352 ON THE DISCOVERY OF LOCAL THEOREMS AND FORISMS. 



note (CD) at one of these points below the axis, until the part 

 below (DF) is equal to the abscissa corresponding to the other point 

 ( AE), then the locus of the extremity of the ordinate (F) thus 

 produced, is always a periodic curve of the second order. Fig. 1 1. 



The questions which we have now considered, appear to 

 me sufficient to point out the nature of that connection be- 

 tween the theory of functions and that of curves, which it was 

 my object to establish. The difference between the properties 

 thus brought to light, and those which have been hitherto 

 known, seems to consist chiefly in two points. The first is, 

 that the families of curves to which they relate are larger : this 

 arises from the arbitrary functions necessarily introduced into 

 the solution of functional equations. The other difference is, 

 that the properties discovered relate to many points of the 

 same curve, only connected by some given law. In the first 

 respect, they in some measure approach to the investigations of 

 M. MoNGE, in his excellent work U Application de V Analyse a 

 la Geometne ; whilst, in the second respect, they bear some 

 analogy to the more general properties of curves, deduced 

 from the theory of equations in the Proprietates Curvarum of 

 Waring : these resemblances are, however, but superficial. 

 The nature of the questions we have considered requires, by 

 the usual methods of analysis, the application of mixed differ- 

 ences ; and, in most of the few instances in which any such 

 problems have been proposed, they have been attempted by 

 that method. 



Devonshire Street, Portland Place, 

 July 1. 1818. 



I 



XXV. 



