3"0 On the Position of Frames in a Sloping Gallery. [Oct. 



in a steep gallery, sufficient to set aside the application of the general 

 rule ; and if in such a case the frames were placed only so much out of 

 the vertical position, as to resist the effect of the first falling of the 

 fragments, I should conceive it abetter position than if perpendicular to 

 the inclination of the gallery. 



When the perpendicular position can be resorted to, as there is less 

 weight on the frame, and the frame is shorter, the scantling may be 

 less, which is a saving of timber, and the frames are lighter and more 

 convenient for use, so that I am far from objecting to the rule, but cases 

 may occur in which it would be found to fail, and I merely offer these 

 remarks for the consideration of such as take an interest in the sub- 

 ject. 



I am tempted to try your patience a little further by a comparison of 

 the pressure exerted on the frame, with that thrown upon the point p. 

 Let a d, a e, af represent other positions of the frame ; a e being sup- 

 posed horizontal. It will be found that the pressure on the frame is to 

 the pressure on p, universally, as the secant of the angle of elevation or 

 depression of the frame is to the secant of the angle of inclination of 

 the gallery. For, in any position of the frame, as a d, the force on a d is 

 to the force on p, in the direction a p } as a d to d b, or as the sines of their 

 opposite angles, thus : 



Force on ad: force on ap :: ad : db : : sine Lab d : sine Ldab\ but 

 the sines of these angles are equal to the cosines of the angles of eleva- 

 tion (or depression) of the frame and gallery. . * . Force on a d : Force 

 on ap : : cosine elevation (or depression) gallery : cosine elevation (or 

 depression) frame, and as cosines are reciprocally as secants of the 

 same arcs or angles, we have force onad : force on ap : : secant eleva- 

 tion (or depression) frame : secant elevation (or depression) gallery. 

 In the vertical position of the frame, as a b, the secant being infinite, the 

 frame bears the whole weight, and there is no pressure on the point p. 

 If the frame slopes at an angle equal with that of the gallery as a d, 

 the secants being equal, the pressure is equal on each. In the horizon- 

 tal position, as a e, the secant becomes equal to radius, which is the least 

 possible, and here the frame has the least possible pressure compared 

 with that at the same time acting on the point p, and the same rule 

 holds good when the frame comes into the position a /, where the secant 

 of the angle of depression is again greater than radius, and the frame 

 is charged with a greater relative pressure than when horizontal ; and 

 so on, until it comes infinitely near to the same inclination as the gal- 

 lery, when the secants are again equal and the pressures also, but at 

 the same time their power to sustain a vertical force at a ceases, as two 

 equal and opposite forces can have no effect but to neutralize each 

 othei-j and all support is withdrawn. 



Your obedient servant, 



A Borer. 



