LIEUT. -COL. A. E. CLAEKE ON STANDAEDS OF LENGTH. 
451 
In order to reduce these comparisons, put 
x, =[b c] — [eg] at 61°*25 Fahr. 
x' =[a c] — \_k (/] „ 
y, = expansion of 1 inch of half klafter for 1° Fahr. 
y — 5 ? foot ,, ,, 
Also for convenience put 4(^— y') = 3y ; then the equations of condition take the form 
x,+ oayj— b t = 0, 
x, J r?ja 2 y—b 2 —0, 
x'-\-4:a' 1 y—b\=0, 
x'+id.yj— #2=0, 
which give, finally, a system of equations of the form 
0=31 x' -\-d(a)y— (b), 
0 = 30 a/ + 4 (a')y-(b'), 
0 = 3 (a)x, -j- 4 (a!)orf + { 9 (a 2 ) + 16(a l2 )}y — S(ab) — 4 (a'b 1 ). 
The substitution of the numerical values gives 
x, = + 4T0 . . . with reciprocal of weight^ 0*04457, 
^ = -1*86 ... „ „ 0*04721, 
y =+0*00043 „ „ 0*00003. 
The sum of the squares of the residual errors of the sixty-one equations is 11*295 ; 
hence the probable error of a single comparison is 
+ 0*674 a 9 5 = +0*297 ; 
and the lengths of the two small spaces are therefore, both bars being at the temperature 
of 61°*25, these : — 
[i b c]=[c ^]+4*10 + 0*063, 
[a c\=\Jcg~\ — 1*86 + 0*065. 
We now know the values of the different spaces on the half klafter, namely, by the 
difference of the last two equations, 
[a b~\ = \]c c]— 5*96, 
P<q=Y w +0-17; 
[a«q=Y„+[*c]-5-79. 
and 
adding these together, 
