FUNCTIONAL DIFFERENTIAL EQUATIONS. 135 



the complete integral. (The latter is, as may readily be seen, 



The demonstration of this proposition is probably familiar 

 to the majority of my readers, and I shall therefore not dwell 

 upon it. Similar considerations apply to equations of higher 

 orders. 



Generalizing the remarks already made, we see that in the 

 equation 



M=fN, 



the function / need not be a given one ; it may be, in any 

 way we please, dependent on the function which, in virtue of this 

 equation, y is of x. In all such cases the equation in question 

 is functional. Nevertheless, Lagrange's reasoning applies as 

 much in these as in other cases. Let us take one or two ex- 

 amples of what has been said. 



The following problem may be proposed. 



Any point P of a certain curve is referred to the axis of x 

 in M, and to that of y in N. MP is produced to Q ; PQ is 

 taken equal to a, and NQ touches the curve. Find its equa- 

 tion. 



Let x, tyx be the co-ordinates of the point where NQ 

 touches the curve. 



a 



and as P is a point in'the curve, 



, or 



(10). 



This is the equation of the problem. Differentiating it, we 

 get 





The former equation gives the complete integral, but, for a 

 reason I shall hereafter notice, leads to no tangible solution of the 

 problem ; the latter corresponds to the singular solution. 



