214 Mr. J. J, Sylvester on Poncelet's approximate 



The following observation, which constitutes a veritable theo- 

 rem, and is presupposed in Prof. Peirce's solution, is very im- 

 portant: — "Any circle being found which, cither passing through 

 three of the given points such that no two of their joining lines 

 form an obtuse angle, or which described upon the line joining 

 two of the given points as a diameter, includes all the rest, is the 

 minimum circle which contains all the points of the given cluster; 

 so that one, and only one, circle exists satisfying the above alter- 

 native condition." 



It may be instructive to proceed to the application of the 

 method now fully explained to some of the more salient cases of 

 inequality, it being understood that these cases are given to 

 afford some general notion of the precision of the method, and 

 by no means as specimens of such as it would be applied to in 

 practice, for which the limits I shall suppose would be far too 

 wide to furnish any useful result. 



Ex. 1. x, y, z unlimited. Here the values of F, G, H, Q are 

 the minor determinants of the matrix, 



1 I 



10 1 



11 



F=G=H = 1,Q= 1, and the linear approximation to\/# 2 -f- y 2 + z 1 



2 

 becomes __^ +&c ^ or ^/ 3 _i)# 4.(^/3 _ 1)^(^3 _j)^ 



by the band passing at the same moment through one other point at the 

 opposite extremity of a diameter to A, or through two other of the given 

 points forming a non-obtuse-angled triangle with A, it will begin to re- 

 volve (always contracting the while) about a tangent at A to its intersec- 

 tion with the sphere as an axis, until it meets a second of the given points, 

 say B. If the line A B is a diameter of the band, cadit qucestio, the pro- 

 blem is solved. If not, the band will go on further contracting, revolving 

 meanwhile round A B as an axis until either A B becomes a diameter in 

 virtue of the contraction of the band's dimensions (and so the problem is 

 solved), or else before this can take place the band is arrested at a third 

 point C, either forming a non-obtuse-angled triangle with A B and so 

 solving the problem, or else an obtuse-angled triangle with A B and 

 lying exterior to the arc A B on one side of it or the other; on the latter 

 supposition the line joining C with the extremity of A B nearest to it, will 

 (it appears to me) form a new axis of rotation for the band, which will 

 quit the further extremity of the old axis, and thus the motion will continue 

 with an intermitting change of axes, until at last the band either finds out 

 for itself an axis which in the course of the contraction becomes a diameter, 

 or else brings the band into contact with a third point forming a non-obtuse- 

 angled triangle with such axis, in either of which cases the minimum peri- 

 phery is attained, the contraction comes to an end, and the problem is 

 solved. 



