1868.] 



Uniquadric nomographics. 



397 



Eg to represent the four planes dfi^c^, djb^c-^, e^b^c^, ^fi-fx respectively ; 

 through the line of intersection of the planes draw the plane P the 



distances of any point in which from and D2 have to each other the 



ratio of to ^-—^ ; through the line of intersection of the planes E ^ 



Ej draw the plane Q which is such that the ratio of the distances of any 



point in it from E^ and E2 is the same as that of to ; find the line 



of intersection mm of the planes P, Q (this line mm will he a tangent to the 

 quadric ), and the point in which it touches the quadric ; in the line 

 c-^a^ find the point 5 conjugate to^. Then if we regard the n given fixed 

 points and the point g as w + 1 points of a series, any point in the surface 

 will be first extremity of a closed 2(7i+l)'gon. Find, by the preceding 

 case, the two reciprocal lines \jy, zz which pierce the quadric in first extre- 

 mities of closed (;^+ l)'gons whose sides pass in order through these w + 1 

 points ; find the line -p' (( reciprocal to pq ; draw the lines xx and ii each of 

 which cuts the four non-planar lines yy, zz, pq, p'q . Then will xx and ii 

 be reciprocals piercing in first extremities of the four inscribable closed 

 w'gons. 



Fourth method. — The following method is applicable in all cases in which 

 n is even. Omit temporarily the wth point o^'of the given n points, and 

 find the line U which pierces the quadric in the two first extremities of in- 

 scribable closed {n — l)'gons whose sides pass in order through the n—\ 

 points ; find the point q in the line U which is conjugate to the omitted nth. 

 point On' Then if we regard the n—\ given points and the point q as 

 forming the n points of a new series, any point in the surface will be the 

 first extremity of a closed 2?i'gon. Find the two reciprocal lines yy, zz 

 which pierce the quadric in first extremities of closed /^'gons whose sides 

 pass in order through the new series of n points ; find the line p'q reci- 

 procal to Onq ;. draw the lines xx, ii each of which cuts the four non-planar 

 lines yy, zz, Onq, p'q > Then will the lines xx and ii be reciprocals piercing 

 the quadric in first extremities of the four inscribable closed w'gons whose 

 sides pass in order through the n given points. But if the inscription of 

 the closed (/i— l)'gons be partially porismatic, and p the point of concur- 

 rence of the closing chords of the inscribable open (w— l)'gons, then will 

 the line xx through On and p, and the line ii reciprocal to xx pierce the 

 quadric in the first extremities of the inscribable closed Tz'gons. 



N.B. And if On be in such case coincident with p, then the problem is 

 fully porismatic, and every point in the surface is the first extremity of a 

 closed w'gon. 



N.B. We may also observe that when the inscription of the closed 

 (/I— l)'gons is non-porismatic, and that the point o^ is situated in the line 

 U, then, by conceiving q coincident with o„ the lines yy, zz will be identical 

 with XX and ii. 



I may observe that the general problem can be completely solved by 



