OF TESTIMONIES OR JUDGMENTS. 629 
diminish as the amount of the error increases, requires that / should be negative. 
We may therefore write,—/’, for h. Whence — 
f(a)=A err , é F : (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 2, will fall between 2 and +02, we 
must give to A the form C dz. 
Thus we get the expression, 
Ce-** Sz 
Lastly, the certainty that the ball must fall at some point for which the value 
of a lies between —ac and o gives us the equation 
vm Claman — i 
whence al =1 and C= Te Thus, the probability of a deviation from the axis 
y to a distance lying between w and «+6 a will be given by the formula 
k 2 
mae SO ee nO (5) 
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+6y from the axis x will be 
aaah — Ky? 
7 
whence the probability that it will actually fall upon the elementary area bx dy 
will be 
ke 2 
dana +) da dy 
Now, this result admits of a remarkable confirmation. For it is manifest that 
the probability that the ball will fall somewhere between the distances 2 and 
+0 from the axis y, ought to be equal to the above expression integrated with 
respect to y between thelimits—cc anda. But that probability has been already 
determined to be o e~** dx; we ought therefore to have 
Hav (= ere ay by= fm ong Ox 3 : (6) 
an equation which is actually true. 
Mr Exuis 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 Iy. 8G 
