BUI 



6ome authors, who give live or six places of de. 

 tor u i-lered that no cotiini >n rrn 



suring cither diitancrs, angles, or weights, can pro- 

 to any tiling m'.ir that nu-i-ty. Y.'t, that we 

 may not ivot satisfied with an approxirrat:oii without 

 shewing what degree of accuracy can be obtained, and 

 espfcuuiy that we may render this mode of conceiv- 

 ing the subject more useful by a more complete so- 

 i of the problem, we proceed to the following 



analytical investigation. 



We have aln-.i.ly shewn that the weights of the 

 eciious must bi proportional to the differences of 

 tbv t ui) nts of the successive angles of abutment. 



This is to be provided for, 



I.) OK. 



I it, By the weight of the arch-stoncs ; here tak..-n 

 as con .taut. 



2d, liy tlu- weight of matter forming tin- roadway, 

 &c. ; here taken as of iinil'orm thickness and varying 

 in effect only as ihe dill'-rcncc of the sines of the 

 ,ccs from the crown. 



3d, By the matter in the spandrel ; which may be 

 made to vary in density, and is rqual in the longi- 

 tudinal section to the versed sine multiplied into the 

 difference of the sines. 



Take: z, the angular distance from the vertex, 3 

 the density in the spandrel, the thickness of the 

 arch or keystone, r the thickness of the road, &c. at 

 the crown. 



Theory. 



The sections are = ( + r) flux. tan. z = (4-r) v 

 The road is =:rX flux, sine z=rz cos. z, wherefore 

 (-|-r)(l+tan. 1 z)zz=nz fur the archstone 

 -J-r cos. z-c for the road 

 -f 3 X cos. z z x V sin. z for the spandrel 

 that is n + r-f f'z(n4- >-)=-(-> cos. z-f- 3 coi zx V sinez 

 and /'z(i7-f-r)4-r V sine z=3 cos. zX V sine z 



t in. * r S s r sec. X tan.' . 



3= if-^-(" + '') + or rw~^ ( a + r ) H or T^ ("+ r ) + r X sec. x 



cos. XV s'n. cos. z cos. 3 V sine cos. V sine 



Which may be thus expresed : 



/;/ an an /i of uniform f/iickni'ss. with a horizontal 

 rnadif: 'he t/iic.difAsrx o/' t'.e arch and road- 



wa:/ ; required lite ncnxifi/ in every pml nf the si and- 

 / if t.'/ew/iu/e may he preserved in cquil'brlo. 



To til-ice tlic log. tangent of thf angul ir . xtance 

 from / it:! / the log. secant, a 'id xubtruct !'>c 



log. versed sine ; take tJie curtespondffig number, and 

 mul:i: (i/ ':>/ the thickness ofcroirn; and arid :o //;.'- the 

 secant multiplied by the thickness of roadway. These, 



being expressed in tcrm^ of the radius, the rcsnUing 

 number gives the density in the tpandrei, or propor- 

 tion which the solid matter in measuring transversely 

 across the arch bears to the whole breadth at the 

 crown. 



Accordingly, we have constructed the following 

 short Table from this formula. The first line shews 

 the multiplier for the thickness at the vertex. The 

 second shews that of the roadway ; and is merely 

 the table of natural secants. 



As an example of the use of this Table, let us take the thickness at the crown =TS "f radius ; and upon the 

 Hum that the roadway, &c. is as thick as the archstones at the crown, we hare, by multiplying as 

 above directed, the following densities, viz. 



And if these densities be compared with those of 

 line 1 1th, the reader will satiiiy himself as to the 

 value of the approximation which is t'len- employed. 



It would not be difficult, upon principles similar 

 to the above, to establish^ a theorem for the elliptic, 

 parabolic, and other curves similar to that we have 

 now given fr the circle. But i. i .. jf j- ,s j^ncral 

 use; and the limits assigned to an article of this kind, 



prevent us from entering at present upon an investi- 

 gation, through which, perhaps, few of our readers 

 would be inclined to follow us. 



Another opportunity may be found of offering 

 this to the public notice. 



But, in the meantime, the reader must at once see, 

 that by this mode of txpresnuig the density in the 

 spandrel, the solution we have given applies to any 



