76 GEOMETRICAL RESEARCHES, 



THEOREM 3. 



If a closed n'gon c c 2 .... c n Cj, ftowe its angular points 

 Cj, c 2 , .... c <w n ^ecZ straight lines L J5 L g .... L^, of which 

 any three L , L , L , taken in successive order meet in one point 

 q : then o p o 2 , . . . . o being the n powts in which any straight line 

 through this point q cuts the respective sides G I c 2 , c 2 c 3 , .... c n GJ 

 o/ the n'gon, we can (if we conceive these points to become fixed) 

 deform the n'gon, so that its sides will continue through these fixed 

 points, and its angular points move on the fixed straight lines. 



8. If all the given straight lines but L pass through one 

 point q ; then, from the general investigation, it is evident the 

 straight line b b _, (or XX), is indeterminate, and may have 

 any position we wish with respect to q. And it is also evident 

 that by giving b l b n __ l any fixed position through q, then will 

 b b be coincident therewith; and o and o will be in directum 



n 1 n n1 



with q. Hence we have the following porism : 



PORISM 4. 



If all the sides of a closed n'gon c x c . . . . c c, pass through 

 given points, and all its angular points except one c move on given 

 straight lines meeting in a point q which is in directum with the 

 points through which the sides containing the free angle pass; then 

 the locus of tliis angular point c n is a determinable straight line. 



9. Secondly. When the data is all given but the straight 

 lines L K _ I and L^ ; to find positions for these two lines so as to 

 render the problem porismatic. 



Here, if we assume a, in the intersection of the lines o o 



nl n 



and Lp it is evident the corresponding point a _, must be on the 

 line o o , and also on the line o a _ : and therefore it must 



2 n 2 



be (generally) their point of intersection. And it follows that 



