CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
331 
S„ and T„ are the expressions referred to in last article : they are the particular 
solutions of the equation 
^+hf+(c-^H- ) ')R=o. 
If we write x=\r, S„ and T„ are respectively equal to 
x n . 
1 d \ n sin x 
x dx 
and x n \ - 
1 d \” cos x 
x dx 
To expi’ess the boundary conditions we shall write 
a=Xa, /3=Xb, S = S(Xa), S / = S(Xb), &c ; . 
we have, accordingly, 
Ca"=AS+BT, «Ca”- ] = x(A~+B|^ j 
Db—>= AS'+BT', - (»+1 )Db-«=B^) j 
(59) 
(CO) 
These equations may be put into much more elegant forms ; but to do so we require 
various properties of the S— and T— functions ; and as these are not very generally 
understood, I shall here digress into a brief sketch of the whole subject, confining 
myself as strictly as possible to those properties which have a direct bearing on the 
present question. 
[.Properties of the “ Spherical Functions .” 
§ 10. If we write x z =t, and work out the differentiations and integrations, it can be 
readily shown that 
/I Y* sin a? . . . /I d \ cos x 
x n [~ — ] -=(—!)" x~ n ~ x 
x" 
x dx) x 
1 d \ n cos x 
x da 
\x dx 
a f 
yX dx 
, . ,. . i 1 d\~ n ~ l sin a; 
(B 3 ) 
The constants introduced by integration must be so adjusted as to make a certain 
number of terms in the second equality coincide : the others will be exactly equal. 
If we write each member of these two equations, as before, respectively S„ and T„, 
and differentiate the quantities on the left-hand sides of the equations, we find 
ci dS rt rp dT n rp 
- no,,, xl„+- [ =x— - n 
1 dx dx 
2 x 
(B) 
MDCCCLXXXI. 
