622 THE SHAPES OF HORNS [ch. 



such an attachment, by subcutaneous connective tissue, to the 

 bony core; and accordingly a torsional strain will be set up in 

 the growing horny sheath, again provided that the forces of growth 

 therein be directed more or less obliquely to the axis of the core ; 

 for a "couple" is thus introduced, giving rise to a strain which 

 the sheath would not experience were it free (so to speak) to slip 

 along, impelled only by the pressure of its own growth from below. 

 And furthermore, the successive small increments of the growing 

 horn (that is to say, of the horny sheath) are not instantaneously 

 converted from living to solid and rigid substance; but there is 

 an intermediate stage, probably long-continued, during which 

 the new-formed horny substance in the neighbourhood of the zone 

 of active growth is still plastic and capable of deformation. 



Now we know, from the celebrated experiments of St Venant*, 

 that in the torsion of an elastic body, other than a cylinder of 

 circular section, a very remarkable state of strain is introduced. 

 If the body be thus cylindrical (whether solid or hollow), then a 

 twist leaves each circular section unchanged, in dimensions and 

 in figure. But in all other cases, such as an elliptic rod or a 

 prism of any particular sectional form, forces are introduced which 

 act parallel to the axis of the structure, and which warp each 

 section into a complex anticlastic surface. Thus in the case of a 

 triangular and equilateral prism, such as is shewn in section in 

 Fig. 321, if the part of the rod represented in the section be twisted 

 by a force acting in the direction of the arrow, then the originally 

 plane section will be warped as indicated in the diagram : — where 

 the full contour-lines represent elevation above, and the dotted 

 lines represent depression below, the original level. On the 

 external surface of the prism, then, contour-lines which were 

 originally parallel and horizontal, will be found warped into sinuous 

 curves, such that, on each of the three faces, the curve will be 

 convex upwards on one half, and concave upwards on the other 

 half of the face. The ram's horn, and still better that of Ovis 

 Ammon, is comparable to such a prism, save that in section it 

 is not quite equilateral, and that its three faces are not plane. 

 The warping is therefore not precisely identical on the three faces 



* St Venant, De la torsion des prismes, avec des considerations sur leiir flexion, 

 etc., Mp'm. des Savants Strangers, Paris, xrv, pp. 233-560, 1856. 



