BY MARTIN GARDINER, C.E. 85 



The numerous problems which may be formed by di- 

 versifying the data give rise to porisms which may be easily 

 evolved. I will not enter on their investigation in this paper ; 

 but the following porism, comprising a multitude of particular 

 cases, can be easily deduced : 



PORISM 10. 



If there be a closed n'gon, having its 1 st and n 1 th angular 

 points resting on given straight lines L I? L n _ 1} and that the 

 nature of the conditions imposed on the n 2 first sides and angles 

 be such that by forming the open (n %ygo-ns according to these 

 conditions, we shall have the given straight lines Lj and L n _ 1 , 

 divided homographically by their extremities : two straight lines 

 XX, YY, can (generally) be found, such that if we look on the 

 points o and o in which they are cut by the n 1 th and n* sides 

 of the closed tigon as fixed points, and that we " deform " this u'gon 

 so that its sides and angles continue subject to the imposed con- 

 ditions, and that its n 1 th and n* sides continue through, the deter- 

 mined points o and o , then ivill the n th angular point of the 

 n'gon describe a determinable circle passing through o , and o 



tt 1 



(see Porism X. in Transactions of the Royal Society of Victoria 

 for 1859). 



20. From the well-known properties of three pairs of points 

 in one straight line, which are in involution (see " Geometrie 

 Superieure " ) we infer the following theorem : 



THEOREM 4. 



If we can form one closed 2 n'gon whose first n sides and whose 

 second n sides pass successively through n fixed points taken in 

 prescribed order, and whose first n angular points rest in succession 

 on n given straight lines taken in prescribed order : then any point 

 whatever in any of the lines is an answerable position for an angular 

 point of a like closed 2 n'gon ; or which amounts to the same we can 

 deform the 2 rigon subject to the imposed conditions, so that its angular 

 points will move along the n straight lines. 



