8o Mr. Gwyther on a 



being arbitrarily chosen are interchangeable) we arrive at 

 the symbolic solution of the differential equation 



where/ and F denote the initial values of ^ and^. 



But if f and F initially belong to a state of motion 



satisfying the differential equation, then 



(^^ V ' + k^--^ lb'' V ' + /^')F = 0. 



and the solution simplifies to 



r J ^ Fsin^/ 

 ^fcoskt + ■ — 



in accordance with an ordinary consideration of a Fourier 

 expansion, and shows that has the form of a function of 



('-*> 



^4* 



The relation between the displacement and rotation at 

 any point of a spherical wave are such that if V and W are 

 the coefficients in the components of the displacement along 

 the tangents to the meridian and parallel at the point, then 

 /W/2 and — />V/2 are the corresponding coefficients in the 

 components of the rotation, the trigonometrical terms 

 having been omitted for brevity. 



Continuing to omit the trigonometrical terms, let us 

 form the expression for the secondary wave from the dis- 

 placement, and then form the expression for the rotation 

 in the secondary wave. Thus we get 



2Y IP = - Y(b^ - a,)x-'yz + Wx'''-'{a„{x'^ + s^) + b,f^ 



2Zlp = - N{pe'-^{aSx'-' +ji/^) + h,z'- + W(/;, - a,)x''-^yz 

 all divided by r'^+2. 



It is more convenient to write these 

 2X// - - W(^,_i - b,,_^ + b,,^-^x''-^y + etc. 

 2Y// = W{(«,_i - A,_i + ^«+i)^" + K_i - /^„_i):r«- V}etc. 

 zZlp = - W(^„_i - b„_-^x""-yz - etc. 



