620 THE SHAPES OF HORNS [ch. 



no such plane of symmetry is manifested in the field of force to 

 which the growdng annulus corresponds or appertains. 



Now we must carefully remember that the rates of growth of 

 which we are here speaking are the net rates of longitudinal 

 increment, in which increment the activity of the living cells in 

 the zone of growth at the base of the horn is only one (though it 

 is the fundamental) factor. In other words, if the horny sheath 

 were continually being added to with equal rapidity all round its 

 zone of active growth, but at the same time had its elongation 

 more retarded on one side than the other (prior to its complete 

 solidification) by varying degrees of adhesion or membranous 

 attachment to the bone core within, then the net result would be 

 a spiral curve precisely such as would have arisen from initial 

 inequalities in the rate of growth itself. It seems highly probable 

 that this is a very important factor, and sometimes even the 

 chief factor in the case. The same phenomenon of attachment to 

 the bony core, and the consequent friction or retardation with 

 which the sheath slides over its surface, will lead to various 

 subsidiary phenomena : among others to the presence of transverse 

 folds or corrugations upon the horn, and to their unequal distribu- 

 tion upon its several faces or edges. And while it is perfectly true 

 that nearly all the characters of the horn can be accounted 

 for by unequal velocities of longitudinal growth upon its different 

 sides, it is also plain that the actual field of force is a very compli- 

 cated one indeed. For example, we can easily see that (at least 

 in the great majority of cases) the direction of growth of the 

 horny fibres of the sheath is by no means parallel to the axis of 

 the core within; accordingly these fibres will tend to wind in a 

 system of helicoid curves around the core, and not only this 

 helicoid twist but any other tendency to spiral curvature on the 

 part of the sheath will tend to be opposed or modified by the 

 resistance of the core within. But on the other hand living bone 

 is a very plastic structure, and yields easily though slowly to any 

 forces tending to its deformation ; and so, to a considerable 

 extent, the bony cpre itself will tend to be modelled by the curva- 

 ture which the growing sheath assumes, and the final result will 

 be determined by an equilibrium between these two systems of 

 forces. 



