178 DR. T. J. I'A. BROMWICH ON THE 



As regards the transformation of (l'l) to (I'll), it is sufficient to note that the 

 gradient of P has the spherical polar components 



3P 1 8P 1 3P 



8r r 30 r sin 30 



and that (r, 0, 0) corresponds to the Cartesian vector (x, y, z). 



To obtain the formulae corresponding to (l'2) we observe that the vector on the 

 right is equal to the vector-product of the two vectors, 



/ N /3Q 8Q 3Q\. 



(x, y, z) and , f -S, JS, J6)i 



\Sx cy 827 



and that these two are represented by 



/ M /3Q 18Q 1 3Q\ 



(r, 0, 0) and ^> - -^> r- 



\ 3r r 80 r sm 30 / 



Thus the vector-product has the spherical polar components 



(o -J-^Q, 3QV 



sin 30 30/ 



Consequently equations (1'2) now become 



d-21) ^1=0 aH - 2 - J- a -Q, M ?l2 = 3Q, 



M 3< 8e sin 30 8 30 



while (l'4) and (l'5) give 



80 



A consideration of these formulae suggests that further simplifications can be 

 obtained by writing 



(re) 



or c at 



which together satisfy equation (l'4l); and then equation (l'5l) leads to the 

 equation for U : 



M K3 2 U 3 2 U 1 3/. .3U\ 1 3 2 U 



^ sin ^ 



c 2 3^ 2 3r 2 r 2 sin 30 V 30 / r 2 sin 2 30 2 



