186 
ME. M. W. CEOFTON ON THE PEOOF OF 
Let us suppose all the infinitesimal simple Errors which it is proposed to combine to 
be successively superposed upon some assumed function of Error y—f{cc) ; then by 
equation (9) the new function arising from the first of them will be, putting D=~, 
If another be now superposed upon this, we shall have 
y= (l-f3D+£ D s ) (l -*D+! D 
and finally the function arising from the superposition of all the given Errors upon the 
assumed Error y=.f{x) will be 
3,= (l- K D + ^D ! )(l-/3D+|D>)(l- y D+^D') . . (13) 
But as a, X are infinitesimals, we have, retaining the square of a, 
Thus (13) will become 
1 _ a D + £ D 2 = 
y — e -^ + l 3 + Y + - • • .)D+i(A- a 2+M-0 2 + - • ■ .) D 
or, adopting the notation (11), 
y = e^ h ~ i)V 'e~ mD f(x) = e i(h ~ i)D f(x — m ). 
9. Let us now take as the assumed function of Error 
. . (14) 
N 
v=A x )=w* e 
(15) 
(where N is the number of observations), and imagine the whole given system of small 
Errors superposed upon it ; the resulting function is 
N 
y $ 7r 
(x—m ) 2 
0 - 
Now by a theorem in the Differential Calculus*, 
1 — k* 2 
e aV>- e - kx 1 '- _ e ~ 1 + 4 ak • 
V 1 +4 ak 
* This theorem, which is new to the present writer, may be proved in various ways. Thus if we put 
u—e a n 2 g— 
and differentiate with regard to a, we have 
du 
again, 
da 
du 
di 
— e aD"j)2 e -kx2 = e aI)2 (4i:V —2k)e~ : 
we thus obtain the partial differential equation 
