52 



J. F. Texm^nt— On Slamhrd Wcir/lits. 



iNo. I, 



determination of its weight will be almost entirely dejiendent on the error 

 generated in the comparisons of the group* of the series, and not on that 

 derived from the starting weight : this renders the choice less important. 



As a matter of fact I have worked both through («) and {h) taking 

 the meau result and in this case. 



O5 = IT) 0x0 + i (^3 + ^2 + *\ + ^lO + e -v/fl" 



O3 = + 2V (8^5 —^'^-1. — 5^3 — x^ — ^x^—^x^') + e 



= T%0,o — J^(8^,— 4^, +I0x^ — Qx^ +2x^ +2x/) ± e v^ff" 



My choice was a matter of accident, but it turns out that the smn of 

 the squares of the j^i'obable errors of all the deduced weights is less than for 

 any one of the single series. 



The other system of weights, which I have in this paper slightly to 

 deal with, is what I shall call the " English grain system." In it the weiglits 

 interpolated between 10 and 1 are 6, 3 and 2. Thus starting from either 

 end of the decad there are four weights to be derived ; but among these 

 "weights alone, only three equations can be obtained. 



P,, = P, + P3 + P, + 

 P, = P3 + P^ + P^ + X, 



P3 = Pg + Pi + ^3- 



To make a definite resect the best plan is to use a second P^ called 

 Pj': P. 1 + P. 3 + P.j from the next lower decad height be used but the 

 equations would not be independent for the separate decads. 



P, = Pi + P/ + and P, = P/ + X, 

 and we now have 5 equations to determine 5 quantities, and the result is 

 definite. Of course by substituting P\ for P^ , we can get 3 more equations 

 like the first three, but the labour would be increased, and the result would 

 still be definite, though slightly more accurate, especially as regards the 

 spare weight P^'. 



From the equations we have ; in ascending (increasing weights) 

 Pi'=Pi-x, ±e. _ 

 P2 = 2 P, — a,-, + ± e s/2 

 P3 = 3 Pi — X, + ic^, + X3 ± e v/3 



• I use the term decad to include tlie -weights from O'l to 1, or from 1 to 10, &c., 

 the last being ten times the first ; and a groiq) of equations consists of those connecting 

 the weights of a decad. 



