316 Mr. Challis on the Theory of the 



its velocity. But as it passes through the medium, it strikes at 

 every instant against fresh particles, and the resistance from this 

 cause is as the square of the velocity. (See Art. 9) Hence upon 

 the whole, the resistance to a projectile is partly as the velocity 

 and partly as the square of the velocity. This is probably a 

 near approximation to the law which obtains in nature. In 

 small vibratory motions like those of chords, we may safely say 

 that the resistance is as the velocity. 



Suppose a series of primary waves to impinge perpendicu- 

 larly on a vibrating chord, the diameter of which is small 

 compared to 2X the breadth of a wave. According to the law of 

 rejection demonstrated in Art. 18. the air in contact with every 

 point of that half of the chord which looks towards the source 

 of the waves, will suffer condensations in virtue of reflection 



alone, proportional to sin — — ; the air in contact with the other 



half being supposed not affected. But it is plain that by reason 

 of the gliding of the particles along the surface of the chord, 

 the condensations will be established all round it. The law they 

 will follow may be thus found. Whether the wave impinge on 

 the chord, or the fluid be at rest and the chord oscillate just as 

 a particle of the wave does, the effect as to the distribution of 

 the condensations will be the same. In this latter case, each point 

 of the surface of the chord will at every instant generate a con- 

 densation proportional to its velocity in the direction of the nor- 

 mal. Hence if ahcd (Fig. 13) be the transverse section of the 

 chord; OA the direction of its motion; hf-Af=l, the conden- 

 sation at i at a given instant, on the scale in which Of^ D the 

 mean density; the density Op corresponding to any angle 9 

 reckoned from OA, will be D + ^ cos 0. In the case of the chord 

 .stationary, Od will be the mean density, and Op, the radius- 



vector of deAe'd, will become D + 2& cos' -; for had it not been 



2i 



