Brown. — Phenomena of Variation. 533 



APPENDIX. 



I. 



The " Mississippi Problem " is of some celebrity, and may 

 with advantage be discussed. It refers to the velocity of the 

 water at different depths in the Mississippi at Carrollton and 

 Baton Bouge. The experiments were made in 1851, and were 

 reduced by the experimenters by means of a parabolic approxi- 

 mation, which they applied according to common-sense 

 principles similar to those of the present writer, except that 

 they apparently did not perceive the bearing of the facts that 

 are fundamental theorems in the graphical calculus (sec- 

 tion 23). Consequently they failed to get such a good approxi- 

 mation to the experiments as is possible, although many 

 engineers may think their approximation quite sufficient. 

 Then in 1877* Professor Merriman, after referring somewhat 

 caustically to " tedious approximative methods," proceeds to 

 give a reduction by what he calls the " strictly scientific " 

 method of least squares. This application is one to which 

 our definition of least squares is strictly applicable. The 

 calculations are given also in Professor Merriman's " Theory 

 of Least Squares," 1900. 



Again, in 1884, Mr. T. W. Wright, " Adjustment of Observa- 

 tions," page 413, reverts to the phenomena, applying both a 

 parabolic and a cubic formula by least squares ; and he remarks 

 that, since the latter formula yields a smaller " sum of the 

 square of the residual errors " — the italics are the present 

 writer's — "the observations are better represented by the 

 formula last obtained." From the graph of deviations ob- 

 tained by the present writer he has no hesitation in saying that 

 the indications are for the application of a discontinuous for- 

 mula, the first section holding from depth to 0*5 or 0"6, and 

 the other from that to 09, the formulae differing chiefly in the 

 constant term. This reduces the deviations to g^oo' about, at 

 most (judging from the graph), against about ^^ with the 

 least-square parabolic. The value of the probable (fortuitous) 

 errors is not given or discussed in either reference, so that it 

 is quite a matter of speculation whether this indication of 

 discontinuity is genuine or whether it is a mere matter of 

 luck. At any rate, we should not attempt to improve such a 

 graph by means of a cubic formula; it evidently would require 

 a formula of a large number of terms to reduce the deviations 

 to as small limits as those of the discontinuous parabolic. It 

 is to be noted that all these considerations are obvious upon a 

 mere inspection of a graph of the deviations which are given 

 by Professor Merriman, and also that it is not suggested that 



* Journ. Prank. Inst., C. iv., p. 233. 



