PHYSICAL CURVES. 95 



The same term is also applied with approximate accuracy to 

 the motion of the common pendulum. The motion of the 

 perfect pendulum is the same as the back and forth motion of 

 any given point on a vibrating cord when the wave form is a 

 true sine curve. 



The Pa?'abola. — The physical relationships of the parabola 

 are probably more numerous than any other curve, and only 

 a few of them will be mentioned. A projectile in a vacuum, 

 shot forth at any angle from the vertical, describes in its 

 flight a parabola. In actual practice, as for example in the 

 case of projectiles from cannons, issuing forth with extremely 

 high velocity against the resistance of the atmosphere, the 

 curve is modified by such resistance. It is then no longer a 

 true parabola, and becomes what is known as the ballistic 

 curve. This curve is to a large extent empirical, although it 

 has for its basis the parabola as modified by air resistance. 

 If a series of projectile parabolas are drawn, in which the 

 projectile is assumed to start with equal velocity and at dif- 

 ferent angles from the same point, there will be formed what 

 may be called a family of projectile parabolas. In this case 

 it can be shown that the envelope of the family of curves, 

 that is, the curve to which all the parabolas are internally 

 tangent, is a parabola itself. 



Closely related to the projectile curves is the curve assumed 

 by a jet of water issuing out of the side of a reservoir or tank. 

 In this case, if atmospheric friction is neglected, the curve of 

 the issuing jet of water becomes a parabola. 



As mentioned elsewhere in speaking of the catenary, the 

 parabola is that curve assumed by a flexible body loaded uni- 

 formly along the horizontal. 



Again, consider a simple beam, such as is used in build- 

 ing, carrying a load uniformly distributed along the horizon- 

 tal. At every point in such a beam there exists a tendency 

 to-break it transversely, which tendency is measured by what 

 is called the bending moment. If the value of this bending 

 moment is plotted at various points in the length of the beam 



