530 Transactions. — Chemistry and Physics. 



simple numbers for convenience in con version),* and then, 

 provided with a scale to measure the constant term, a 

 straight-edge to produce linear terms and drawn curves of the 

 standard parabolic and cubic, and with a protractor or pro- 

 portional compasses, we proceed to build up a curve the 

 ordinates of which are added proportions of each of these 

 four constituents, till we get a curve that is as nearly like 

 the given one as possible. It will be quite obvious when 

 this is done that we have a most clear idea of the prospective 

 advantages of any other cubic formula whatever, and that we 

 can arrange the deviations in any desired way — for instance, 

 to arrange them for the application of a quartic formula, if 

 it appears that such a course is advisable. We shall also 

 develope a decided opinion on the subject of the application of 

 common-sense to the resultant curve of deviations, both for 

 interpolation and for extrapolation, and for residual causes or 

 for error in the theory of the formula. In cases where the 

 accuracy of the figures is great it may be necessary, after 

 a rough analysis, to replot the deviations to a larger scale, 

 so as to get over the limited accuracy practicable in a 

 graph. 



32. Many details will become obvious if a trial is made, 

 and we need not pause over them ; but it may be mentioned 

 that the process is obviously applicable to all such formulae 

 as are made up of sums of terms each of which is linear in 

 and contains only one constant. So the process might be 

 arranged for harmonic analysis, or for the solution of simul- 

 taneous equations, and so on. The process is approximative, 

 so that it is of indefinite accuracy, and is limited solely by 

 the power of the judgment to indicate what alterations are 

 desirable. The vast difference between this procedure and 

 that of least squares will be apparent from the fact that 

 we may be led to apply formulae which have more terms 

 or constants than there are datum points. This is due to 

 the part we are allowing the judgment to play in controlling 

 the interpolation. 



33. The process is evidently susceptible of mechanical 

 treatment, and the writer hopes to be enabled to construct 

 a machine for this purpose. 



34. With respect to the higher-power formula, there is a 

 point which seems of theoretic interest in simplifying mathe- 

 matical formulae which are to be applied for a definite range 

 (the converse of our experimental range), for, as is noted in 



* It is to be noticed that in these formula the adjustment of the 

 scale by introducing q is a perfectly simple matter of arithmetic; but 

 to alter the zero poinc (p) is more troublesome, and should be avoided 

 when possible. Trie Z functions of the Appendix afford an alternative 

 range of - 1 to + 1, using the same curves. 



