NOTES. 



NOTE I., pp. 18, 23. 



THE demonstration of the XXVIIIth Lemma, Principia, lib. I., 

 has been generally admitted to be inconclusive ; there being 

 many curves which can be squared and rectified returning into 

 themselves, and not falling within the exception in the Lemma, 

 of curves having an oval, with infinite branches. Thus the 

 whole of the figures whose equation is 



y m = n m (aj ( "- I)m ) x (" #*) 

 are quadrable when m is an even number ; for 



f y dx = f n x n ~ l (a" a?") 5 " d x 



is integrable, because the power of x without the radical sign 

 is one less than the power within ; and yet the curve can 

 have no asymptote, because there is no divisor ; while it is 



plain that the root of a" x n is impossible when either 



4- x or x is greater than a, n and m being both whole even 

 numbers. Therefore the curve returns into itself; and, as 

 y = both when x = and when x = + a, or a ; therefore 

 the curve consists of two ovals touching at the origin. These 

 are quadrable ; for the integral y d x is 



m m+l 



c- "(*-*) .~. 



n (m + 1) 



