﻿290 On the Plane Representation of a Solid Figure. 



and then the projections of each of the required lines are T' = T', 

 a common tangent of 2)', W 3 and T"= T", the corresponding com- 

 mon tangent of X", 2". 



It is material to remark how the construction is simplified 

 when there is given one of the lines, say, M, which meets lt v L 2 , 

 L 3 , L 4 . Here M is a common directrix of the two hyperboloids ; 

 we may for the hyperbolas £' and %" consider, instead of L l5 L 2 ,L 3 

 and two new generating lines, the lines L 1? L 2 , L 3 , M, and a 

 single new generating line N ; and similarly for the hyperbolas 

 X', %" the lines L l5 L 2 , L 4 , M and a single new generating line N. 

 2/, %' have thus in common the three tangents U lt L' 2 , M', and 

 therefore only a single other common tangent, T'= T' ; and simi- 

 larly 2", S ;/ have in common the three tangents L'^, L" 2 , M", 

 and therefore only a single other common tangent, T"= T"; and 

 we have thus the other line cutting the four given lines. 



I take the opportunity of mentioning the following theorem : 

 "If in a given triangle we inscribe a variable triangle of given 

 form, the envelope of each side of the variable triangle is a conic 

 touching the two sides (of the given triangle) which contain the 

 extremities of the variable side in question." 



We have thence a solution of the problem {Principia, Book I. 

 Sect. Y. Lemma XXV1L), in a given quadrilateral to inscribe a 

 quadrangle of given form. The question in effect is in the tri- 

 angle AB C to inscribe a triangle a/3 y of given form ; and in 

 the triangle A D E a triangle &} /3' 7' of given form such that the 



sides a y, cJ <y' may be coincident. The envelope of a y is a conic 

 touching AD, A E, and the envelope of a! 7' a conic also touch- 

 ing AD, A E : there are thus two other common tangents, either 

 of which may be taken for the position of the side a y 

 and the problem thus admits of two solutions. 



«Vj 



