PROPOSITION 95.

Problem.

856. If a body moving on a surface is acted on by any forces, to define the normal forces, evidently the normal force pressing [on the surface], the deflecting force [in the tangent plane normal to the curve], and the force along the tangent, with all arising from resolution.

Solution.

Whatever the forces acting should be, these can be reduced to three forces, the directions of which are along the three coordinates x, y, z [at M]. Now let the force at M (Fig. 92), drawing the body parallel to the abscissa PA, be equal to E, the force drawing the body parallel to the direction QP to be equal to F, and the force drawing the body along MQ to be equal to G. Hence the forces are each to be resolved in turn into three parts, clearly a normal pressing force, a normal force of deflection [in the plane of the tangent], and a force along the tangent [to the curve at M]. Moreover since these three directions are normal to each other, from each of these forces E, F et G, the normal and tangential forces themselves can be produced, if these are taken by the cosine of the angle that the directions of these forces make with these [other directions]. We may begin with the force along the tangent, the direction of which is MT, for which the equation arises [from Prop. 93] : and

Hence the cosine of the angle QMT, that the direction the force G makes with the force along the tangent, is equal to : [p. 478] If the force G is multiplied by this, then the tangential force due to G produced, equal to

Now the cosine of the angle that MT makes with the direction of the force F, which is parallel to QP , is equal to Hence the tangential force from F that has arisen is equal to Again the cosine of the angle that the direction of the force E makes with MT, is equal to

…

and thus the tangential force arising from E is equal to [Thus, the direction ratios of Mm are ( dx , dy , dz ), and the components E cos α , F cos β ,G cos γ lies along this element, where cos α = dx , etc] dx 2 + dy 2 + dz 2

The normal pressing force is considered (Fig. 91), and the direction of this is along MN, with the equations or

From which it follows that Therefore the cosine of the angle that the direction of the force G makes with MN is MQ

MN 1 (1+ P 2 + Q 2 ) and thus the normal force arising from G is equal to G . 2 2 (1+ P + Q ) Again the cosine of the angle that the direction of the force F, which is parallel to QP, makes with MN, is equal to Hence the normal pressing force that has come from the force F is : − FQ (1+ P 2 + Q 2 ) . And in a like manner the normal pressing force arising from the force E (Fig. 94) is equal to : − EP . (1+ P 2 + Q 2 )

Finally, since the force of deflection is in the direction MG : as then the cosine of the angle that MG makes with the direction of the force G , is equal to [p. 479] [MQ/MG in Fig. 94]:

whereby the deflecting force arising from the force G is equal to :

Again the cosine of the angle that MG makes with the direction of the force F, is equal to: [as PG + EG is the projection of the horizontal component QG on the y-axis] on account of which the deflecting force arising is equal to :

Then the cosine of the angle that the direction of the force E makes with MG is equal to : [-PE is the projection of QG on the x-axis; note that all the positive forces act along the negative directions of their axis. ] Hence the deflecting force arising from the force E is equal to :

Moreover since previously we have called the tangential force T, the normal pressing force M, and the deflecting force N, we have reduced the three proposed forces E, F, and G to these; namely and and Q.E.I.

Corollary 1.

857. Therefore if the body is acted on by three forces E, F and G, on putting v for the height corresponding to the speed at M, then (849), if in place of T there is put the tangential force arising from the resolution of the forces E, F and G. [As this is the only force that does work, as we now understand the physics.]

Corollary 2

858. If in addition, the body is moving in a resistive medium and the resistance at M is equal to R, then (850)

Corollary 3.

859. If, in the equation (851) found, in which the effect of the deflecting force N has been determined, in place of N the deflecting force arising from the resolution of the forces E, F and G is substituted, then there is produced :

Corollary 4.

860. Therefore if the two equations are solved together with the elimination of v, the equation is produced, which joined with the local equation for the surface dz = Pdx + Qdy , determines the path described on the surface by the body.

Corollary 5.

861. Moreover the force by which the surface is pressed along its normal, both by the normal pressing force M as well as the centrifugal force that has arisen, is equal to: (845), on substituting the value found in place of M.

Corollary 6.

862. Now from the equation in corollary 3, it has been found that in which with the value substituted there, the total pressing force is equal to [p. 481] : Scholium.

863. Since the three forces E, F, G have directions normal to each other in turn, the equivalent force from these is equal to ( E 2 + F 2 + G 2 ) . Now we have found the three forces M, N and T to be equivalent to these three forces, the directions of which also are normal to each other in turn ; whereby from these three, the equivalent force is equal to ( M 2 + N 2 + T 2 ) . On account of which, if in place of M, N and T the values found from E, F and G are substituted, there must be produced ( E 2 + F 2 + G 2 ) ; which is the thing to have checked in setting up a calculation. Moreover this serves the part of a criterion, however involved the calculation, by completely resolving the question that the solution has been rightly or wrongly set up. Now from this criterion these formulae are found to be correctly established.