Another and more elegant theorem, relating to the directions, is, that the directions of all the three com pound forces, if produced both ways, will pass through the same point ; and if the point be within the triangle, they tend directly all to it, or all from it ; but if with out the triangle, two tend directly towards it, and the third from it, or the reverse.
That all three pass through one point, is thus de monstrated: In any figure, from 13 to 18, which ex hibits all the different cases above mentioned, let the force of C be that which enters the triangle, and let the other two HA, QB meet in D. The force be longing to C is also directed by D. Let CV, Cd be the composing forces, and having drawn CD, let VT be parallel to CA, meeting CD in T : if it be shown that it is equal to Cd, the thing is proved, since, by drawing dT, we have dV a parallelogram. Its equa lity will be seen by considering the,ratio of CV to Cd, as compounded of the five ratios CV : BP, BP : PQ; PQ, or BR : AI; AI or HC : Cd. The first, by calling A, B, C masses, of which these are the centres of gravity, is, from the equality of action and reaction, the ratio B : C. The second sin. PQB or ABD, to sin. PBQ or CBD ; the third A :"B ; the fourth sine HAC or CAD, to sine GHA or BAD ; the fifth C : A. The three ratios of the masses compose the ratio B x A XC: B xC x A, a ratio of equality. There remains the ratio sin. A BD X sin. ACD, to sin. CBD x sin. BAD. For sin. ABD and sin. BAD, put AD and BD proportion al to them ; and for sin. CAD and sin. CBD, put sin.ACD X CD sin. BCD x CD and equal to these BD by trigonometry ; and we have the ratio sin. ACD CD : sin. BCD x CD, that is, sin. ACD or CTV (equal to it sine VI, CA are parallel,' to sin. BCD or VCT, or which is the same, the ratio of CV : VT. 'Therefore CV : C d :: CV : VT, or Cd=VT; and therefore CVTD a parallelogram. Q. E. D.
Cor. Should two of the forces be parallel, the third must also be parallel, and the middle one has the op posite direction of the other two.
Cor.. If the directions of two forces be given, the third may be found, being drawn through their point of concourse.
' Let us next compare the magnitudes of the forces —there immediately occurs this theorem : The acce lerating forces of any two masses are, in the ratio, compounded of the direct ratio of the since of the angles, which the line, joining their centres, makes with the lines joining the same centres with the cen tre of the third,—.the inverse ratio of the sines of the
angles, which the directions of the forces make with the same lines joining them to the third,—and the in verse ratio of the masses.
For BQ is to AH as BQ : BR, and BR : AI and AI : AH. The first ratio is that of the sines QRB, or CBA, to the sine BQR, or PBQ, or CBD ; the • second as A : B ; the third sin. IHA, or HAG, or CAD, to the sin. HIA, or CAB : these ratios, changing the order of antecedents and consequeuts, are the ratios of sin. CBA : sin CAB, which is the first direct ratio ; sin. CAD : sin. CBD, which is the second or inverse ratio, and of the mass A to B, which is the third and inverse ratio. The demon stration is the same, if BQ or AH be compared ; and in this demonstration the angles, or their supplements, having the same sines, may be taken indiscriminately.
From this proposition, a number of elegant corol laries are derived ; but as they cannot easily be abridged, we refer our learned readers to the work of the author. We shall only observe, that the properties of the lever, and of the equilibrium of forces acting in the same plane, are derived with facility, independent. of the usual, but unphilosophical, supposition of in flexible connecting destitute of all force but cohesion. With equal ease, he derives the proper ties of the centres of Oscillation, conversion, and per cussion. But ere we take leave of this part of the subject, we cannot refrain from offering to the atten tion of the reader, the solution of the following pro blem, respecting the equilibrium of two masses con nected by two other points, since all that relates to momentum and equilibrium in the lever is compre hended in -it.
Let there be any number of points of matter in A, which call A, and any number in D, which call D. Let points be solicited in the directions AZ, DX, parallel to the given straight line CF, however different may be the forces. Let in C and B two points, which mutually act on each other, and on the points situated in A, B, and by these actions, .hinder all action of 'the forces in A and and all mo tion of the point B ; the motion of C being prevent ed by the contrary action of some fulcrum upon which it•acts, according to the direction compounded of all the forces it has. Required the ratio of the sum of the forces at A and D must have to this, that the equilibrium may exist, and likewise the magnitude and direction of the force exerted on the fulcrum at C.