398 On Undevelopable Uniquadric Homographics. [May 14, 



" methods of reduction," amougst whicli the following is perhaps the most 

 obvious and simple : — 



Let S be the quadric, and o^, O3, 0^, . . . o„ the n given points. Put xx 

 for the line through 0^ and 0^. Instead of 0^ and 0^ we can substitute the 

 point ^2 in which the line xcc is cut by the plane o^o^o^, and another point 



determinable in the same line xx. Then instead of the four planar points 

 p^, O3, 0^, 0. we can (see theorem 38) substitute two other points ^3, p^ in 

 the same plane ; and therefore instead of the series of n points, we can sub- 

 stitute the series of n—2 points p^, p^, o^, o-, . . . o„ and the inscribable 

 (n — 2)'gons, closed and open as may be, whose sides pass in order through 

 these points will have extremities identical with the extremities of inscri- 

 bable w'gons. And thus step by step we can reduce the number of sides, 

 until at length we find three points or four points, according as n is odd or 

 even, snch that the extremities of all inscribed 3'gons or ^'gonswhose sides 

 pass in order through such points are identical with extremities of inscri- 

 bable w'gons whose sides pass through the original n points ; and therefore 

 to solve the problem all we have to do is to inscribe the closed 3'gons or 

 closed 4'gons as may be. 



And in respect to this method we may observe,— 



(1) If any four consecutive points of any of the series be colinear and 

 such as to render the inscription of closed 4'gons real, we may omit such 

 points altogether from the series. 



(2) When n is odd, and that we reduce the problem to the inscription 

 of closed 3'gons whose sides pass through three known points, then should 

 such points be coHnear or form a conjugate triad, the problem will be par- 

 tially porismatic. 



(3) In the case in which n is odd, it is easy to perceive how the pro- 

 blem can be reduced to the drawing of a line through a known point to cut 

 two reciprocal lines (which point will be on 07ie of the lines when the pro- 

 blem is partially porismatic). And when n is even, it is easy to see how 

 the problem can be reduced to the drawing of the two lines which cut two 

 pair of (determinable) reciprocal lines. 



(4) The following method of finding the line in the plane of four points 

 which pierces in first extremities of closed 4'gons is obvious : — Let o^, o^, 

 O3, 0^ be the four planar points. 



Find ^ the point of intersection of the lines 0^0^, 0^0^ ; in the line 0^0, 

 find the point 7n such that 0^0^, mp, and the pair of (real or imaginary) 

 points in which 0^0^ pierces the quadric, will form an involution ; in the line 

 o find the point n such that the pairs of points OgO^, p/i, and the points in 

 which 03©^ pieces the quadric, form an involution. Then will the line mn 

 pierce in first extremities of closed 4'gOES. 



