354 
PROFESSOR A. R. FORSYTH ON 
and therefore from the above 
r=p +1 
X $ V 
bf 
,= 1 P+r H + r 
—o, 
equivalent to one of the two equations 
X=0 
or 
r=P +1 bf 
u '+'bu„r°- 
The latter, taken with the equation 
/K+n • • •> « 2 p+i) = 0, 
implies that there is a homogeneous relation between the quantities u p+r ; this we 
may reject. The former leaves f arbitrary or non-existent, and so there would be 
only p equations to determine p +1 quantities, a difficulty, however, obviable at any 
time by assigning some new equation to make up the requisite number ; but X=0 
simplifies the resulting equations in which it occurs, and therefore this is selected. 
Let us assume as our new equation 
v u 
p+l 
<-p + 2 
+ 
Uc, 
b b — a p+1 b — cc p+2 
2P+1 
b—ct, 
2P+1 
where v is a quantity which may have a definite value assigned to it at any time, if 
desired. Thus we have 
bV s = p 1 *>U 
-= £ 
and therefore 
b^p -i -r s= l CC S ( f pbU s 
b\J 
al 2 p+r = 1 + 
bUp + r 
. (14') 
similar in form to (14). 
18. Let us obtain the new u’s explicitly from the above equations. Writing 
we have* 
g(z) = (z—b) 1 P(z) 
Wp+r = 
_ g(((p+r) I Q(«i) ?q , , Q(gp)_, Q (b) b 
■f ...+^ 
and 
Q!{upv)\j\ a \) «i-«p+>- 1 ‘ * ' ' 9'( a p) ap-flp+r 9\b) b-ap +r J 
Now make b infinite, so that the assumed equation takes the form 
u p+l -\-u p+2 -{- . • . -\-u 2 p +1 =v 
* Cf. Scott’s ‘ Determinants,’ l.c. 
