268 PEOFESSOE K. PEAESOX ON THE MATHEMATICAL THEOEY 
Now, (7^, cr^,, cr„ and are all known. Hence the correlation of the absolute 
displacements will be known as soon as we find 
But Xy/X^ = xjxp, 
or, — hxphn^^. 
Hence squaring and summing we find : 
v~ 
+ 
V-x 
-Vp 
- 2t’x ttx, 
X 
Ap 
n I o 
= vv + 
V:r r 
But since X^, = X'^, + ’5, SX^, = SX'^,, whence we have at once ;— 
<^Xp — cr x'p, 
and 
Thus we find :— 
{ n , n 
^’XpX. - 
O-'X', 
+ 
cr-Y' 
-0'x',P~ K'pAx'pXy 
(wx-„+ -5)- (wx'p + ‘5)^ (»jx-„ + -oKwtx'p + -5) ?• • (xviii.) 
'Cx, '^'xp 
Here is given by Table X., vix', ux' and rx-^xv are all entered in Table I., so that 
can be found. Hence from (xvii.) we find ?Vp'.r', the correlation of the absolute 
displacements. 
Substituting the numerical values Ave easily find the folloAving results :— 
Table XIT. 
'A.n — 'Olio 
= -9359 
>'X,X2 = '9358 
/• = -3596 ± -0263 
= -1242 ± -0297 
= -2223 ± -0287 
There are thus seen to l)e substantial correlations between the errors in the absolute 
displacements, not reduced to the length of the bisected line as unit. 
To find the correlations betAveen the relatiA^e displacements not reduced to the 
length of the bisected line as unit, Ave haAm to find first 
(T 
— o-A- + cr 
using Tables XL and XIL, and thence find 
P 1. -’3 , , (j , , 
(^x\x'3 
There results, the standard deviations being in half-inch units : 
