346 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



This equation is one of those which I noticed in a paper in 

 the Philosophical Transactions for 1817, p. 211. I there sta- 

 ted, that it appeared impossible for any function to satisfy the 

 equation, unless it contained a radicle, and unless different 

 roots were taken in different parts of the equation. This expla- 

 nation will perhaps be rendered more satisfactory by the ap- 

 plication to geometry. The general solution of the equation 

 in question, is 



. - , 9 X . \/c % (a. X — x)~ 



<p x -f- 9 a x 



where the upper sign must be used in one part of the equa- 

 tion, and the lower sign in another. From this equation we 

 learn, that for every value of x there are two values of v, 

 equal, but extending in opposite directions ; or that the curve 

 is symmetrical with regard to the axis of the x's : and since we 

 must use different values of the radicle, it appears that the two 

 points B and C cannot be in the same branch of the curve, and 

 on the same side of the axis, but that one point being situated 

 above the axis, the other must be placed in the corresponding 

 branch, which exists below the axis. In fact, it seems to fol- 

 low, from the very nature of the equation (a), that no curve 

 possessing this property can have both the two points B and 

 C situated in the same branch. These considerations render 

 it necessary, in some measure, to limit the generality of the 

 function ? ; and it may be stated, that no form of p is admis- 

 sible, which takes away the double sign placed before the 

 whole of the value of y. 

 Thus, if we take fx=zux + \/c 8 — (a, x — x) 2 , we have 



»=Y +|*/c 8 -(«*-*) 2 



but 



