METHODS OF WEIGHING. 
763 
In the comparisons of weights, I at first employed the method of weighing invented 
by the Pere Amiot, more commonly known as Borda’s*. The counterpoise was in- 
variably placed in the left-hand pan, and the weights to be compared alternately in 
the right-hand pan. The reading of the divided scale was noted at the end of each of 
three consecutive oscillations. One-fourth of (first reading-!- third reading-!-2 second 
reading) was taken as the reading of the scale in the position of equilibrium of the 
balance. 
In all the more important weighings the reading of the scale was noted at the end 
of each of four consecutive oscillations, of which the last three only were used in find- 
ing the reading corresponding to the position of equilibrium of the beam. The first 
reading is apt to exhibit small irregularities, especially when it follows very soon after 
the interchange of the weights. Hence the employment of it in finding the position 
of equilibrium would not be likely to increase the accuracy of the result. The obser- 
vation of an additional reading is not, however, without its use; for by comparing 
the first with the third, as well as the second with the fourth, the error of an integer 
in either of the readings, if it occurred, would be instantly detected. 
Let P, Q be the apparent weights in air of two bodies P, Q, either of which in the 
right-hand pan is nearly in equilibrium with the counterpoise C in the left-hand pan; 
(C, P), (C, Q) the scale readings in the position of equilibrium of the balance when 
P, Q respectively are in the right-hand pan ; and let m be the weight equivalent to 
one part of the scale, the readings increasing with an increase of the weight in the 
right-hand pan. Then Q=P-!-m[(C, Q) — (C, P)]. 
Subsequently I used the method attributed to Gauss'I'. Let P, Q be the apparent 
weights in air of two bodies P, Q, and (P, Q) the reading of the scale in the position 
of equilibrium of the balance, when P is in the left-hand pan, and Q is in the right- 
hand pan. Now let P be placed in the right-hand pan, and Q in the left-hand pan, and 
let P, Q become R, S respectively, by the addition of small weights, in order to bring 
the balance nearly into its former position of equilibrium. Let (S, R) be the read- 
ing of the scale in the position of equilibrium of the balance, when R is in the right- 
hand pan, and S is in the left-hand pan. Then, m being the weight equivalent to one 
division of the scale, the reading increasing with an increase of weight in the right- 
hand pan, Q-|-S=P-f-R-f-7/^[(P, Q) — (S, R)]. 
When the weights P, Q are very nearly equal, the balance may be so adjusted by 
placing a small constant weight in one of the pans or hanging it on the beam, that, 
on interchanging the weights P, Q, the position of equilibrium may still be near the 
middle of the scale. Supposing the balance to be so adjusted, let (P, Q) be the read- 
ing of the scale in the position of equilibrium of the balance, when P is in the left- 
hand pan and Q is in the right-hand pan ; and let (Q, P) be the reading of the scale 
* Peclet, Cours de Physique, p. 48. 
* Steinheie, Denkschriften der K. Akademie der Wissenschaften zu Munchen flir die Jahre 1844—46, 
B. iv. S. 222. 
5 H 
MDCCCLVI. 
