361 



THE CIVIL ENGINEER AND ARCniTECT'S JOURNAL. 



[Septembeb, 



convexity that the rise at the centre was one foot {x = 1). 

 Then, from the formula, it appears that the centrifugal pressure 

 would be one-fourtli tlie weight, or that only three-fourths the 

 weight pressed on the beam. 



It may be remarked that this law lias most important effects on 

 such stupendous structures as the tubular bridges for the Chester 

 »nd Holyhead Railway. The C'onway-bridge, after it was con- 

 Ktructed, sank eiglit inches at tlie centre by its own weight ; this 

 depression was anticipated and corrected by a previous upward 

 convexity of the tul)e. 15ut in this, and all analogous cases, a rise 

 or convexity, considerably exceediny the natural depression, would 

 tend greatly to security : because, the curve being ccmvex, an in- 

 crease of the velocity would diminish, instead of increasing, the 

 pressure of a given load. It may therefore be safely asserted, 

 that it contributes to the security of girders to give their tipper sur- 

 faces as great a convexity as is consistent with other practical require- 

 viaits. 



General Conclusions. 



At the close of these investigations, it may be convenient to 

 recapitulate the general conclusions derived from them. Tliey 

 comprehend the following laws for the motion of very heavy loads 

 at practical velocities over horizontal girders. 



1. If the girder be perfectly elastic, the pressure exceeds the weight 

 on the girder by a fraction of the weight, not more than one- 

 fifth the actual deflection (in parts of a foot), divided by the square 

 of the number of seconds in which either end of the load traverses 

 the girder. 



2. In compound and imperfectly elastic girders this fraction is 

 increased. 



3. The influence of the inertia of the beam on its deflection is in- 

 considerable. 



4.. In all girders, a convexity of their upper surface, or rise of the 

 rails from end to centre, may be made to materially diminish the 

 pressure. 



These conclusions will, it is believed, furnish a tolerably accurate 

 idea of the influence of moving loads upon railway girders. The 

 only subject on which no definite investigation has been here at- 

 tempted, is the defect of elasticity in jointed girders. The modes 

 of construction of compound girders are so numerous, that to es- 

 tablish any general law respecting them is obviously impossible — 

 no accurate knowledge can be derived of the law of elasticity or 

 deflection in these cases, but by direct expei-iment. 



No pains have been spared to render the views here expressed, 

 correct. They have occupied many months of reflection, and 

 have been subjected to the careful revision of the author's ma- 

 thematical friends. As the great object has been to exclude all 

 operose mathematical investigations, it will be readily understood 

 that the subject, by constant corrections and simplifications, has 

 assumed an entirely different shape to that originally given to it. 



ON .MR. CLARKE'S SURVEYING PROBLEM. 



The problem proposed and solved by Mr. Clarke, in last month's 

 Journal, p. 230, is by no means so new as he appears to think. 1 he 

 form under which the problem is most usuaOy presented, is : — 



Given the base and the angles at the base ; to find the perpendicular, 

 and the segments into which it divides the base. 



Mr. Clarke's angles $ and e are the complements of the angles at 

 the base of the triangle CAD, as is obvious, {See his figure, p. 

 230.) 



The following investigation of the question is taken, almost 

 literally, from the 12th edition of " Huttoii's Course" ; and it wjll 

 be at once perceived to be more brief and simple than Mr. Clarke s, 



■B ' D A B '' A O 



Let C D be the perpendieular from the vertex to the base, and 

 denote the angles of the triangle, as usual, by A, B, C, respectively. 

 Then, by right-angled triangles, we have— 



C D cot B + C D cot A = c, or 

 c c si n A sin B _ c sin A sin B 



^^= cot B + cot A ~ sin (A + B) — sin C 



This value of CD is often required in problems of this class, 

 giving (in .Mr. Clarke's illustrative example) the /ion>on(n/ distance 

 of the point of observation from the observed object. It likewise 

 as frequently occurs in determining the height of an object, as a 

 hill, upon a horizontal |dane. 



From substituting the above value of C D in the equations, 

 A D = C D cot A, and B D = C D cot B, we obtain 



. _ c cos A sin B , „ r> c sin A cos B 



A D = 1 — = — , and B D = . — . 



sm C sin C 



I have left theie expressions in sines and cosines, instead of 

 changing the denominator into the factor cosec C. There is no doubt 

 that the better form of working, when C does not contain set^ondt 

 (with the ordinary tables I mean, for surveyors seldom use tables 

 to seconds), is tlie form which Mr. Clarke has adopted : but in tlie 

 other case it is somewhat questional)le. 



As, however, this is a mere question of experience — perhaps, too, 

 of liatiit — every one should adopt the plan he can most easily use. 



This mode of treating such problems is, in fact, the same with 

 finding the co-ordinates of the point of observation, referred to the 

 horizon and the vertical object observed. I have often been led to 

 tliink, that if the greater part of the problems (if not all) which occur 

 in surveying were systematically treated, according to the calculus 

 appropriate to the co-ordinate system, the processes of computa- 

 tion would be considerably improved. Even were the actual work 

 not materially lessened, the systematising of the entire class of pro- 

 blems would be in itself a great practical advantage. 



When, however, we confine ourselves, as Mr. Clarke has done, 

 to finding the difference of levels, a still shorter method of operating 

 may be used, for it requires one reference less to the tables. It 

 may be thus investigated. 



Let M be the middle of A B ; denote A B by 2 a, and M D by * . 

 Then (fig. 2) A D = a; — a, and B D = x -\- a; whence 

 A D tan A =; C D = B D tan B, becomes 

 (x — a) tan A = {x ■\- a) tan B, or 



K — a 

 » -1" a 



tan B 

 tan A 



; or again, a? = — . 



a sin (A -|- B) 

 ^in~(A — B)' 



A corresponding form, adapted to the case represented in (ig. 1, 

 is deducible in the same way ; but further notice of it here is unne- 

 cessary. A formula so simple, and so easily derived, can scarcely 

 be new. Still, I do not recollect to have noticed it elsewhere. 



Another variation of the same general problem is often useful. 

 It is, where the segments of the base A B, and the angle C, are given, to 

 find the perpendicular C D, 



PutDA = a;D B = b; AC B = 2-y; let C D bisect BC A ; 



and denote N C D by e, and C D by x. Then we have 

 BCD = e-|.7, and ACD= + (8 — 7); 

 anda = + a; tan (fl — 7) ; 6 == j; tan (fl -t- 7). 

 Wherefore, 



^ + tan (8 — 7) _ sin( e— 7 ) cos ( 9 -f 7 ) _ ^ sin 2fl — singy 

 b=^t^le + V) ~-cos(e+7)sin(e-7) -sin2e + 8in2y 



a — b . „ 

 and hence, sin 2 9 = + j-qj^ sin 2 7. 



Whence the angle becomes known, and is very easUy com- 

 puted ; and hence the perpendicular C D is obtained from either 

 of the preceding equations, 



X = +acot(e — y),x = 6 cot (fl -f 7)- 

 —But I need not further dilate on so simple a subject. 



UsKlffE. 

 Sth Augutt, 1848. 



