spherical Wave of LigJit. 83 



axes be <:cos0, rsin0cos0, rsin0sin0, and imagine new axes 

 as before to be drawn through Q, then 

 X = ^{cosacos0 + sinasin0cos0} 

 y = ^{(sinacosQ - cosasin0cos(^)cos/3 + sin0sin0sin/3} 

 z = <:{(sinacos0 - cosasin0cos^)sin/3 - sin0sin(^cos/3} 

 and if 2U-1, ^U-i, ^U_i, denote the coefficients of the 

 trigonometrical function in the components of the displace- 

 ment at Q resolved along the axes through Q, 



r,U_i = A,i^'"~^{cos/3cos0 - sin/3sin0cosa}/^'* 



— B,i::c"~%sinasin(/)/^^' 

 ^U_i = - x\„A;"~^{cos/3cos0sin0 + sin,/3(sinasin0 + cosacos0cos0) }/V' 

 + B,,a;""%(cosasinO - sinacos0cos0)/^", 

 where I have retained x and z for brevity. 



We are to regard these now as expressing the coefficients 

 of the displacements in a wave converging to the origin, 

 and each element of this wave is to be replaced by a 

 secondary wave in the same way as the original wave was 

 replaced by a secondary wave. 

 Write 



X = <:-pcos0, 

 Y - psin0, 

 Z = 0. 

 ^2 =, x2 + Y2 + Z- = ^2 _ 2^pCOS0 + p2 

 r=^-pcos0. 

 Hence the components of the secondary wave arising 

 from the element at Q resolved along the axes through Q 

 are 



^1 = S,U_i«,X^-iYsin/|^(T + /, + /)- ^ + r + ^|;V^-"' 

 7?! = 2,U_i^,X^-sin/|/^(T 4- 4 + /) - r 4- r + ^|/r^+^ 



.^ = 2.U_i(«,X^ + 4X^-2Y2)sin/>|/^(T + /, 4- /) - ^+ r + j|/^^+\ 



where I have written the first term only of the expansion 

 and k has the same ranee of values as n and aj, = K^, b^, = ^k' 



