562 REPORT — 1894. 



6. The Asymmetric ProbaUlity Curve. By F. Y. Edgeworth, M.A. 



The asymmetric probability curve is the general form of the law of error. It 

 may be obtained by solving a system of partial diHerential equations, which is tbe 

 generalisation of the system given by Mr. Morgan Crofton for the symmetrical 

 probability curve ('Encyclopedia Britaunica,' article on Probabilities, p. 781, 

 equations 45, 46). The generalised system may be written : 



(l)2/ + .r^ + 2;tf + 3/5( = 

 ^ ^ -^ dx dk ■' dj 



p^ 'hf ^ ^iJJ 

 ^'^' dk 'J.dx' 



f^\ dy _ 1 d.<ii 



^^^ dj^'Qdj^ 



where y is tbe frequency with which any error .v occurs ; x is measured from the 

 centre of gravity of errors ; /.: is the sum of squares of errors measured from that 

 point; / the similarly measured sum of cubes. The solution of the system is a 

 series of ascending powers of x, each term of which consists of a series of ascending 

 powers ofj'-f-Zti If?' is put =0,the curve being treated as symmetrical, the series 

 reduces, as it should, to the ordinary probability curve 



1 



y = 



\/7r ^-^k 



lij-^hi is small, the curve being only slightly asymmetrical, the series reduces to a 

 curve which is indicated by Todhunter as being related to the ordinary probability 

 curve as a second is to a first approximation (Todhunter, 'History of Probabili- 

 ties;' Laplace, art. 1002, p. 568). This slightly asj'mmetrical probability curve 

 may be used to correct the theory of correlaiion investigated by Messrs. F. Galton 

 and H. Dickson ('Proc. Roy. Soc'.,' 1886), and by the present writer ('Phil. Mag.,' 

 November and December i8il2). Whereas, according to the first approximation, 

 the most probable y corresponding (or ' relative ') to any assigned (or 'subject') 

 value of X lies on a right line passing through the origin, according to the second 

 approximation the locus of correlates is a parabola. 



7. On the Order of the Growps related to the Anallagmatic Displacements 

 of the Regidar Bodies in n-Dimensional Sj^ace. By Prof. P. H. 

 ScnoUTE. 



1. The groups related to the regular bodies in ordinary space have been amply 

 studied by F. Klein in his • Vorlesungen iiber das Ikosaeder' (Leipzig, Teubner, 1884). 

 There in every case the order of tlie group has been found by enumeration of the 

 possible positions; so tbe remarkable fact that this order is always twice the 

 number of edges is not observed. 



I now wish to publish a simple general principle by means of which tbe order 

 of the group may be easily determined. This principle will prove to be capable of 

 immediate extension to 7)-(1imensioual space. _ 



2. General Principle. — The manner of coincidence of the regular body ABCD 

 with the given jiosition PQRS . . . (say the orinitntion ofABCD . . . with 



respect to PQRS) is de'ermined if the position of the vertex A and that of the 

 edge AB have been as-i'/iied. 



Therefore the order of the group is the product of the number of vertices by the 

 number of edges passin^r through a given vertex. 



This product is evidently the twofold of the number of edges. 



Fiiur-dimensional Space (S^). 



3. General Trinciph' o.rinvled. — The oiientation of the cell ABCD . . . with 

 reference to the given positum PQRS ... is determined, if the position of the 

 \'p,rtex A, that of the edir^ AB through A, and that of the face ABC through AB 



