CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
340 
47 Tinw 
\~= - .i, and if we put cr=Rc, and treat R as finite, we must suppose X 3 c finite, 
and therefore Xc is a small quantity of the order c 4 , while ^ is also of the order c 4 . 
The expression 
S T' / T Q' 
w+l -L V V 
/ T r/S \ / r/ 2 T \ 
= S» +1 T f —T» + 1 S,,+\c ^ T„ +1 -^ j+T»+i"^fj + • • • 
If we write 
q dfR_ m dPS v 
we can readily find a formula of reduction in u p . 
S„ and T„ both satisfy the equation 
£C 2 y|-+ 2Xy^{cr — l)}?/ = 0. 
(i cC 
By Leibnitz’ theorem 
x z y {p+2) +2 (p +1 )xy (rp+1} -J- (cc 3 -|- (p—v) (p -\-v +1 ))y° 5) + 2py (p 1} J rp(p — 1 )y (p 2) =0, 
wherein 
whence 
'll 
2(^ + 1) 
y+a 
l Wi+ M 
„o»=*2. 
j dxP ’ 
(g-y)(jp+ y +i)\ .. , 2 p „ ■ ffQ-LL 
tq+^^-i+ ^— ^8= °> 
in which we have to write a for x in the problem before us. We conclude that u p+% 
will usually be of the same degree in - as u p , and when u 0 and u x are given we can 
readily find the remaining values of u p . But as each of them is multiplied by ever 
increasing powers of Xc, it will be only the first one or two terms which will give finite 
terms when w T e treat - as infinitely small. Let us now r take up cases (ci) and (b) of 
cl 
last article. 
(a.) From formula (26) and remembering (B 6 ), we obtain 
C 
<d + A 
S„ +1 Tk H — T„ +1 S'„ + 1 _ 
271 + 1* S re+1 Tk—T b+1 S'„ ’ 
2 z 2 
