30 
ON THE PLANE REPRESENTATION OF A SOLID FIGURE. 
[423 
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. V. Lemma 
XXVII.), in a given quadrilateral to inscribe a quadrangle of given form. The question 
in effect is: in the triangle ABC to inscribe a triangle a/3y of given form; and in 
the triangle ABE a triangle d$'y of given form, in such wise that the sides ay, ay' 
I) 
may be coincident. The envelope of ay is a conic touching AD, AE, and the envelope 
of a'y a conic also touching AD, AE: there are thus two other common tangents, either 
of which may be taken for the position of the side ay = a'y' ; and the problem admits 
accordingly of two solutions.