340 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



ply some name to the series of ordinates thus found : I shall 

 therefore call them Corresponding Ordinates. These curves 

 may be divided into orders, according to the value of the in- 

 dex n\ if n zz 1, the first order consists of only one curve, 

 namely, y zz -^ x zz x t which is the equation of a right line, 

 making an angle of 45° with the axis. 



When n zz 2, and ^ is determined from -^ * x zz x, 



yzz^x 

 represents all periodic curves of the second order, and so on. 



The equation -^ n x zz. x has been solved by me in the Philoso- 

 phical Transactions for 1816, and numerous examples are gi- 

 ven in another paper, published in the following year. This 

 solution has, however, been much improved by Mr Horner 

 (see Annals of Philosophy, October 1817), who has shewn that 

 the algebraic equation to which my method of solution leads 

 in all cases, admits of a ready solution. The form which is thus 

 assigned to the function -^ is 



a + bx 



b* — 2 b c cos 1- c s 



■^ x zz <p 1 A c 



2 (1 + cos — J a 



X 



Amongst those included in the second order of periodic 

 curves, will be found the right-angled hyperbola referred to 

 co-ordinates parallel to its assymptotes, and also circles of all 

 orders, where equations are x n + y n zza n \ these curves pos- 

 sess many singular properties. All those in which n is even are 

 re-entering curves, without any infinite branches, whose form is 

 nearly that of the subjoined figure, Fig. 2. In this curve, if 

 we take any abscissa, and ordinate CD and BD, and if we turn 

 the triangle CBD, into such a position that B shall coincide 

 with the axis CF, then the point C will coincide with G some 



point 



