72 GEOMETRICAL RESEARCHES, 



2. When the data is so related that the points a n+l , & n+1 , w+1 

 fall on the respective points a , Z^ , Cj ; then evidently the problem 

 is porismatic ; for equation (1) assumes the form 



which holds when p^ has any position whatever in L X . 

 Hence we may announce the following theorem : 



THEOREM I. 



If there be n straight lines and n points in a plane, and that 

 any three closed logons can be described, such that each one of them 

 has its n angular points in the n respective lines, and its n sides 

 passing in order through the n points, then ivill any point in any 

 of the fixed lines be an answerable position for the prescribed 

 angular point (resting on that line) of a closed u'gon fulfilling the 

 like conditions with the three others. 



3. Theorem 1 will enable us to arrive at some interesting 

 porisms and theorems. 



Firstly. Suppose we were given all the data but the two 

 points o n _ 1 and o n , and that it is required to find such positions 

 for these points as will render the problem porismatic. 



Here our object is to form 3 closed ft'gons ^ a ... a a v b 



^2 ' ' ' ^ n ^i' i C 2 * c n c 'i> ( su ^J ect to ^6 imposed conditions 

 as respects the given data) whose sides a n __ J a^ b n _ l b n , c n _ l c^ 

 will pass through one point, and whose sides a n a , b b , c c , will 

 pass though another point : for these two points would evidently 

 be answerable positions for c^__ and o n . If we take b 1 in the 

 intersection of Lj and L W , we can find the corresponding point 

 b on L . If we take a . in the intersection of L and L 



n 1 n 1 n 1 1 n t 



we can find the corresponding point a 1 on L r And if we take G I 

 anywhere in L A , we can find the corresponding point c n _ 1 on the 



