4

[4]

[5]

COMPOSITION OF FORCES.

[Mec. Cel.

Let us now determine the angle ${\displaystyle \theta }$. If we increase the force x by the differential ${\displaystyle dx}$, without varying the force y, that angle will be diminished by the infinitely small quantity ${\displaystyle d\theta }$ now we may conceive the force ${\displaystyle dx}$ to be resolved into two other forces, the one ${\displaystyle dx'}$ in the direction z, and the other dx" perpendicular to z ; the point M will then be acted upon by the two forces ${\displaystyle z+dx'}$ and dx", perpendicular to each other, and the resultant of these two

forces, which we shall call z will make with ${\displaystyle dx''}$ the angle ${\displaystyle {\frac {\pi }{2}}-d\theta }$ we shall

thus have, by what precedes,

${\displaystyle dx''=z'\cdot \varphi \left({\frac {\pi }{2}}-d\theta \right)}$

consequently the function ${\displaystyle {\frac {\pi }{2}}-d\theta }$ is infinitely small, and of the form ${\displaystyle -\ kd\theta }$,

${\displaystyle k}$ being a constant quantity, independent of the angle 5 ;t we shall therefore have

• (5) The resultant of the forces x, y, is, by hypothesis, in

the direction A Z, and, by increasing the force x by dx, the forces become equal to z in the direction A Z, and dx in the direction A X, and the resulting force z', must evidently fall be- tween A Z, AX, on a. line as A G, forming with AZ an infinitely small angle ZAG, represented by dL Then the force dx, in the direction A X, may be resolved into two forces, the one doif in the direction AZ, the other dec" in the direction AE, and as

this last force is inclined to AX by the angle ${\displaystyle XAE={\frac {\pi }{2}}-d\theta }$, we shall have as above <math>\frac{dx}{z'} = - k d\theta or by substituting the preceding value of (p( 2 — V"^^^' ^~~z~'

f (6) This angle is equal to GAE.== '^—dd ; and if the force z' in the direction A G is resolved into two forces in tiie directions A Z, AE, the last will (by tiie nature of the function cp) be represented hy z' .cp( -^ — dd.

X (7) Because (p r- — d& contains only die quantities ^, <?^, but does not explicitiy con- tain ^. Moreover, the function (p ( 1^ — di being developed m the usual manner, according