( H5() ) 



It is lo be proved lliiit the locii.s of the poles of tlie two other sides 

 with respect to a" is likewise the tistroid just foiiud. To that end 

 we construct the jtole of /A'. 



If we take as centres of the projective pencils of rays generating 

 A' point / and the point of IX at intinily, then on account of the 

 former reasoning the pole of IX lies on the j)arallel through li to 

 IF; at the same time this pole lies on a parallel drawn through 6' to 

 IX; so the point of intersection D of the latter right lines is the 

 demanded pole. As D is symmetric to 6' with respect to / }", it 

 also belongs to the astroid. In the same way we can prove that 

 the pole of lY is likewise a point of the astroid. 



By projective transformation the above problem can be put as 

 follows : 



Given a conic and a polar triangle of it. To determine the locus 

 of the poles of the sides of that triangle with respect to the system 

 of conies passing through the vertices and touching the orginal conic. 



If we regard this as a problem by itself we arrive at the following 

 algebraic solution : 



Take the polar triangle as triangle of coordinates; then for the equation 

 of the given conic can be written: 



a^.r,'' -f a,.r,' 4- a,.r,- = , ...... (1) 



and for that of the conic described about that polar triangle : 



Pv^,-U I P.-'^'j-^i + 7^3-'-i'^, ~ (2) 



If we introduce the condition that (2) touches (1) then the coefti- 

 cients of the latter satisfy the relation: 



The jH)le of one of the fundamental sides, e.g. of .r, =0, is found 

 by substitution of 



Vx — —Vx ;;',= - /^« - • 



By this the equation of the locus of these poles becomes: 

 which can also be written in the form: 



1 2 J^ 2 2. ^ 



«I '^"i -r «., X. 4- «, •'•, = ^ • 

 We recognize in this the form of the astroid on oblique coordinates; 

 the curve itself is a [>rojective transformation of the common astroid. 

 The locus of the poles of the other sides gives the same result. 



