XIII] OF SHEEP AND GOATS 619 



In this case then, and in most other trihedral or three-sided 

 horns, one of the three components, or three unequal velocities of 

 growth, is of relatively small magnitude, but the other two are 

 nearly equal one to the other. It would involve but little change 

 for these latter to become precisely equal ; and again but little to 

 turn the balance of inequality the other way. But the immediate 

 consequence of this altered ratio of growth would be that the 

 horn would appear to wind the other way, as it does in the 

 antelopes, and also in certain goats, e.g. the markhor, Gapm 

 falconeri. 



For these two opposite directions of twist Dr Wherry has introduced a 

 convenient nomenclature. When the horn winds so that we follow it frem 

 base to apex in the direction of the hands of a watch, it is customary to call 

 it a "'left-handed" spiral. Such a spiral we have in the horn on the left-hand 

 side of a ram's ?iead. Accordingly, Dr Wherry calls the condition homonymovs, 

 where, as in the sheep, a right-handed spiral is on the right side of the head, 

 and a left-handed spiral on the left side ; while he calls the opposite condition 

 heteronymous, as we have it in the antelopes, where the right-handed twist 

 is on the left side of the head, and the left-handed twist on the right-hand side. 

 Among the goats, we may have either condition. Thus the domestic and 

 most of the wild goats agree with the sheep ; but in the markhor the twisted 

 horns are heteronymous, as in the antelopes. The difference, as we have 

 seen, is easily explained ; and (very much as in the case of our opposite spirals 

 in the apple-snail, referred to on p. 560), it has no very deep importance. 



Summarised then, in a very few words, the argument by which 

 we account for the spiral conformation of the horn is as follows : 

 The horn elongates by dint of continual growth within a narrow 

 zone, or annulus, at its base. If the rate of growth be identical 

 on all sides of this zone, the horn will grow straight; if it be 

 greater on one side than on the other, the horn will become curved : 

 and it probably ivill be greater on one side than on the other, 

 because each single horn occupies an unsymmetrical field with 

 reference to the plane of symmetry of the animal. If the maximal 

 and minimal velocities of growth be precisely at opposite sides 

 of the zone of growth, the resultant spiral will be a plane spiral ; 

 but if they be not precisely or diametrically opposite, then the 

 spiral will be a spiral in space, with a winding or hehcal com- 

 ponent; and it is by no means hkely that the maximum and 

 minimum will occur at precisely opposite ends of a diameter, for 



