882 THE SHAPES OF HORNS [ch. 



growth, the resultant spiral will be a plane spiral ; but if they be not 

 precisely or diametrically opposite, then the spiral will be a gauche 

 spiral in space ; and it is by no means likely that the maximum and 

 minimum will occur at precisely opposite ends of a diameter, for no 

 such plane of symmetry is manifested in the field of force to which 

 the growing 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. 



Fig. 4,*i5. Marco Polo's sheep: Ovis FoU. From Cook. 



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 bhe bony 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 probable that this is an 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 distribution upon its 

 several faces or edges. And while it is perfectly true that nearly all 

 the characters of the hotn ca n be accounted for by unequal velocities 



