84 GEOMETRICAL RESEARCHES, 



positions as will render the quadrilateral porismatic, then will the 

 problem of the forming of the complete %'gon be porismatic. 



But it is evident (from a well known porism) that if we 

 assume any two points in the straight line r n _ 4 * n _ 4 as positions 

 for o , and o , then we can find a position for L so as to render 



1 n' n 



the problem of the forming of the quadrilateral porismatic. 

 Hence we may announce the following porism : 



PORISM 9. 



If a closed n'gon have its first n 2 sides passing through n 2 

 given points, and have its first n 1 angular points resting on n 1 

 given straight lines : a straight line can be found, such that if we 

 look on the two points in which it is cut by the two last sides of the 

 j^gon as fixed, and that we deform the n'gon so that its sides will 

 continue tlirougli the n points composed of the n 2 given ones and 

 the two determined ones, and that its first n 1 angular points move 

 on the n 1 given straight lines, then will the locus of the n th angular 

 point of the n'gon be a determinate straight line. 



This porism is evidently derivable from Porism 8 by 

 reciprocation : 



19. It is evident from the properties of homographic pencils 

 and divisions (see Chasles' " Geometric Superieure "), that we 

 can solve the more extended problem, in which all or any 

 number of the entities o 1 , o a , .... o n may be replaced by conies 

 to be touched by sides of the closed w'gon, provided these 

 conies touch the respective pairs of given straight lines on 

 which the extremities of the touching sides are to rest. And all 

 the data but the conic o n being given, the method of finding this 

 conic so as to render the problem porismatic is evident. 



And if instead of requiring all the angular points of the 

 closed %'gons to rest on straight lines, we were to have all or 

 any number of them rest in given circles, or other given conies 

 passing through the pairs of given points through which the sides 

 of the w'gon forming such angular points pass, then also we 

 can solve. 



