OF TESTIMONIES OR JUDGMENTS. 629 



diminish as the amount of the error increases, requires that h should be negative. 

 We may therefore write, — #*, for h. Whence 



/(4»)±iAe- w ( 4 ) 



To apply this result to the case in which the ball is supposed at some point 

 on the plane which, projected on the axis of x, will fall between x and x + dx, we 

 must give to A the form C Sx. 



Thus we get the expression, 



Ce- k * xS Sx 



Lastly, the certainty that the ball must fall at some point for which the value 

 of x lies between — oc and oo gives us the equation 



L 



Ce- W 3* = l 



whence C—^- = l and C = 7/=-- Thus, the probability of a deviation from the axis 



y to a distance lying between x and % + 8 x will be given by the formula 



le 



V 



= €- k ** 2 8x .... (5) 



7T 



an expression which agrees with (1). 



In like manner, the probability that the ball will deviate to a distance greater 

 then y and less then y + 8y from the axis x will be 



-7=6-** 



V 7T 



whence the probability that it will actually fall upon the elementary area 8x 8y 

 will be 



P 



~ € -^(^+y 3 )8x8y 



Now, this result admits of a remarkable confirmation. For it is manifest that 

 the probability that the ball will fall somewhere between the distances x and 

 x + 8 x from the axis y, ought to be equal to the above expression integrated with 

 respect to y between the limits- oc and oo . But that probability has been already 



k 



determined to be -j= €~ k ^ 8x; we ought therefore to have 



an equation which is actually true. 



Mr Ellis considers this as showing, that the principle from which the demon- 

 stration sets out, viz., that the actual deviation of the ball from the mark may be 

 regarded as a compound event, of which the two independent components are the 

 deviations from the axes, involves either a mistake or a petitio principii. But 

 consistency of results can never be a proof of mistake in the principles from which 

 they are deduced ; and alone, it offers no adequate ground for the suspicion of a 



VOL. XXI. PART IV. 8 G 



