182 
MR. M. W. CROFTON ON THE PROOF OF 
dx dx„ each point corresponding to a compound error, will be 
yy l dx dx l or f{x l )<^{x)d^>, 
dS being the element of the area. Draw through P a line UV equally inclined to the 
axes, then x J r x l is constant along this line; put f;=#+#i=AV, take Vv=d%, and draw 
uv parallel to UV ; take K k=dx, then the number of points within the elementary 
parallelogram PQ pg will be 
f(Z—x)$(x)d% dx. 
Hence the whole number of points between the parallels UV and uv (that is, the number 
of compound errors whose magnitudes lie between \ and % -j- r/|) will be 
dJ~ij f(%—x)<p(x)dx. 
The total number of compound errors thus obtained will be Nn ; however, for uni- 
formity, we will suppose the number of observations taken, affected with the compound 
Error, to be N, the same as for (5). This will oblige us to divide by 
/»a 
n—\ (ft(x)dx. 
Thus if we represent the compound Error by a curve whose coordinates are (|, ;?), it 
will be 
n- 
M-x)${x)dx 
-b 
<p(x)dx 
( 7 ) 
Thus if we wish to find the Error resulting from the combination of the two Errors 
whose equations are 
N (*-<o 2 
:e 91 , 
W 
y=—-7-e & 
J <t vn 
y s \/ 7r l 7 ^ ■ 
we have from formula (7) (N, N' denoting the numbers of observations), 
whence 
]y p ® (|— x— a) 2 (*-p ) 2 
" e ~ cJx ' 
N 
(|-a-p)g 
-Q ' »-+<P 2 . 
V (fl' 2 + <p 2 ) It 
TIence it is easy to see that if any number of Errors of the forms 
N Q-°) 2 N' Q-0) 3 N" o- y) 2 
y=wi e > y=^u e *' > y=w~* e *’ • &c - 
> y=f^ e 
be combined, the resultant Error will be 
N 
y— 
(x -a-p-y -....) 2 
q e 2 +<p 2 +>p+. . . . 
V 7 r(0 2 + 4> 2 + \f/ 2 + ) 
Expanding /(| — x) in formula (7), it becomes 
* =m - *m + \m - hf"® + &c - 
(3) 
