| dA1 = - 4 sin θ.dA = - | 4 l | ydA; |
whence
| A1 = - | 4 l | ∫ydA = - | 4 l | Ay, |
where A is the area of the given figure, and y the distance of its mass-centre from the axis XX. But A1 is the area of the second figure F1, which is proportional to the reading of W1. Hence we may say
Ay = C1w1,
where C1 is a constant depending on the dimensions of the instrument. The negative sign in the expression for A1 is got rid of by numbering the wheel W1 the other way round.
Again
dy2 = - 3l cos θ {4 cos² θ - 3} dθ = - 3 {4 cos² θ - 3} dy
| = - 3 | 4 l² | y² - 3 | dy, |
which gives