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4 ON THE DISTRIBUTION OF ELECTRICITY 
where the term in { } is 
= (X m+1 — 1) [— a 4 + a?c (c — #)] + (X n — X) [a 2 (b 2 — c 2 ) + a 2 c (c — x)~\; 
= a 2 {(X n+1 — 1) (c 2 — a 2 — cx) + (X n — X) (6 2 — cx)}; 
= a 2 {(X n+1 — 1) (yx + S) + (X n - X) (yx - a)}, 
whence the relation in question. 
The proof of the second equation is a little more complicated. We have 
1 /a + d^" 1 
[706 
viz. this is 
or it is 
c n a 2 + d n (c — x) 
{(X n+1 — 1) [ca 2 + d (c — #)] + (X n — X) [ca 2 — a (c — #)]}, 
X 2 — 1 \X + 1/ 
where the term in { } is 
= (X n+1 — 1) [— ca 2 + (c 2 — b 2 ) (c — x)~\ + (X n — X) [— ca 2 + a 2 (c — a?)]. 
Comparing this with 
1 /a + 8\ n 
cc n+1 x + /3 n+1 = {(X”+ 2 - 1) (ax + /3) + (X ,l+1 - X) (- 8x + /3)}, 
where the term in { } is 
= (X”+ 2 - 1) [b 2 (c - x)] + (X 5l+1 - X) [- c (c 2 - a 2 - b 2 ) + (c 2 - a 2 ) (c - x)\ 
it is to be observed that the quotient of the two terms in { } is in fact a constant; 
this is most easily verified as follows. Dividing the first of them by the second, we 
have a quotient which when x = c is 
(X n+1 - 1) (- ca?) + (X n - X) (- ca 2 ) _ a 2 (X w+1 - 1 + X" - X) a 2 (X + 1) 
(X n+1 — X) {— c (c 2 — a 2 — 6 2 )} ’ “ (X n+l - X) (c 2 - a 2 - b 2 ) ’ “ ( c 2 - a 2 - b 2 ) X ’ 
and when x = 0 is 
(X n+1 - 1 )c(c 2 -a 2 - b 2 ) (X n+1 - 1) (c 2 — a 2 — b 2 ) c 2 -a 2 - b 2 
(X n+2 - 1) b 2 c + (X n+1 - X) b 2 c ’ " (X n+2 -1 + X m+1 - X)b 2 ’ ” b 2 (X +1) : 
these two values are equal by virtue of the equation which defines X; and hence the 
quotient of the two linear functions having equal values for x = c and x = 0, has 
^2 qZ . ^2 
always the same value; say it is = + • Hence, observing that a + d = a + S, 
= c 2 - a 2 -b 2 , the quotient, c n ci 2 + d n (c - x) divided by a n+1 x + /3 n+1 , is 
X -+• 1 c 2 — a 2 — b 2 1 
c 2 -a 2 -b 2 ' b 2 (X + 1) ’ b 2 ’ 
or we have the required equation 
c n a~ + d w (c x) = p (a n+1 ic + /3 n+1 ). 
WBm