524 Proceedings of the Royal Irhh Academy. 



14. If a, /3, y, and S are unit vectors, the equation 



p = U{xaa -{- yh^ -T zcy) 



is that of a sphere anharmonically cliTidecl. The unit triangle is 

 ABC and the unit point is S = JJ{aa + hjB -^ cy). The sphere may 

 be considered to be divided by the projection on it from its centre o£ 

 any one of the planes 



xaa -r yh^ -r %cy 



P = 



xl + ym + zn 



where /, «», and n are arbitrary. If however I = m = n = 1, the unit 

 point on the plane is the mean point of the unit triangle in it. So in 

 order to divide a sphere anharmonically with respect to a given unit 

 triangle (a, ^, y) and a given unit point (8), it is only necessary to 

 resolve S along a, (i, and y, to di-aw a plane through the points thus 

 determined on these lines, and to construct in this plane a net having 

 for unit triangle the points on these lines and for unit point the mean 

 point of the triangle. 



15. The quaternary coordinates^ for a plane, x, y, z, and w, where 



xaa + yhfi + zcy 4- wdh 

 xa' + yh' + %c' + wd' ' 



and = ffa + J/3 + cy -f dh, 



and = o' + J' + f' + d', 



may also be used for a sphere. The vectors being still unit, 



p - V'{X(ia + ybfi + zcy + zvdB), 



with the condition 



aa + bj3 + cy ^ d8 = 0, 



is a symmetrical form which expresses that the sphere is anharmoni- 

 cally divided with respect to the four points represented by a, ft, y, 

 and 8, or by the symbols 



(1000), (0100), (0010), and (0001). 



1 A particular case of the Qtiinary coordinates for space. See "Elements 

 of Quaternions," Art. 70. 



