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Transactions Tennessee Academy of Science 



fixed, and P or C describing the curve; and in the second case (nega- 

 tive cell) , we attach the free end of the extra link to or 0', C or P 

 being fixed, and 0' or describing the curve. 



By having two fulcra instead of one, the cell is made more com- 

 plete, as sometimes we are thus enabled to describe more easily cer- 

 tain discontinuous curves. As an example of this use of the quad- 

 ruple cell, Mr. Hart, in the Messenger of Mathematics for 1875, 

 gives the construction of the Ovals of Cassini, a special case of which 

 is the famous Lemniscate of Bernoulli. 



The consideration of this quadruple cell leads us to an inven- 

 tion of Mr. Hart which gives us a five-link straight line motion in- 

 stead of the seven-link motion of Peaucellier. I will close my paper 

 by giving a brief description of this apparatus. 



If we take an ordinary parallelogrammatic linkwork, in which 

 the adjacent sides are unequal, and cross the links so as to form 

 what is called a contra-parallelogram. Fig. 8, and then take four 

 points on the four links dividing the distances between the pivots in 

 the same proportion, those four points have the same properties as 

 the four points of the double cell (what we called the quadruple 

 A B 



D E 



cell I . That the four points always lie in a straight line is seen thus: 

 considering the triangle ADE, since 



AO : on :: AP : PE 

 therefore 01* is paiallcl lo Dl^. and ihe peipcndirular distance be- 

 tween the parallels is lo the height of the triangle ADE as OD is to 

 AD; the same reasoning a|)plies to the straight line CO', and since 

 AD : OD : : BE : O'E, and the heights of the triangles ADE, BDE, 



