HoGBEN. — On Earthquake Motion. 591 



and the other distortion and rotation without dilatation.* 

 That is to say, we may discuss the normal and transverse 

 vibrations separately with some hope of getting an approxi- 

 mation to the actual facts of earthquake motion. If we con- 

 sider the normal vibrations of an earth-particle at some dis- 

 tance from the origin of disturbance we may take the mode of 

 motion to be wholly irrotational, and it has been shownf that 

 the displacement potential ^ then satisfies the equation 



OV</> + ^'<^ = (i.) 



Q being independent of i, and, as will appear presently, equal 

 to the velocity of propagation ; and %^ a function of the initial 

 impulse and of the elasticity moduli and the density of the 

 system of particles for the series of vibrations in question. 



It is also true that the form of this equation (and hence 

 also the form of its solution) remains unchanged whatever be 

 the value of i for any particular series (provided that the series 

 be homogeneous and isotropic). Now, in the case under con- 

 sideration we may choose the axis of x in the line joining the 

 origin and the particle considered, and neglect all vibrations 

 except those parallel to the axis of x — in other words, if u, v, 

 w be displacements parallel to the co-ordinate axes, we may 

 put 



d(t> 

 U = -^, V = 0, 10 = 0. 



So that <^ is a function of x only, and equation (i) reduces to 



"'S + ^'^ = ^ ("-^ 



every solution of which is of the form 



^i = A sm.— + 13 cos. — 



for any particular i series. 



The corresponding partial solution for ^ may be written 

 in the form 



Qsin.^ {X -m- Si) -f Di sin.^ (a; -f Qt-T^ 



where Q, Di, S;, Tj, i, fi are unaltered as long as the same 

 series of waves — that is, due to the same impulse at the 

 origin — propagated through the same isotropic solid, are being 

 considered. 



The displacement potential has the same value when we 



put i = o, ^ = -v-,f = — , &c., X being constant — i.e., while we 



* Ibbetson, " Elastic Solids," p. 286. 

 t IbbetsoD, I.e., p. 288. 



