152 Prof. F. Y. Edgeworth on the 



of small errors which had arbitrary laws of frequency of their 

 own" *. " Laplace does not seem to have regarded an error 

 in this light" |j as Grlaisher states correctly : " Nowhere does 

 he assume that' if one observation only is made its law of 

 facility is e~ h ~ x ~. ,} X Laplace evidently did not contemplate this 

 deduction when in the important section dealing with the 

 application of inverse probability to errors of observation 

 he speaks of " the complete ignorance in which we are with 

 respect to law of errors for each observation" §. 



(13) Laplace's proof of the law-of-error derives strong- 

 support from Poincare's theorem that the frequency-function 

 pertaining to the sum of numerous variable elements is such 

 that all its mean powers of deviation are approximately equal 

 to those of the normal error-function ||. 



Further confirmation is obtained from other proofs. To 

 some of them, in particular the proof by way of partial 

 differential equations, it may be objected that they assume 

 the existence of a final state, an ultimate law of frequency, 

 to which the continued superposition of elements must tend IT. 

 But (14) the assumption is surely not very arbitrary. It is 

 of a piece with the assumption countenanced by Tait** 

 that the velocities in a molecular medley tend to a final 

 distribution. 



It is pleasant to believe that these views are not so much 



* The aptly- worded predicate of this proposition is borrowed from 

 Prof. Sampson (Phil. Mag. p. 349; ; but not the proposition itself. 



f Memoirs of the Royal Astronomical Society, vol. xxxix. p. 106 

 (1872). Cp. Monthly Notices of the Society, vol. xxxiii. p. 397 : " he 

 (Laplace) did not himself so apply it (the law)." 



X Memoirs, loc. cit. p. 108. 



§ Theorie Analytique, Liv. II. cap. iv. art. 23. 



|| See Phil. Mag. vol. xxxv. p. 426 (1918). This proof is confirmed by 

 the solution of Stieltfes' problhne des moments, as presented by Prof. 

 O. H. Hardy in the 'Messenger of Mathematics/ vol. xlvi. (1917) p. 175; 

 regard being had to the rapidity with which the error-function tends to 

 zero (as the variable increases). 



% Article on "Probability," Encyclopedia Britannica, 9th edition, 

 p. 393. The objection does not applv to the proofs given by Morgan 

 Orofton, Phil. Trans. (1870). 



** "Everyone therefore who considers the subject from either of these 

 points of new (ordinary) statistics or the theory of probabilities, must 

 come to the conclusion that continued collisions among our set of elastic 

 spheres will, provided they are all equal, produce a state of things in 

 which the percentage of the whole which have at each moment any 

 distinctive property must (after many collisions) tend towards a definite 

 numerical value." The proposition is presently extended to sets of 

 spheres " no one of which is overwhelmingly more numerous than 

 another, nor in a hopeless minority as regards the sum of others." — 

 ■" Kinetic Theory of Gases." Poval Society of Edinburgh, vol. xxxiii. 

 j>. 225. 



