1880.] J. F. Tennant— Standard Weiglts. 51 



^ = 10 Oj + S^Tj + 4a^4 — 2xj^ + 3a;3 + ^1 ± e v^34 from («) 

 j s 10 Oi + 2.r5 + — 3a?3 + 4vr2 + cc{ ± e v/46 from {h) 

 C s 10 Oi + 2^5 + 4^,j^ — SiPg + Sit^o + x{' ± e \/39 from (c) 



which equations give the ascending series ; and it is important to note, that 

 if the probable error of the observations be alike, there is a disadvantage in 

 using any comparison but («), and that even if (h) and (c) be observed as 

 checks, they should not be used in computing, as they will lower the weight 

 of OjQ, on the accuracy of which we are dependent for continuing the up- 

 ward series ; thus the mean value of O-y^ from (a) and (c) will be 



Ojo = 10 Oj + i (4^x^ + 4iX^ — + BiCg + x-^^ + x{') ± e -s/^- 

 and if the series (J)) had been involved the loss of probable accuracy would 

 have been greater. 



Next as to descending/ or decreasinj series from TF^q. 



1st. Descending through (a) 



O, = ^0,0 + fl^Zf^ ± e V'lT 



O,. = ^ + (2^5 + 4.r, — 2x, — 2x, — 4^x,) ± e 



O3 = + ^ (4.r, — 2x., — 4.^3 + X,— Sx,) ± e 



O, = -3^ 0,0 — -jiy (4^, — 2x,. + Gx, — 4^, + 2x,) ± e s/iT 



Oi = To 0,0 — T& (2iP6 + 4<iP,. — 2^3 + H- x^) ± e -s/liT 

 Again descending through (5) 



O5 s -5^0,0 + i(^3 -^lO ± e 



O, = 0,0 + ^0 (2^5 + 4^4. + — — 4.r,0 ± e ViT 



O3 = -rV 0,0 + iV (4^5 — 2^, — ^3 — 2^, — 3a^') ± e 



O, = T% 0,0 — (4^5 — 2.-^.,. + 4.^3 — 2x, + 2*-,') ± e . 



O, = 0,0 — 1^ (2^^% + 4^', —3^3 + 4^', + ^,0 ± e -^/^ 



Also descending through (c) 



.r, + .r., — x" , 



O, = 0,0 + ^ ^ ± e 



O, = ^ 0,0 + (2a;, + 4^.,. + 2.^3 — 2x^ — 4^,") ± e s/'^ 

 O3 = 0,0 + (4^5 — 2.t;, - 0:3 + .r, — 3^,") ± e 

 O, = 0,0 — (4;i^5 — 2:r, + 40:3 — 4.r, + 2.^,") ± e s/ ^ 

 O, = tV 0,0 — To (2^*'5 + 4.-r^ — ^^-^3 + + x^') ± e 

 If we were to be guided here by the same consideration as before we 

 should absolutely prefer the use of series («) alone, but it is easy to see, 

 that as the probable error of 0, involves only Jg- of that of 0,o; the 



