The author has deter mined the line of resistance of a great number of actual arches, and has frequently found arches that had given no signs of failure, which were unstable according to Scheffier's theory; but he has found none which were unstable by the rational theory. Of course, an arch may be made so heavy as to be stable by an approximate theory; but the fact that some arches were unstable by Scheffier's theory, and stable by the rational theory, seems to show that the latter is the more accurate.
Rankine's theory of the arch recog nizes the existence of the horizontal component of the earth pressure, and in the mathematical work leaves nothing to be desired; but he does not make it clear that his mathematical theory can be applied in practice. He employes Navier's principle (§ 1214) to find the crown thrust; and determines the position of the joint of rupture by means of a differential equation; but it is not clear that the relationship between the several variables in this equation can be stated for any practical case in a manageable mathematical form. Rankine does not work out any numerical example; but in an illus tration of the method of determining the stability of any proposed arch he virtually assumes that the portion of the semi-arch above the joint of rupture is acted upon by only three forces—the crown thrust, the weight of the arch, and the reaction at the joint of rup ture.* This is erroneous (a) because it neglects the horizontal com
ponents of the external forces; and (b) because it finds a new value for the thrust at the crown which, in general, will differ from that employed in finding the position of the joint of rupture. At best Rankine's theory determines the line of resistance within limits at only two points; and hence gives no definite information as to the degree of stability against sliding, overturning, or crushing at these points, and gives no information at all for other points.
Although this theory has long been before the public, it is com paratively little employed in practice. This is probably due, in part at least, to the fact that Rankine's presentation of it is not very simple nor very clearly stated, besides being distributed throughout various parts of his books—" Civil Engineering" and "Applied Mechanics."