78 GEOMETRICAL RESEARCHES, 



10. If, in the investigation to the preceding porism, we 

 consider the particular state of the data in which all the given 

 points , o . . . . o are in one straight line, it is obvious that 

 the intersection of the lines o . o and o a _ becomes in- 



n I n n 4 n a 



determinate, as also that of o n _i an( i \ o However, it is 

 evident that if we fix on any point in the straight line o c> 2 ... , 

 as the point through which L must pass, then there corres- 

 ponds another point in the same line through which L must 

 pass. And we have the following porism : 



PORISM 6. 



If all the sides of a closed n'gon pass through given points 

 which lie in one straight line, and that all its angular points 

 except one move on given straight lines : then will the locus 

 of the free angle be a determinable straight line (see Mulcahy's 

 Geometry, page 75). 



11. Porisms 5 and can be easily derived from those which 

 precede them by the usual method of reciprocation ; and other 

 particular theorems and porisms can be deduced from these, 

 &c. However, I will not enter more into details in the present 

 paper, as my chief object is to get at the more general relations. 



12. Thirdly. Given all the data but the point o and the line L 

 to find positions for these which will render the problem poris- 

 matic. 



Let a 1 a 2 a n _ l a^ and & x & 2 b n _ : b^ be the two 



known closed (n l)'gons, having their sides passing in the 

 prescribed manner through the n 1 given points, and their 

 angular points on the n 1 given straight lines. If we 

 put b for the point in which the straight line & w _ 1 b 1 o n _ l cuts 

 L/ n ; then evidently (p n having any position not in w _ 1 b^ f| _ 1 ) 

 we must have b n coincident with 6 , or, in other words, we must 

 have L passing through 6 , the first angular point of one of the 



