192 
MR. ARTHUR SCHUSTER ON THE 
Case II. n = 2, cr = 2. 
sm 
1_ ^ ^ = fy (-i/V + i/ig 1 ) cos (X-a) - (2i/i 2 2 + fy> 3 2 ) cos (2X-a) 
~iy^ 3 3 cos (3X—a). 
Our next step must be to find the expression resulting from the terms containing S 
in the elimination of Pt. 
We shall begin by assuming S to be a spherical harmonic of the form Q/, and 
shall again take the parts depending on y, y separately. Independently of both 
these quantities, we have 
Jo sin 0 + -J-Q = -n.n+1. sin 0Q n " 
da da sm a dk 
■ (2 n 
As factor of y we have 
d • a a dQn , + o d?Qn 
— sm # cos 6 —Af- + cot 6 - 7 y? . 
da da «X' 
The value of this has been obtained under (25). 
Finally, as factor of y, we have 
— sin 2 0 cos X —+ y- cos X —. 
dd dd d\ d\ 
This is identical with the expression (19), the result of the operation being given 
by (22). 
If we collect our results, by adding the right-hand side of (27) to (25) and (22) 
after applying the appropriate factor, we shall have obtained an expression for 
d 
dd psm6 dd 
dQr? + 1 
_ d_ dQ / 
sin 6 dk ' P dk 
(28). 
It will appear that S can be expressed in the form of a series 
S = k„° cos aQ n ° -|- X {«■/ cos (o-X— a) + jji/ cos (a-X + a)} Q/ . . . (29), 
cr = 1 
where a is determined by the phase of the velocity potential xjj n a sin (o-X — cr), which 
rules the flow of matter in which the electric currents are induced. We shall avoid 
the labour involved in the consideration of special cases if we write (29) in the form 
<7 = + CO 
S = X k/Qs cos (o-X—a) . 
(30). 
