332 
PROFESSOR C. RIVEN OR THE INDUCTION OF ELECTRIC 
Differentiating: the riodit-hand members we obtain 
o o 
—a; ^ + (71+1)8,, — ad\,_ T =x^'-f (a + l)T„ .... (B 5 ) 
These are the fundamental formulae of reduction in these functions : if in the latter 
we write n+1 for n, and substitute the values of S„ +1 and T„ +1 from the former, we 
can show easily that S# and T„ satisfy the differential equation 
7?+-^+(i-^)r=o.(B) 
ax ■ x ax \ x“ ) 
and are therefore the two solutions of it. If we eliminate from the above x -- -, and 
fix 
x ~ . we nnd 
ax 
(2??-f-1) — d-S w+1 +S«_ 1 =0 | 
. ; ^ .w 
(2n+l) -f+T„ +1 +T„_ 1 =0 
* J 
whence also 
S« +1 T # —T fl+1 S*=S 4 T B _ 1 —S^T,,.(B 7 ) 
and, by successive reductions, 
S, + ,T,-T, +1 S„=S 1 T 0 -T 1 S 0 =f,.(B s ) 
and thus also, by (B 6 ), 
S„ 1 T„_ 1 -T„ +1 S„_ 1 =-^.(B a ) 
We obtain also directly from equations (B^t, 
rp dS H q aT n _rn m Q _I_ /T> \ 
•*-» ( f r —+ A»+lO w - . \°10/ 
a well-known result.] 
11. If we now turn to equations (GO), we find by elimination 
^( T f- s f)=- a ”- IC (“f- BT ' 
XB ( T f 
XA ( T f- s f)=- b - 2D ^l+(' l + 1 ) T ' 
XB (Tf -S'f)= +b—D +(., +1 )S' 
