THE SHAPES OF VERTEBRATE SKELETONS. 203 



be left out of the account (since for similar masses moving 

 with equal velocities the resistances increase but little faster 

 than the squares of the dimensions, which is the rate at which 

 the sectional areas of the notochords increase) we see that aug- 

 menting bulk, taken alone, involves but a moderate residuary 

 increase of strain on each portion of the notochord ; and 

 this is probably the reason why it is possible for a large slug- 

 gish fish like the Sturgeon, to retain the notochordal struc- 

 ture. But now, passing to the effects of greater ac- 

 tivity, a like dynamical inquiry at once shows us how rapidly 

 the violence of the actions and reactions rises as the move- 

 ments become more vivacious. In the first place, the resist- 

 ance of a medium such as water increases as the square ol 

 the velocity of the body moving through it ; so that to main- 

 tain double the speed, a fish has to expend four times the 

 energy. But the fish has to do more than this it has to 

 initiate this speed, or to impress on its mass the force implied 

 by this speed. JNow the vis viva of a moving body varies as 

 the square of the velocity ; whence it follows that the energy 

 required to generate that vis viva is measured by the square 

 of the velocity it produces. Consequently, did the fish put 

 itself in motion instantaneously, the expenditure of energy in 

 generating its own vis viva and simultaneously overcoming 

 the resistance of the water, would vary as the fourth power 

 of the velocity. But the fish cannot put itself in motion 

 instantaneously it must do it by increments ; and thus it 

 results that the amounts of the forces expended to give itself 

 different velocities must be represented by some series of 

 numbers falling between the squares and the fourth powers 

 of those velocities. Were the increments slowly accumulated, 

 the ratio of increasing effort would but little exceed the ratio 

 of the squares ; but whoever observes the sudden, convulsive 

 action with which an alarmed fish darts out of a shallow into 

 deep water, will see that the velocity is very rapidly gener- 

 ated, and that therefore the ratio of increasing effort probably 

 exceeds the ratio of the squares very considerably. At any 



