no 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Apiiii,, 



11 = T- 



d«2 >~ 



(C) 



Which eqiinfinn r/ires the tension at any point of the chain in terms of 

 its curvature and length. 



Before prnceedinsr to show the application of this result to as- 

 certain the tension of a suspension hrid^e, deflected from its posi- 

 tion of equilibrium, it may he remarked tliat the equation applies 

 in all cases where the velocity is zero at every point. The case of 

 actual equililirium ought, therefore, to be included by the equa- 

 tion. And this will be found to be the case: for the curve of 

 equilibrium being the common catenary, we have from the known 

 formulpD for the catenary, m and c being constants — 



rf-T _ mc? 

 T = m{c^-+sf. :. --, - (^,^^,)j 



Also, — = 



r' {c'+s^y 



and T - — 



(C2-1-.S')3 



So that the equation (C) is satisfied. 



The integration of the equation (C) will involve two constants, 

 which are to bo determined by the conditions that as the two ex- 

 tremities of the catenary do not move, the accelerations at those 

 points are zero. To ascertain the tensions at the extreme points, 

 we must recur to the equations (B). Multiplying the first of these 



dy 



rf,r 



equations by . , and the second by ^, putting the accelerations 



di 



equal to zero, and subtracting, we find that at the extremities of 

 the chain, 



jd-y dx d-x dy\ __ _dy 



V 



0=.,g; or,T=..,2 



(D) 



\ds'-ds ds^ds' ""ds' 

 Whence the following simple rule : — 



When a uniform chain fixed at its extremities is in n position of 

 instantaneous rest, the tension at either extremity is to the weiyht of 

 the chain as the lenrjth of the radius of curvature at that point, multi- 

 plied by the cosine of the angle of horizontal depression of the curve at 

 the same point to the length of the chain. 



For example, if the whole chain weigh 500 tons, and the radius 

 of curvature 1)e three times the length of the chain, and tlie angle 

 of horizontal depression at the point of contact = 15°, of which 

 angle -96592 is the cosine, the tension will be 500 tons X 3 X 

 •96592 = 1448-98 tons. 



We will now proceed to determine the value of the chain when 

 the curve of instantaneous rest is a circle, and r therefore a con- 

 stant in equation (C). In this case the complete integral of that 



equation is 



_« « 



T = ce ^ -\- c'i^ ('^) 



where c, c are the two constants of integration. 



When « = let T' be the tension. When s = S the total length 

 of the chain, let T" he the value of the tension. T' and T" may 

 be at once determined by the rule given above. Substituting suc- 

 cessively these values of T in the last equation, we have two equa- 

 tions for determining the two constants ; and substituting their 

 values so determined in the equation for T, the value of the ten- 

 sion at any point will be completely determined. 



If the tvvo extremities of the chain be in the same horizontal 

 line, the values of T' and T" become equal. It may be easily 

 shown that in that case the tension at the lowest point is a maxi- 

 mum or mininn>m. Also at the same point, «=gS. Substituting 

 in equation (E), we have the tension at the lowest point 



= 2T'(e27 + r-v)~ =2T'(e"+« ) 



Where a is angle between radii of curvature meeting respectively 

 the centre and extremity of the arc S. And substituting this 

 value in equation (E), it will be found that 



T = T'(e9-i-j-e)(£a-|-£-o)-i, 



whei-e 6 is the angle between the radii meeting the extremity of * 

 and the centre of S respectively. From this equation the tension 

 of the chain at every point may be immediately determined. 



To illustrate these results by a numerical ex.amplc, let us sup- 

 pose that they are applied to a suspension bridge of which the 

 semi-s])an is 338-25 feet, and the deflection 50 feet, which are the 



dimensions of Hungerford Bridge. By the geometrical properties 

 of a circle, 



{semi-chordy + {versed sine)" (333-25)= + 50= 



radius - 



= llG9feet. 



2 . versed sine 2 x 60 



The tension at the highest point is determined by ecjuation (D). 

 The cosine of the angle of horizontal depression of the curve at 

 that point is equal to (radius — versed sine)-^ radius = ;; .'.g 

 = -95808 = cos 16° 39' nearly. Therefore the value of T in (D) 

 is 1119^3, or the tension .at the highest point is to the total weight 

 of the chain as 1119 feet to the total length of the chain, a =: 

 16° 39' = -290597. Hence by the tables, e" lies between 1-33 and 

 1-34, and e'"" lies between 1-78 and 1-79. It follows that the ten- 

 sion at the lowest point of the chain lies between ^f|T' and ir^T'. 

