79 



been described from a centre (if we may so speak) not within 

 itself. This property of Peaucellier's invention has been put 

 to practical use. 



The curve which D describes when c is fixed and d is moved 

 is a curious one (fig. IV.) ad infinitum in each direction, always 

 approaching but never reaching a straight line at right angles 

 to the axis of the curve, the axis being 2 C. The figure is under 

 the curve. In the negative cell its inclination is reversed. The 

 conjugate is 



a 2 -b* 



2C 



If, in the positive cell, the transverse and conjugate be made 

 equal, the two arms of the curve will meet in a cusp ; and if the 

 conjugate be the less, they will form a loop. I have been unable 

 to find an expression for the ordinate without using a variable. 

 The law of the curve is : — 

 Abscissa : axis : : ordinate 2 : ordinate 2 -f (abscissa + conju- 

 gate—transverse axis) 2 .* 

 It remains to describe the five-linked cell. If two pairs of 

 bars (a a and b b) cross one another as in fig. V., at whatever 

 angle they may be, the product of the distances AB and CD is 



*The ordinate, using t as a variable, is 



V(2Q 2 — eXa 2 ~i> 2 -e 2 ) ; 

 zee 

 the abscissa 



(zciW) 



2C 



squaring the ordinate and dividing, we obtain 



( fl 2_ p— e 2 f ord 2 



2« 2 abs 



which is equal to 



diag of rhomb 2 

 axis 

 But an examination of the figure (VII.) will show that the diagonal of the rhombus 

 is equal to the square root of the square of the ordinate -f- the square of the abscissa -f- 

 conjugate — transverse. If the conjugate be less than the axis of the curve — that is 

 if D be nearer to A than C is, this last quantity may become minus j but this does not 

 affect the law of the curve, as it must be squared. 



