IflS 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[JUNH, 



fg ::= CC X sin T =^ - cosec 6 sin t, 



c c 



and - cosec 9 sin t x sin 6 r=— sin t 1=1 fl; zn ih 

 a a 



EA = (1— - sinr) 



c r -^-e 



tanIEK 



a 



cot'e + sin T 



cosec 9 cos t 



••• (1 ■ 



sin 



.(^ 



c ^r -\-e 



cot- fl -f" sin r 



cosec 6 cos t 



) 



of the tangent at li above E. 

 Equating these we have 



e 



(1 sin 



a 



..(^ 



cot' 8 + sin T 



cosec 6 cos T 



) = 



= the altitude 



cosec e cos T 



Whence we obtain 

 sin T = 



The vahies of t for tlie several cases of obliquity given in my " Essay " 

 are here computed by the formula now given, and for the sake of 

 comparison, the former values are also inserted, as follows : 



By formula now given. 



When e = 65° then t = SS'^ 55' 



55 „ T = 38 (5 



45 „ T = 36 35 



35 „ T = 17 50 



25 40',, T= 



r-\-e 



As before given. 

 27^ 17' 

 25 13 

 21 47 

 15 38 

 



If the ratio ' " were omitted as suggested by B. H. B., the ex- 

 pression would become 



(2-c^-^«-fa) (B) 



Now in this equation whatever may be the value of e, the value of 

 T rem.iins the same, and when the arc is a semicircle 



sin T = _ =r 



a 3-1416 



= -63661 = sin 39° 32' 24". 



This result is in accordance with the speculations of B. H. B., but 

 it is entirely at variance with practice and with correct theory, and so 

 will any formula into which an expression for the thiclcness of the arch 

 does not en'.er. The formula A now given contains it, and will be 

 found correct. By this formula sin t continually approximates to, but 



2 

 never reaches , and vanishes when 6 ^z either 25° 40' or 90° as it 



7r 



ought to do. 



B. H. B. says, " in finding a term for CO, I would reject the thick- 

 ness of the cylinder, and consider the point O as that to which the 

 tangents of the small curves, which show in the face of the arch tend: 

 this is more correct, Ijecause the joints of the voussoirs being segments 

 of curves there can be no point on the face of the arch at wliich a ball 

 would roll down the bed in a line exactly parallel to the face; this 

 may be considered too minute for observation, but besides being more 

 correct it will simplify the question much." 



Here, I beg to observe B. H. B. is again wrong, and for this reason ; 

 these curves of tlie joints in the face of the arch are all in a vertical 

 plane, and if the thickness of the arch be rejected, they must be re- 

 garded as lints merely, and a ball would consequently roll down any 

 one of them, or down the chord of any one of them. 



My investigation proceeds upon the supposition that the chord of 

 tlie small curve forms one side of a triangle, the tangent of the intra- 

 dusal spiral another side, and a line at right angles to the face of the 

 arch, the third side ; this triangle must be supposed to exist in the 

 thickness of the arch, and to be parallel to a tangent plane at the 



point sought, and therefore this is one amongst many reasons 

 why the thickness of tlie arch sliould not be rejected, even if it were 

 attended with the advantage stated by B. II. B., namely, that " it 

 will s.iiiiplify the question much." But instead of simplifying, B. H. B. 

 has |iroduced an equation without explaining how it is obtained, and 

 wliich he has not been able to reduce to a form for direct solution. 



He infers from his equation, "that in all arches of a moderate skew, 

 the point t is about 40^ above the level of the axis of the cylinder;" 

 but 1 have herein shown that when the thickness is omitted, the point 

 is independent of 0, and always 3'J- 32' 24" above the axis. 



Now, although B. H. B., with much complacency, has informed 

 your readers that my last chapter " had better have been omitlcd al- 

 together," I remain of a dift'erent opinion. That chapter commenced 

 as follows. " It will naturally be asked to what extent of obliquity is 

 it safe or practicable to construct an arch on the principles herein 

 given ? This question we will attempt a solution of, or at least to 

 throw some light upon it." How far 1 have succeeded it is for others 

 to decide. I am well aware that the subject is not exhausted, inas- 

 much as I have pursued it further since the publication of the essay, 

 but I have herein confined my remarks to tlie matter contained in 

 B. H. B.'s communication. 



It may be proper to observe, that in all this investigation friction is 

 not taken into account ; but friction is an important element in bridge : 

 building, indeed, no arched bridge of masonry would stand without it- 

 if, then, an expression for friction were to enter into the equation, the 

 value of sin t would be very much diminished. And for this reason, 

 my first equation, as given in the " Essay," though not strictly accu- 

 rate, is practically better than the amended one now given. 



Let B. H. B. take up the subject involving friction in his con- 

 ditions, and he may have an opportunity of rendering considerable 

 service to the engineering profession. 



Your obedient servant, 



Manchester, May, 1840. Geo. W. Buck. 



ON LMESTONE IN IRELAND. 



,/lit Account of the White Limestone which lies along the Coast of the 

 County of Antrim, in Ireland. By William B.\ld, F.R.S.E., M.R.I.A., 

 &c., June 1837. 



What is the white limestone on the Antrim coast? 



It is of the same geological composition and formation as the chalk 

 strata in England; but it possesses a characteristic difference in being 

 of much greater induration than in general the English chalk strata ; 

 the dynamic unit of the force of crushing, and fracturing it by weight 

 may be taken as equivalent to nearly that under which the Scotch 

 Craigleith sandstone moulders into ruin. 



The white limestone lies under the basaltic rock, and in contact 

 with it, it is generally alIo%ved to differ from the chalk of the south of 

 England only in its being of superior induration ; the white limestone 

 assimilates to it in the nature and arrangement of the flints, and organic 

 remains which it contains. The flints as mentioned in a former paper, 

 are dark and grey, some of them of a reddish tint. The large nodules 

 of flint are sometimes from eight to twenty inches long. Organic re- 

 mains occur in the flints; belemnites of the real kind are common, and 

 generally petrified by spar of a calcareous nature and sometimes 

 ammonites. 



The white limestone rests on the mulatto, a rock consisting of grains 

 of sand, with specks of green earth. This mulatto rock corresponds 

 with the green sandstone found under the chalk strata in England ; it 

 also contains fossil remains. 



Under the mulatto rock lies a bluish limestone containing much 

 clay ; this rock is analogous to the lias limestone of England, it abounds 

 in animal remains. 



Under tlie lias or blue limestone are beds of marl containing much 

 clay, and in which are beds of gypsum or sulphate of lime (alabaster), 

 ami the rock underneatli consists of sandstone of a reddisli colour. 



I have now traced the comparison between the strata connected 

 with the white limestone in Ireland, and the chalk strata in England, 

 so as to leave no doubt whatsoever of their entire and perfect identity 

 with each other. Besides, my friend Dr. Smith, the father of English 

 geology, whom I have known for more than twenty-two years, and 

 who has been iu the north of Ireland, and is acquainted with the An- 

 trim limestone, agrees in the description which I have here given of 

 it; further Dr. Smith informed me that the Antrim white limestone 

 was rock of the same formation as Flamborough Head, in England.* 



• " Carbonate of Lime. — Almost all the varieties uf marble and common 

 limestone, together with thuse earthy concretiuns that take place in many 

 natural springs and caverns, as also the numerous class of substances called 