 When the radius of curvature is variable, the equation (C) must 

 in general be integrated by a series from which the tension at every 

 point may be ascertained with any required degree of accuracy. 

 The tension at the fixed points will be immediately and exactly 

 ascertained from equation (D). 



THE PHILOSOPHY OF NATURE AND ART.* 



(Continued from page 7 7 -J 



There is too little known of Assyria and Babylon to justify us 

 in discussing Mr. Fergusson's remarks, and more particularly as 

 they have been written in anticipation of the publication of Mr. 

 Layard. The subject of Mr. Fergusson's theory of the site of the 

 Temple of Solomon has been very lately before the Institute of 

 British Architects, so that we are exempt likewise from that. We 

 cannot, however, dismiss Phenicia without expressing our surprise 

 tliat Mr. Fei-gusson should assert that the Pheniciaii alphabet is of 

 Pelasgic invention, when the characters and their names are sig- 

 nificant in Hebrew, and some of them have been traced to the 

 Egyptian, as the Eye [o^»i], and the ^V^ater [mew]. The beth, too, 

 is the Egyptian plan of the House. It seems much more reason- 

 able to suppose that some one Phenician or Hebrew, perhaps Mo- 

 ses, availed himself of the Egyptian phonetic system to construct 

 a new alphabet. Perhaps this alphabet had more than one phone- 

 tic for the same sound; and perhaps in writing the pentateuch, 

 pictorial emblems were used for woi-ds, where easily understood. 

 The waving m for Water [^niem'}, the Ox's head for a \_alephl, the 

 Camel for g [gimel^, the House for b [icM], the Hook for v [i'ai']^ 

 the Hand for v [i/ori], the Eye for o [^'lyinl, are all common objects, 

 easily remembered and alliterative in Hebrew, and aiford a curious 

 confirmation of the legend of the Cadmean introduction from Phe- 

 nicia, for the Greeks took the name and the form without under- 

 standing the allusions. Alpha, beta, &c. have no significancy in 

 Greek. The practice of writing from the top to the bottom would 

 very naturally be applied to a mixture of phonetics and hiero- 

 glyphics. The ^ovaTporp-nSoi', or bull-plonghing up-and-down line is 

 only an attempt to get continuity evolved from the other practice 

 of writing in single columns — namely, having gone down one co- 

 lumn, to join on and go up the next. BouffTpo^ijSoK was doubtless 

 written horizontally as well as vertically, and would be more con- 

 venient to read than when written vertically. From horizontal 

 /3ouiTTpu<pi?5oy, the next step would be to writing in horizontal lines, 

 either from the right hand or the left. 



Mr. Fergusson's remarks on the Lycian and Halicarnassian mo- 

 numents include a very ingenious, and as it seems to us very jus- 

 tifiable restoration of the celebrated Mausoleum at Ilalicarnassus, 

 in which he has availed himself of his Indian experience. 



The Third Chapter of the First Part brings us to Greece, a sub- 

 ject of particular interest to our readers. 



The writer takes a very candid view of the influence of Pugin- 

 ism on Greek art. The Pugiiiists have taught the public to ad- 

 mire styles produced in our hitherto proscribed climate by our 

 English race, which we agree with him has done good. Mr. Fer- 

 gusson does not, howevei-, think that the absurd copying of medie- 

 val examples can hold its ground any more than the copying of 

 Greek examples, and in the end the field will he left clear. In the 

 meanwhile, Greek art is left to its own merits and demerits ; not 

 believed in as the sole faith in art, and as the sole vehicle of beauty, 

 but candidly acknowledged in its beauties as in its deficiencies. 

 Thus we shall reach a fair and right standard of criticism for all 

 styles of art. 



* "An Historical Inquiry inlo the True Principles of Beauty in Art, more especially wiiii 

 ref.Tence to Archllecture." By JAMKS FEBGUSSON, Ksq.. Architect, author ol "An 

 Essay on the Ancient Topogrnphy of Jerusalem." " l-.^■lure^que lUustratious of Ancient 

 Architecture in Himlustuu," i'urt the Fust. London; Longuiaus, lS4y. 



