But in the case of the unequal rafters represented in the figure, the determination becomes more com plicated, and we must first find the direction of the mutual thrust of the rafters, which must evidently be such, that the perpendiculars falling on it from each end of the tie beam may be in the inverse propor tion of the motive powers of the weights of the raft ers, that is, of the products of those weights into the horizontal distances of the centres of gravity from the respective fulcrums, or into the segments of the tie beam made by a vertical line passing through the summit, which are proportional to these distances; and if we produce the base of the triangle, and find in it a point, of which the distance is to the length of the tie beam, as the smaller product to the di rence of the products, a line drawn from the sum mit to this point will show the true direction of the thrust ; and its magnitude may then be readily de termined, by dividing either of the products by the respective perpendicular falling on this line.
Where, however, there is a king post supporting a heavy tie beam, it is necessary to determine the centre of gravity of the half roof, together with this addition; and the distance of the centre of gravity from the middle will then be to the half span, as the weight of one of the rafters with its load is to the weight of the whole roof, including the tie beam and . eieling ; and' if we erect perpendicular passing through the centre of thus found, and equal to the height, the oblique thrust on the abutment will be in the direction of the line joining the upper end of this perpendicular and the end of the tie beam.
l)D. P. 634. In order to obtain a distinct idea of the operation of the forces concerned in this expe Iiinent, we must have recourse to Proposition C of -this• article, and substitute in the formula for the e deflection d — 1 a=1, TA" 16f M a 840' M = 1 900" 000pounds, the specific gravity of fir being .56,f = 6720, and c = 6, the middle of the pillar being considered as the fixed e point : we then find 6f a = 1.427, which.is the M length of an arc of 81° 45°, and the tangent becomes 6.9, we have d 4.2 X 6.g = .0345, or somewhat more than the thirtieth of an inch the strength must have been reduced in the proportion of 1.207 to 1. (Art. BRIDGE, Prop. E.) But, considering how near the arc thus deter mined approaches to a quadrant, it is obvious that any slight variations of the quantities concerned in the calculation must have greatly affected the mag nitude of the tangent : so that the loss of strength may easily have been considerably greater than this, as it appears to have been 'found in the ex pnriment. It would however scarcely been expected that such a pillar, however supported, could withstand the pressure of 90 hundred weight, since Emerson informs us that the cohesive strength of a pillar of fir an inch in diameter is only about 35: but supposing the facts correct, the coincidence tends to show the near approach to equality of the forces of cohesion and lateral adhesion, as explained in the Introduction to this article.