70 GEOMETRICAL RESEARCHES, 



relations in as complete a manner as forms definable by words 

 or equations. 



In addition to the theorems given in the preceding investiga- 

 tion, I would direct attention to the following general theorem. 



THEOREM. 



If a rigid tangent of fixed length perform (as tangent) a 

 'movement round any given complete curvilinear plane figure so that 

 the point of contact continues at the extremity of the tangent ; then 

 if m denote the number of right loops, and n the number of left loops 

 which constitute the given figure, the rectare of the track of the out- 

 ward extremity of the tangent will differ from the rectare of the 

 given figure by (m n) times the area of a circle having a radius 

 equal in length to the tangent. 



NOTE. The porisms evolved in this paper cannot fail to be 

 interesting to geometers who apply themselves to questions in 

 Speculative Astronomy. 



PAPER No. 2. 



Researches concerning n'gons inscribed in other n'gons by 

 MARTIN GARDINER, C.E . 



[Read 9th July, 1862.] 



Given n straight lines L L , L 2 , ...... L^ and the like 



number of points o v o 2 , o n in a plane ; to describe a closed 



n'gon p l p 2 p n p^ which will have its points p v p 2 , .... jp w? 



on the straight lines L p L 2 , L M taken in order, and its 



successive sides p l p 2 , p 2 p 3 , P n P\ P ag sing in order through 



the n respective points o v o , .... o . 



