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UNIVERSITY OF VIRGINIA PUBLICATIONS 



Form the product of the two determinants on the left and in the result 

 put ^ = 0, the result is the determinant in (45) in which s = 0. Hence 

 on dividing out s- in (45) the result is a quadratic in s-. The actual 

 solution is effected more simply by reducing the determinant in (45) by 

 subtracting rows from rows and columns from columns in such a manner 

 as to transform it into 



1 



h-C-a" 



c-a-p- 



c~a-p^ 







l-h-y- 



a-b-y^ 







= 0. 



(46) 



Whence on expansion 



s* — s^ 2 (6- + c- — a-) a- = "${0- -\- b- — C-) cV/3- — 2 b-c-a*. 

 Actually solving this quadratic we find 



s- = i 'S.ib- + c- — a-) a- ± 88A, 

 the same result previously obtained in (23). In the algebraic work of 

 reduction wherever equations have been multiplied or divided by o-, s or 

 T y z the assumption is made that these numbers are not zero, and there- 

 fore circumcircle solutions have been thrown out. It is not worth while 

 solving (44) for x^ y, z. 



None of the solutions of the problem thus far presented serve in a 

 satisfactory way for higher dimensions. We now indicate a process which 

 is applicable in all dimensions, furnishing the solution by means of quad- 

 ratic equations. 



18. We shall make use of some known results, letting the demonstra- 

 tions subsequently for n dimensions serve for particular cases. The de- 

 terminant 



the vanishing of which expresses the mutual relation existing between 

 the distances between any four points in a plane, we represent by 



