304. | Method of tracing Oval Arches. 
The following conditions are (fig. 1.) laid down by the French Engi- 
neers, in order to avoid the indetermination of the problem. 
1st. The number of centres shall be uneven. 
2nd. The sum of the arcs shall be 180°. The two ares of 
smallest radius shall have their centres on the transverse diameters, 
and the arc of greatest radius shall have its centre in the line of the 
conjugate. 
3rd. pe, the distance from the centre of the oval to the centre of 
the middle arc, shalk be three times pa, the distance from the centre 
of the oval to the centre of the arc of smallest radius. 
Ath. pa, the distance above mentioned, shall be divided by the ra- 
dii of the other arcs into parts which are to each other in the ratio of 
the natural numbers 1, 2, 3, 4, &c. In all cases deduct one from 
the number of centres to be employed, and half the remainder will 
give the parts into which ap must be divided. 
5th. The distances between the prolongations of the radii meas- 
ured on the conjugate axis produced, shall be equal. 
The following method of determining the point a, was discovered 
by my friend Benjamin Aycrigg, Civil Engineer, and is I believe the 
best and most simple. 
He supposed the concentric curve ao, (fig. 2.) to be drawn commen- 
cing at the point a, with the radiustza. ‘Then it will be perceived on an 
examination of the figure that we have sufficient data to calculate all 
the angles contained within the curve a 0, and all the radii and lines 
in terms of ap. Suppose this done and the line op to be found. 
Then because the curve ao is concentric to the curve 6d, the lines 
ab, ah, rg,od, &c. are allequal. Let each of them=za, and in or- 
der to construct an arch .of any given rise and span, we have this 
proportion, (op+x) : (pa+«)* ‘rise : semi-span; whencethe value 
of x=ab=od, is easily found in terms of ap, so as-'to give pd, and 
ob, in the required proportion. Having this we can of course find 
the value of the radii and other required lines in terms of the rise 
and span. 
In these ovals it will be observed that the value of all the lines and 
angles within the curved line ao, are general, and when once calcu- 
lated will serve for allarches. I shall now proceed to give these gene- 
ral calculations for curves of 5, 7, 9, and 11 centres. That of 
7 was computed by Mr. Ayerigg, the others by myself. ‘Their ac- 
curacy has been abundantly tested in various constructions. ‘The 
Alleghany Portage Rail road, on which I am at present engaged, has 
