Law of Weights

LAW OF WEIGHTS

From   the   method   of   least   squares   the      following   laws   of   weights   are established:                   

(i) The weight of the arithmetic mean of the measurements of unit weight is equal to the number of observations.                  

For example, let an angle A be measured six times, the following being the values:

A       Weight        A       Weight

30 o 20? 8'   1        30 o 20? 10' 1

30 o 20? 10' 1        30 o 20? 9'   1

30 o 20? 7'   1        30 o 20? 10' 1

Arithmetic mean           

          = 30 o 20? + 1/6 (8' + 10' + 7' + 10' + 9' + 10')

= 30 o 20? 9'.

Weight of arithmetic mean = number of observations = 6.

(2) The weight of the weighted arithmetic mean is equal to the sum of the individual weights.

For example, let an angle A be measured six times, the following being the values :

A       Weight                  A                Weight

30 o 20? 8'   2        30 o 20?        10'    3

30 o     20? 10'        3        30 o     20?     9'      4

30 o     20? 6'          2        30 o     20?     10'    2

Sum of weights = 2 + 3 + 2 + 3 + 4 + 2 =16

Arithmetic mean  = 30 o 20? + 1/16 (8'X2 + 10' X3+ 7'X2 + 10'X3 + 9' X4+ 10'X2)

= 30 o 20? 9'.

Weight of arithmetic mean = 16.

(3) The weight of algebric sum of two or more quantities is equal to the reciprocals of the individual weights.

For Example        angle A = 30 o 20? 8', Weight  2

                   B = 15 o 20? 8', Weight  3

Weight of    A + B =     

(4)  If a  quantity  of  given  weight  is        multiplied  by  a  factor,  the  weight  of  the result is obtained by dividing its given weight by the square of the factor.

(5) If a quantity of given weight is divided by a factor, the weight of the result is obtained by multiplying its given weight by the square of the factor.

(6) If a equation is multiplied by its own weight, the weight of the resulting equation is equal to the reciprocal of the weight of the equation.

(7) The weight of the equation remains unchanged, if all the signs of the equation are changed or if the equation is added or subtracted from a constant.