884 THE SHAPES OF HORNS [ch. 



But there is yet another and a very remarkable phenomenon 

 which we may discern in the growth of a horn when it takes the 

 form of a curve of double curvature^ namely, an effect of torsional 

 strain; and this it is which gives rise to the sinuous " lines of growth,"' 

 or sinuous boundaries of the separate horny rings, of which we have 

 already spoken. It is not at first sight obvious that a mechanical 

 strain of torsion is necessarily involved in the growth of the horn. 

 In our experimental illustration (p. 880), we built up a twisted coil of 

 separate elements, and no torsional strain attended the development 

 of the system. So would it be if the horny sheath grew by successive 

 annular increments, free save for their relation to one another and 

 having no attachment to the 'Solid core within. But as a matter 

 of fact there is 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 shp 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 instan- 

 taneously converted from living to solid and rigid substance; Init 

 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 



♦ St Venant, De la torsion ties prismes, avec des considerations sur leur flexion, 

 etc., Mem. des Savants J'jtrangers, Paris, xiv, pp. 233-560, 1856. Karl Pearson 

 dedicated part of his History of the Theory of Elasticity to the memory of this 

 ingenious and original man. For a modern account of the subject see Love's 

 Elasticity (2nd ed.), chaj), xiv. 



