NEWTON S PRINCIPIA. 73 



being touched by branches of infinite extent. But it does 

 not even apply to all cases of ovals returning into them 

 selves, and unconnected with any infinite branches. There 

 is, for example, a large class of curves of many orders, 

 those whose equation is y m = ri n x (n ~ l ^ m x (a n x n ) ; and 

 when m is even these curves are quadrable ; and in 

 every case where m and n are whole positive even num 

 bers, it is the equation to a curve returning into itself. 

 This is manifest upon inspection : forfy d x =^fn x n ~ l 



(a n x n )* d x is integrable because the power of x with 

 out is one less than that of x within the radical sign ; and 

 because there is no divisor there can be no asymptote; 



while it is plain that the - - root of a n x n is impossible 



when either -\-x or x is greater than a, n and m being 

 both whole even numbers. Wherefore the curve re 

 turns into itself; and as ?/ = 0, both when x = 0, and 

 when x = + #, or a, therefore the figure consists of two 

 ovals meeting or touching in the origin of the abscissae. 

 These two ovals admit of a perfect quadrature; the in- 



m 



m + 1 



tegral being C - n(m + l \ (&quot; ~ *&quot; ) m Tn &quot;s if 



m = n 2 the area is C J (a 2 ^ 2 ) |, the latter 

 quantity being one-half of an area that has to one-third the 

 rectangle of the co-ordinates the same proportion which the 

 difference of the squares of the diameter and abscissa has 



J5 



to the square of the abscissa ; for f (a 2 x 2 ) 2 = ^ x y x 

 2 -* 2 



The particular inquiry respecting motion in the ellipse 

 did not perhaps require the proposition to be proved in 

 the very general form in which Sir Isaac Newton has 



