640 DR WALLACE ON A FUNCTIONAL EQUATION. 



as an axis, the hanging rods PQ, &c. will be ordinates to that axis : Let B 

 be the lowest point of the curve, and BC the shortest ordinate : Let CQ, 

 the distance of any ordinate PQ from C be denoted by x, and PQ, the or- 

 dinate, by/(cr) : Similarly, x^ and x^ being any values of x, whose sum is 

 x^+Xj and difference x^-x,, let the ordinates corresponding to these values 

 of .r be denoted by 



and let the shortest ordinate BC, or/(cr=;0), be denoted by a; then shall 



a 



This property is altogether similar to a property of cosines of angles ; also 

 to the lines in an ellipse and an hyperbola, which are analogous to cosines. In 

 the case of angles, it is known that the radius or cosine of zero, being denoted by 

 a, and any two angles by x^ and x, , 



2 cos a^;, cos a^, / . \ . / ^ 



— cos (iT, + X,) + COS {X^ — X,). 



a 

 In the ellipse and hyperbola, if there be four sectors, 



the third and fourth of which are the sum and the difference of the first and se- 

 cond, and if these be contained between the semi-transverse axis and other semi- 

 diameters, from the vertices of which ordinates are drawn to the conjugate axis, 

 these ordinates being expressed by a like notation, viz. 



ord (x^) , ord (x,) , ord (x^ + a-,) , ord (x^ - x,) ; 

 then, in lioth curves, putting a for the semitransverse axis, 



2 ord (x„) ord (x) j / . ^ . j / \ • 



^^-^ 5^-^ = ord (ir„ + xj + ord {x^ — x) .* 



a 



27. From the perfect identity of the relations between the semiordinates of 

 the circle and ellipse, also the hyperbola, and the ordinates of the curve we 

 are now considering, it must follow that all the consequences deducible from the 

 formulse which express the relations of the semiordinates of the conic sections 

 may, without farther investigation, be enunciated as properties of the equili- 

 brated curve. 



Thus, putting n x instead of .r, , and x instead of x,, also y instead of/ (a?) 



or/(Av) ; we have 



^i^-^>fl =/{(« + 1)^1 .,./{(«-l).r}: 



and hence again, 



/{(»» + 



See iny paper in this volume, page 430. 



/{(w + l).x}=-^/(«^)-/{(«-l)j^} 



