EARTHQUAKES. 



115 



perpendicular to the surface at a, and become more inclined until it 

 finally becomes parallel with the surface at an infinite distance. The 

 velocity of its transit will be infinite at a, and then gradually decrease 

 until, if we regard the surface as a plane surface, at an infinite distance 

 it reaches its limit, which is the velocity of the spherical wave. Between 

 these two extremes of infinity at a, and the velocity of the spherical 

 wave at infinite distance, the velocity of the surface-wave varies in- 

 versely as the cosine, or directly as the secant, of the angle of emergence 

 x b a, x c a, etc. 



For, if a a, b b, c c, d d, etc. (Fig. 96), be successive positions of the 

 spherical wave, then the radii x a, x b, x c, would be the direction both 

 of propagation and of vibration. Now, when the wave-front is at by 

 while the spherical wave moves from V to c, the surface-wave would 

 move from b to c ; when the spherical wave moves from c' to d, the sur- 

 face-wave moves from c to d, etc. If, therefore, b c, c d, etc., be taken 

 very small, so that b b' c, c c' d, may be considered right-angled tri- 

 angles, then in every position the surface-wave moves along the hypote- 

 nuse, while the spherical wave moves along the base of the small tri- 

 angles b V c, c d d, etc. Letting v = velocity of the spherical wave, and 

 v' that of the surface-wave, and E the angle of emergence (x ba, x c «, 

 etc., Fig. 96), we have the proportion — v : v' : : 1 : sec. E, and v'= v. sec. 

 E, or if v is constant v' a sec. E. Therefore, at a, the point of first 

 emergence, E being a right angle and sec. E = infinity, v' = infinity. 

 At an infinite distance from a the angle E becomes 0, and the secant 



Fig. 97. 



= 1, and v'= v. 1 = v. That is, at the point of first emergence the ve- 

 locity of the surf ace- wave is infinite ; from this point it decreases as the 

 secant of the angle of emergence decreases, 

 until finally at an infinite distance it becomes 

 equal to the velocity of the spherical wave. 



On a spherical surface (Fig. 97) it is evi- 

 dent that E never becomes 0, and therefore 

 v' never reaches the limit v. If we conceived 

 the wave to pass through the w T hole earth (Fig. 

 98), then the velocity of the surface-wave 

 would decrease to a certain point where E is 

 a minimum, say about c, and then would again 

 increase to infinity on the other side of the 

 earth, p, where ^becomes again a right angle. 



If x be near the sur- 



