326 Mr. R. L. Ellis on an alleged proof of 



of a deviation in another, whether the two directions are at 

 right angles or not. Some notion of an analogy with the 

 composition of forces probably prevented the reviewer from 

 perceiving that, unless it can be shown that a deviation y oc- 

 curs with the same comparative frequency when x has one 

 value as when it has another, we are not entitled to say that 

 the probability of the concurrence of two deviations x and y 

 is the product of the probabilities of each. Without this sub- 

 sidiary proof, the rest of the demonstration comes to nothing. 

 The conclusion to which it leads is in itself a reductio ad ab- 

 surdum. Of the above written functional equation the solution 

 isf(x 2 ) = e mx2 , m being a constant, so that the probability of 

 an error of the precise magnitude x is a finite quantity; and 

 I need not point out to you that it follows from hence, that 

 the probability of an error whose magnitude lies between any 

 assigned limits is equal to infinity, — a result of which the in- 

 terpretation must be left to the reviewer. He may have 

 thought that the exponential factor is the essential part of the 

 expression 



— -=e- h2 x 2 dx 9 



and that the others might, for the sake of simplicity, be dropt 

 out. But whatever his views may have been, his conclusion 

 is unintelligible. 



The demonstration may, however, be amended so as to 

 avoid this difficulty, and we will suppose that the reviewer 

 meant something different from what he has expressed. Let 

 f(x 2 )dx be the probability of a deviation parallel to the axis 

 of abscissae, of which the magnitude lies between x and x + dx. 

 Then f(y' 2 )dy is similarly the probability of a deviation parallel 

 to the axis of ordinates, and lying between y and y-\- dy. Thus 

 the probability that the stone drops on the elementary area 

 dxd?j, of which the corner next the origin has for its coordi- 

 nates x and y, seems to be fix^^y^dxdy^ and as all devia- 

 tions of equal magnitude are equally probable, this probabi- 

 lity must remain unchanged as long as the sum of the squares 

 of x and y remains the same ; so that we have for determining 

 the unknown function the equation 



/(^)/(/)=/(0)/(* 2 + 2/ 2 ), 

 of which the solution is 



f(x 2 )=Ae mx2 l 

 and as the deviation must of necessity have some magnitude 



