1124 



Mathematics. — -'The theory of Bravais {on errors in space) for 

 poll/dimensional space, nnth applications to Correlation." By 

 Prof. M. J. VAN UvEN. (Communicated by Prof. J. C. Kapteyn). 



(Communicated in the meeting of March 28, 1914). 



Ill the original treatise of Bravais : "Analyse niathématique sur les 

 probabilités des erreurs de situation d'im point" ') as well as in the 

 articles that have afterwards appeared on this subject ') the problem 

 of the distributioji of eiTors in space has only been investigated for 

 spaces of two and three dimensions. Only Prof. K. Pearson has also 

 treated the case of four-dimensional space '). 



It may be interesting to treat this problem ^.Iso for a space of an 

 arbitrary number of dimensions, not so much with a view to the 

 geometrical side of the problem, as in connection with the subject 

 of correlation. If we consider the problem from this point of view, 

 it comes to this : 



A number (a) of variables u^, u^, . . . m, are given, each of which 

 follows Gauss's exponential law : 



and consequently may assume any value between — cc and -)- go. 

 Further we have a certain number ((>) of linear functions .r'l, .t,, ... .r^ 

 of the variables Ui, viz., 



X\ :=. a\\U\ -f- «12^2 + •••• + «I'tWj, 

 X-2 =1 «21 Wl + «22^2 + ••••+ «2(7^1, 



AV = «;lMl + «;2«2 -!-••••+ «;:r?<J. 



The probability that .Vj ranges between c^ and ^j -\- (fij (J — 1,2,.. .q) 

 is then expressed by the formula 



1) A. Bravais. "Anal. math, etc.'.' Paris: Mémoires préséntés par divers savants 

 a 1' Académie royale des sciences de I'lnstitut de France; T. 9 (1846), p. 255. 



~) E. GzuBER. Theorie der Beobacblungsfeliler. Leipzig, 1891, Teubner; p. 350. 



M. d'Ocagne. Sur la composition des lois d'erreurs de situation d'un point; 

 Gomptes Rendus T. 118 (1894), p. 512; Bulletin de la Soc. math, de France, 

 T. 23 (1895), p. 65; Annates de la Soc. scientif. de Bruxelles, T. 18 (1894) p. 86. 



S. H. BuRBUKY. On the Law of Error in the case of correlated variations ; Report 

 of the British Assoc. (65th m.) (1895), p. 621. 



V. Reina. Sulla probabilita degli errori di situazioni di un punto nello spazio; 

 Atti della R. Accad. dei Lincei, serie 5a, T. 6, sera. 1 (1897), p. 107. 



3) K. Pearson. Mathematical contributions to the Theory of Evolution : Regression; 

 Phil. Trans, vol. 187 (18951, p. 253. 



