326 
MR. AIRY’S ACCOUNT OF THE BARTON EXPERIMENT 
N 
For convenience, call tiiis fraction x 
Differentiate with respect to ic,, and make the differential coefficient =0 ; 
N 
Sx (2'iCi— ?tJ 2 )~N= 0 , or 
Differentiate with respect to and make the differential coefficient =0 ; 
N 
Sx ( 2 ic 2 — tCg)— N=0, or 
N 
Similarly 
N 
N 
2w„_, — zc„_2— ic„=-g 
« N 
2w^—w^_, =g- 
Let ^=26, the value of h being at present unknown or perhaps arbitrary. Then 
= Wi = Wi 
W 2 =2Wi — 2b —2iv ^ — 2b 
Ws =2?V2 — Wi — 2b=3iv ^ — 6b 
=2ic 3— iCa — 2b — Aw ^ — 126 
{n — \)w^ — (n — \){n — 2)b 
iv„ = nw^ — n{n—\)b. 
Substituting’ the two last in the equation 2ic„ — ^c„_, = 2^, we obtain 
w^=nb, 
and, substituting' this in tlie other expressions, 
W2—{2n— 2)b 
Ws={3n— 6)b 
w^={An — 12)6 
w^=(bn—20)b 
&c. 
of which the law is evident. The second difference is constant, and =—26. 
Substituting these in the expressions for N and S, we find (after all reductions) 
N=— .w .M+ 1 . w+2 
S .w+2. 
N 
and the equation g=26 becomes identical. Therefore b is arbitrary. For conve- 
