302 Mr. T. Tate on the Construction of 



ing the resulting equality for Lf,_<, we get 



x{l-(^)f(i-^)}. • (5) 



Neglecting (t i — t)'p (-, — oi) as being very smali compared 

 with 1, we have 



where h tl _ t is proportional to t x — t. This expression is an ex- 

 ceedingly near approximation to the true formula. 



Let the space L^_j, expressed by equation (5), be divided into 

 / 1 — t equal parts, then each part will be the length of one 

 degree on the scale, that is, 



Length of 1 degree on the scale rsL^x A 



.\ Length of t 9 degrees on the scale =L^_f x a . 



But the true length of a degree will be derived from equa- 

 tion (5) by making f f — tss 1, that is, 



True length of 1 degree =L 2 , and 



True length of / 2 degrees =s L, 3 ; 



."'. Error in t 2 degrees = L, 2 — L^* x j^—.- 



t\ — i 



Putting arNfe the error in t^ degrees taken as a proportional 

 part. of the length of one degree on the scale; then 



a' = 



T 1 



^■^vf6^) (7) 



This expression gives the correction for any temperature t + t q 

 or t — t 2 , that is, for any temperature t 2 degrees above or below /. 



The only dimensions of the instrument involved in this ex- 

 pression are the lengths of the columns A K B and B C. 



