206 Mr. J. J. Sylvester on Poncelet's approximate 



surface R = 1 cut off by the plane Ax + By + Cz = 1 . Suppose, 

 then, that Lx -f My 4- Nr is the linear form sought. The pro- 

 portional error is Lx + My + Ns — 1 j so that if we draw the 

 plane Lx + My -fN*— 1=0, the error is expressible geometri- 

 cally (paying no attention to sign) as the quotient of the per- 

 pendicular upon this plane from any point x, y, z in the seg- 

 ment, viz. — - y — -_ , divided by the perpendicular from 

 the origin in the same plane, viz. Hence, then, 



the geometrical question to be resolved is simply to draw a plane 

 for which the greatest value of this quotient, restricted to points 

 within the segment, shall be the least possible. From this it is 

 immediately seen to follow, that the portion of the surface cut 

 off by the plane Lx + My -f Ns— 1 =0 must be a portion of the 

 segment cut off by the given plane Ax -f By + Cz — 1 = 0. And 

 its actual position may be determined by means of a principle 

 generally known, but which, as it will occupy but a few words, 

 it may be well to deduce from first principles. 



Suppose there are (r-f 1) quantities, each containing the same 

 system of r parameters ; for greater brevity, say three quantities, 

 p, q, r, each functions of the same two parameters X, /j, : let 

 us call the greatest of the quantities p } q, r, corresponding to 

 assigned values of X, /jl, the dominant ; so that, according as we 

 change X, //,, the name of the dominant is liable to change ; and 

 that we wish to find M the minimum value of the dominant 

 upon the supposition that the variations of p, q, r in respect to 

 X or /n are never simultaneously zero, and may be made positive 

 or negative at will ; then M will be found from the equations 

 M =p = q=:r. For if we had M =jt? and p> q, p>r y by vary- 

 ing at will X or /j, we could make Bp negative; and consequently 

 since by hypothesis p differs sensibly from q and r, the domi- 

 nant of^-f Sj», q -f 8p, r -f Br would necessarily be less than that 

 of/?, q, r, and thus M would not be the maximum dominant. 



In like manner, if M=p = q, p>r,vre could by means of the 

 equations 



so determine B\, Bp, as to diminish simultaneously p and q ; and 

 thus the dominant of p — e, g— rj, r + B?' would, as before, be less 

 than that of p, q, r. The same reasoning applies to any number 

 (r+1) functions of r variables. And if the number of func- 

 tions should exceed r-f 1, it would still serve to show that when 



