1840.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



271 



S, Z, which equilibrate each other. It follows, the resultant of two 

 or more of these forces must be equal and directly opposed to the re- 

 sultant of all the others, and if, therefore, A B, F E, be produced and 

 intersect at O, the resultant of the forces F, Q, R, S will pass through 

 the same point. Consequently, if the resultant be represented in mag- 

 nitude and direction bv the line O T, and a parallelogram be constructed, 

 whose diagonal is this line, and whose sides, N O, M O, are drawn in 

 the directions of B A, E F ; NO, M O, will represent the directions 

 and magnitudes of the forces to which the extreme points of the sys- 

 tem, A and F, are subject. To proceed now to the case where P, Q, 

 R, S, are parallel, fig. 1 . The proportions we have before obtained 

 will obviously apply here also, but in this case the supposition of equi- 

 librium involves another condition, which was not before essential ; all 

 the forces must be situated in the same plane. For, as three forces 

 are in equilibrium around the point B, they will necessarily be situated 

 in the same plane, and the same can be asserted of C, D, and E ; but 

 B P, C Q, being parallel, B P, B C, and C Q are in the same plane 

 (Euc. 7, 11) ; and consequently, all the forces acting at B and C are in 

 one plane. By extending this reasoning to the points D and E, we 

 observe that all the forces of the system will be situated in the same 

 plane. Referring to the proportions already established, and remark- 

 ing that sin. P B C ^ sin. B C Q ; sin. Q C D = sin. R D C, S;c. we 

 have : — 



V : W : ; sin. B C Q : sin. A B P 

 W : X : : sin. R D C : sin. B C Q. 



And so of Y and Z. From this it appears, that when a number of 

 parallel forces act upon points flexibly connected, the forces developed 

 in the directions of the connecting lines, are inversely as the sines of 

 the angles made by these lines witli the parallel forces. These forces 

 are therefore inversely as the cosines of the angle made by the sides 

 with lines at right angles to the directions of the parallel forces; or 

 denoting the angle by if>, and calling I the force thus developed ; 



1 



t a a sec. <f) 



cos. ip 



When B C is at right angles to the parallel forces, we obtain the 

 relation of the force acting in tlie direction C B to the force acting in 

 the direction E F, by supposing, as before, that the intermediate lines 

 C D, D E, have become rigid. B C and F E being produced will in- 

 tersect at O, through which will pass the resultant of Q, R, S, equal to 

 their sum and parallel in direction. Let tliis be called m, and denote 

 by a the force acting in C B ; then t being the force in E F, and cf tlie 

 Z made by its direction with the direction of a, we have 



1 



I : a : 



cos. (f> ; 



And I \ n :\ \ : sin. <f, 



.*. w ■=. <sin. (f!. 

 It is also evident from these proportions, that 



n : a : : sin. <f ; COS. ifi; from which m ■=. a 



sin. <P 



COS. l(> 



.'. n =1 a tan (p. 



In order to compare the forces P, Q, R, S, let the angles formed by 

 A B, B C, C D, fig. 8, with lines at right angles to the directions of the 

 forces be called a, jS, y, 5. If therefore A B be produced, the / C B b ^ 

 o — $, and in the same manner Z DCc=:fl — 7. Adopting this nota- 

 tion, we have these proportions: — 



P ; W : : sin. ABC (sin C B 6) : sin. P B A 



: : sin. (a— (8) : cos. a 



P sin. o cos. |8 — cos. o sin. $ ^ ,^ , „n 



— = cos. ;8 (tan o — tan 0) 



' W cos. a. 



Again, W ; Q* : : sin. Q C D 



: : COS. 7 



w 



cos. 7 



cos 7 



sin. BCD (sin. D C e) 

 sin. (fi — 7) 



1 



sin./3 — 7 sin. J3 cos 7 — cosj3sin7 cos.j8(tan5 — tan7) 

 Finally, by multiplying these equations we have : — 



P tan. o — tan. p 



Q tan. j8 — tan. 7' 

 The other forces will be found to be related in a similar manner. Let 

 C D become perpendicular to C Q or D R, then tan 7 = Q, and P : 

 Q : : tan. a — tan. p ; tan. fi ; also, P + Q : Q : : tan. a ; tan 8. 



