60 REMARKS ON AN ALLEGED PROOF OF 



rally speaking, equal to/(a; 2 )/(/) dxdy ; so that the equation 

 for determining the form of the function, namely, 



is not legitimately established. 



To illustrate this, let TZ (xy) dxdy be the probability that the 

 stone falls on the elementary area in question ; then the con- 

 dition that the probability of a deviation of given magnitude 

 is constant will be expressed by 



cr (xy) = <sr (\^T7 2 . 0) ..................... (A). 



Moreover, we shall plainly have 



and 



and in order that the demonstration may be valid, we must have 



or 



/.+ .+ 



w(xy)dy\ v (xy) dx = r (xy) ............... (B). 



J -oo J -oo 



If this be true, then, and then only, equation (A) may be 

 replaced by 



But in order that (B) may be true, CT (xy) must evidently be 

 the product of two factors ; one of them a function of y only, 

 and the other of #, and the integral of each factor taken be- 

 tween infinite limits must be equal to unity. Combining this 

 conclusion with (A), we find that 



and consequently 



ay) - 



/(**)= A ,- 



Consequently the equation for determining the form of / results 

 from a tacit predetermination of that function. 



