 the next. The constants involved might then be determined 

 by the method of least squares, so that the function should 

 give the best representation of the observations. But the as- 

 sumption of the form of function would be arbitrary, and the 

 process very laborious. 



On the whole it will be more convenient and equally satis- 

 factory, as far the result is concerned, to proceed empirically 

 from the first, remembering that the main object is to exclude 

 ripples of short period. 



The method here suggested is one which I believe is used 

 in some form or other by meteorologists ; but I am not aware 

 that its merits have been discussed, or that it has been extended 

 to the smoothing of surfaces and of functions of three or more 

 independent variables, as I here propose to do*. 



Empirical Rule. — The observations are supposed to be 

 equidistant and to be functions of only one independent vari- 

 able. The method may be most easily explained geometrically, 

 and the transition afterwards made to its pirithmetical equi- 

 valent. It will be convenient also to speak of the deviations 

 of the several observations from the principal part of the com- 

 plex event which those observations represent as errors. 



It is proposed to substitute for each pair of consecutive 

 points A, B, a point P which bisects the straight line A B. 

 The points P then lie on a series of ordinates halfway between 

 the original ones. 



If the errors of A and B are of opposite sign, P is a 

 better point than either of them ; if they have errors of the 



* Since this paper lias been in the hands of the printer, I have learnt 

 that M. Schiapparelli has written a work entitled Sul modo di ricavare la 

 vera ex];)ressione delle Icggi della natura dalle curve ernpiriche ; (Milan, 1867), 

 and that M. De Forest has written on the subject in the ^Annual Reports 

 of the Smithsonian Institution ' for 1871 and 1873. and in the ' Anal} st 

 (Iowa) for May 1877. 



B2 



