674 
ME. AIEY’S ACCOST Or THE COXSTEUCTIOX OF 
(and consequently — y the increase when Bronze 28 is Inside), y being independent of the 
Observer and of the Bar. 
Then the equations formed from the means of Mr. Sheepshaaks’s two sets of compa- 
risons of Bronze 10 with Bronze 28 are, — 
— 1'382 
ir2--^= — 1T31, 
E E 
these equations being subject to th.e probable errors 
Or ' ^/78..2■2+\/78.?/:=-~^/78x 1-382 
\/ 64.^2“'%/ 64.?/= --Ay64x 1-131, 
where each equation is subject to the probable error E. In this shape, the equations 
are fit for use in determining the most probable values of &c., and y. 
But we shall afterwards have need to determine the probable errors of the values of 
X 2 , ^4, &c.,;?/, so found. For this purpose, it will be most convenient to introduce here the 
consideration of the actual error in each equation. Let then p,, e .2 be the actual eiTors 
in Mr. Sheepshanks’s first and second mean result. Then the follo’^ving equations are 
strictly correct : — 
-s/78 . ^2 +\/^ -2 /= ” n/ 78 X 1 - 382 +x/T8 . c, 
-s/M -\/64 X 1-131 +-s/M. C2 
and similar equations for the other observers and the other bars. The first and last pairs 
of equations for the comparison of Bronze 10 -with Bronze 7 T\fill be — 
x/ 28.^2 — \/28..r44+s/28. 3 /= — s/28 X 2-711 +-s/28x6'5i 
\/ 24.^2 ■~\/24.^44— s/24.?/=— s/24x2-142-f \/24 x^52 
\/8 .^10— \/8 .^46+\/8 -2/=— \/8 x0-350+>/8 x^ss 
s /8 .^, 0 — \/8 . A46 — x/8 .?/=— ^8 xl - 550 - f ^8 x ^ si - 
And from all this assemblage of equations, new equations equal in number to the num- 
ber of unknown quantities are to be formed according to the rules of the Calculus of 
Probabilities. 
The equations are for the most part very simple, those only which relate to x. 2 . x^^. 
^46? and y, presenting any complexity. I shall only give details of the following, as 
specimens of the form and the treatment of the whole. 
By solution of the third and fourth equations, x^ is found = — l-78-|--5(?3+^C4. This 
expression is strictly accurate, but it cannot be used immediately because e.^ and (the 
E E 
actual errors) are not known. But the probable values of e.^ and e^ are — : the 
E E 
probable values of ^e^ and ^e^ are therefore = and — ; and therefore (by the theo- 
V 80 V 80 
rems of Probabilities) the probable value of the combination of these is 