* See WliewcU's Elements of Mei.hanics. 



The principles we have now been considering have been established 

 with regard to a polygon, acted on by given forces, but they may re- 

 ceive a more extended application, by imagining that the equal sides 

 of the polygon become continually diminished until they are less than 

 any assignable quantity, when, it is evident, we obtain a cane, or in 

 other words, a polygon, the number of whose sides is infinite. This 

 curve will vary in its nature, according to the magnitude and position 

 o the forces by which it is generated ; if, for examiile, the forces be 

 iqual, and radialt from the centre of the ordinate, the curve will be a. 

 semi-circle ; if the forces are parallel, equal, and equally distributed 

 along the cam, we obtain the catenary, and if, while equal' and parallel, 

 they are equally distributed along the 0/-duiaie, the parabola is the 

 curve produced. The nature of the forces employed in the production 

 of the serai-circle has already been shown ; and with respect to the 

 catenary, it is clear that this curve being defined, as the form which a 

 flexible thread or chain assumes when freely suspended from its ex- 

 tremities, we shall obtain the same curve, if we replace the forces of 

 gravhy by others which are equal and parallel, whether their magni- 

 tudes be less or greater than the forces they have supplanted. The 

 pi eduction of a parabola by equal and parallel forces uniformily dis- 

 tributed along the ordinate. I have succeeded in proving in the fol- 

 lowing manner : — In the first place, it is clear from what has been said, 

 that whatever be the form of the curve, if we denote by w the sum of 

 the forces acting upon the arc included between the vertex and a given 

 point, and denominate (p the angle farmed by the tangent and ordinate, 

 IV OC tan. (p. If then we assume PAR, fig. 5, to be a parabola gene- 

 rated by the action of parallel forces, we have — • 



NT 2x 



tan.^^.^=- 



But y- 



- pj; or X 



P 



, and by substitution, 



tan. ^ =: 



P 



2i/ 

 Hence, w d — O. y. 



r 



An attempt has thus been made to exhibit in the most simple and 

 intelligible form, some elementary principles, which must tend to sys- 

 teraize and illuminate our ideas upon the nature and mode of action of 

 the several forces to which a suspension bridge is subject. In the 

 composition of this paper, I am much indebted to a chapter in Poinsot's 

 "Traite de 8tatique-," but a somewhat different view of the subject 

 has here been taken, and some new matter has also been added, which 

 it is hoped will not be thought uninteresting. 



CANDIDUS'S NOTE-BOOK, 

 FASCICULUS XVIL 



" I must have litjerty 

 AVithal, as large a cliarter as the winds. 

 To blow on whom 1 please." 



I. Much as has been said and written about styles of architecture 

 the Consumptive Gothic has hitherto escaped notice, and consequently 

 animadversion. This must not be confounded with so-called Carpen- 

 ter's Gothic; for it is frequently correct as to outline, but nevertheless 

 quite otherwise as to execution, the mouldings and details being 

 terribly attenuated, whereby a disagreeable meagreness and insipidity 

 take the place of relief and boldness,and instead of appearing carved, the 

 ornaments look as if they had been stamped with a butter-print. Al- 

 though its design may be exact as to mere pattern, yet if its mullions 

 and transoms be pared away, as not unfrequently happens, to about half 

 their due proportions, as regards the spaces between the former, a 

 Gothic window becomes deficient in that w hich gives character to one- 

 Nor is it a little strange that while architects affect as they do, to be 

 scandalized at the slightest deviations from the proportions of Greek, 

 and Roman columns, thev make no scruple whatever of deviating alto- 

 gether from those proportions upon which the effect of Gothic archi- 

 tecture very materially depends; but because greater latitude and 

 freedom are allowable 'in that style, with regard to composition, con- 

 sider themselves at libeity to disregard what may fairly be called its 

 constitutional principles. 



II. Now that Brummagem silver, and other Brummagem productions, 

 are distinguished by the name of ' Victoria,'— which, by the bye, is a. 



2 2 



