Solution, by the Method of Ordinary Homographic Division, of the Problem, "To inscribe in a given Ruled Quadric a Polygon of any given order whose sides shall pass in any prescribed order of sequence through an arbitrary system of given points in space"
Proceedings of the London Mathematical Society
any given order whose sides shall pass in any prescribed order of sequence through an arbitrary system of given points in space" By Rev. RICHARD TOWNSEND,M.A., F.R.S., Fellow of Trinity College, Dublin. CASE 1.-If the number of points be even. Assuming arbitrarily any three generators A, B, C of either (but of the same) system on the surface, draw through them and through the first of the given points P x a first triad of planes intersecting the surface again in three first derived generators A
... 1} B n C x of the opposito system ; through these latter again and through the second of the given points P, draw a second triad of planes intersecting the surface again in three second derived generators A" B a , C, of the original system; through these latter again and through the third of the given points P s draw a third triad of planes intersecting the surface again in three third derived generators A 8 , B s , C 8 of the opposite system; continue the same process until all the points, taken in the prescribed order, are exhausted, which will give three w th derived generators A", B n , C n of the original system; the two double generators M = M" and N = N fl of the two homographic systems determined by the three pairs of correspondents A and A n , B and B n , C and C" of the original system will be those of that system on which the first vertices of the required polygons lie. The same process, commenced from any three arbitrarily assumed generators A', B', C of the opposite system, aud continued in the same manner until all the given points taken in the prescribed order are exhausted, will give the two generators M'= Mj, and N'= N' n of that system on which they also lie. The four vertices P, Q, R, S of the skew quadrilateral formed by the two pairs of double generators M and N, M' and N' will consequently be the required vertices themselves ; and the problem accordingly admits in this case of four solutions, which are evidently all real or all imaginary together. CASE 2.-If the number of points be odd. Assuming arbitrarily, as before, any three generators A, B, C of either (but of the same) system on the surface, derive from them, as before, two triads of final generators A B , B", C n and A_ n , B_ n , C_" of the opposite system, by application of the same process as before to the given system of points, taken first in the prescribed order P x , P 2 , P 5 , and P n , and then in the opposito order P n , P n _i, P n _ 2 , and Pi; the two double generators * A very elegant construction, at once simple, direct, and general, has recently been given for the case of n odd, by Mr. M. Gardiner, C.E., late Scholar of Queen's College,