The mathematical principles of mechanical philosophy, and their application to elementary mechanics and architecture, but chiefly to the theory of universal gravitation / [John Henry Pratt].
- Pratt, John H. (John Henry), 1809-1871.
- Date:
- 1842
Licence: Public Domain Mark
Credit: The mathematical principles of mechanical philosophy, and their application to elementary mechanics and architecture, but chiefly to the theory of universal gravitation / [John Henry Pratt]. Source: Wellcome Collection.
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![STATICS. SINGLE PARTICLE. represent the magnitude of the resultant, and suppose AR is the direction in which it acts, this line being in the same plane as AP, AQ, and lying between them: let 6 be the angle be¬ tween AP and AR. Draw a line P2AQ2 in the plane of the forces through the point A, and perpendicular to AR. Now let us imagine that P is the resultant of two forces Pj and P2 acting in the directions AR, AP2; and that Q is the resultant of two forces Ql and Q2, acting in the directions AR and AQ2. Then (Art. 14.) R — P\ + Qi . (i), and 0 = P2 — Q2 P1 and P2 are functions of P and 0 ; and Ql and Q2 are similar functions of Q and a — 0. Since P, P]5 P2 are merely the numerical ratios which the corresponding forces bear to the unit of force, and since the relation they bear to one another must manifestly be independent of the unit we choose to adopt, the relation between P and P2 must be of the form P\ P = function of 0 = f(6) suppose; and '** ~ = f(i> 7T We have, then, to determine the form of f(0). mark off' from CD portions equal to these, and let K be the last division, this evi¬ dently falls between D and E; draw GK parallel to AC. Then two forces repre¬ sented by AC, AG have a resultant in the direction AK, because they are commen¬ surable : and this is nearer to AG than the resultant of the forces represented by AC, AB, which is absurd, since AB is greater than AG. In the same manner we may shew that every direction besides AD leads to an absurdity, and therefore the resultant must act along AD, whether the forces be com¬ mensurable or incommensurable. 2. To find the magnitude of the resultant. Let AB, AC be the directions of the given forces, AD that of their resultant: (fig. 4.) take AE opposite to AD, and of such a length as to represent the magnitude of the resultant. Then the forces represented by AB, AC, AE balance each other. Complete the parallelogram BE. Hence AC is in the same straight line with AF: hence FD is a parallelogram : and therefore AE — FB — AD. Or the resultant is represented in magnitude as well as in direction by the diagonal of the parallelogram.](https://iiif.wellcomecollection.org/image/b29286700_0048.jp2/full/800%2C/0/default.jpg)
