74 



UNIVERSITY OF VIRGINIA PUBLICATIONS 



where a, p, y, 8 have the values given in (58). This equation corresponds 

 to the Ptolemaic equation of a circular arc in the plane, and for the 

 position of Q chosen as above is 



a(ch + bg — af)ri + &(c^'' + */ — &^)^o + c(&(7 + fa — ch) r^ = 2alcr^, 



(70) 

 and along this line s has a zero value and a pseudo-minimum. 



IV. 



25. The processes and results of the previous section are true in n 

 dimensions. It is thought not out of place to briefly deduce them for n 

 dimensions, especiallj' as these results can be more simply reached than 

 the writer has seen them attained elsewhere.^ 



A simplicissimum is a figure formed in space of n dimensions by w + 1 

 points, no three of which lie in a straight line, no four in a plane, no five 

 in a h}^er-plane, etc. It was thus defined by Sylvester. 



Designate the ra -|- 1 corners of a fixed simplicissimum (which we shall 

 call briefly the simplex) hj A, B, C, ... . Choose an arbitrary system 

 of orthogonal cartesian coordinates and let 



Xr, 2/r, Zr,... (r = 1, 3, ...,«,+ 1) 



be the coordinates of the corners of the simplex. 



Let mr, (r = 1, 2, . . ., n -|- 1) be any assigned numbers. Let G be a 

 point of mean position whose coordinates are 



M 



y = ir ^^-rVr 



1 



2Av2, 



M ^ '"" M 



where M = 2mr. The numbers m-r are called the ratio (mass, charge, 

 barycentric) coordinates of G. Let It-c = tUr/'M, or what is the same 

 thing let M = 1. Then 



X = 'S.krXj., y = %k^yr, z = ^IctZt, ■ ■ ■ 

 Solving for the A-'s, there result 



. 1 



, 2/1, . . ■ 1 



fcr 



■!'n, 2/n 



. 1 



(71) 



'Special reference is made to a paper by Mr. W. J. Currau Sharp, proceedings of 

 the London Mathematical Society, Vol. XVIII, p. 325, read April 7, 1887. On the 

 Properties of Simplicissima (with special regard to the related Spherical Loci). 



