348 Dr. T. Craig on Steady Motion in 



motion is steady. If we write 



dS dW dY 

 dx dy dz 



d±, dU_ctW 



l ~dy + dz dx> V 



dz dx dy ' J 



the equation of continuity 



du dv die ( 



dx dy dz 



will be satisfied, provided <j) satisfies the condition 



V> = (19) 



The remaining functions U, V, "W must, as is well known, 

 satisfy the following conditions: — 



V 2 U== _ 2 £ v 2 V=-27? ? V 2 W=-2r 



JU + ^V + ^VT ==a [ • (20) 



c&e dy dz ' ) 



The origin of coordinates is to be taken at the centre of the 

 sphere ; then, if r denote the distance from the centre to any 

 point x } y, z, we have for the transformation to polar coordi- 

 nates, 



x = r sin % cos i/r, 



y = r sin ^ sin yjr } 



£ = ?'cos%; 

 the equation V 2 </> = now becomes 



<P(r<t>) , 1 dr. <ty\ 1 d 2 tf> 



The most general solution of this in spherical harmonics is 

 cj) = a Y + a 1 rY 1 + u 2 r 2 Y 2 + . . . 



or 



*=|(^ + Jl)y 



Yt being a surface spherical harmonic of degree ?'. Write 

 <£,- = «,.?-■* Y,-; <f> { is a solid spherical harmonic of degree i; this 



<f> = 2(l + 



w,) )* (*1) 



