\ 
348 
PROFESSOR A. R. FORSYTH ON 
and in 
_ j P (x)dx 
,IX ~~ 2 a = (x—a^y/^x) 
Now the constants in M and N can be so chosen that the roots of the equation 
My-NV=0 
shall be cq, « 3 , . . . , ci p each occurring thrice; for the conditions are that any one, 
say a k , shall satisfy the three equations 
Mj—N¥=0 
M ~ - N 3 zf - - z 2 N "7=0 
dx dx dx dx 
mt +K")- n M^T =o 
where T, U are functions of x. In order that a k may satisfy the first of these 
N=0 
when x=a k ; that it may satisfy the second the additional condition is that 
M=0 
and therefore M will contain x — a k as a factor. Moreover, y~ contains x — a k as a 
factor and therefore in Mh/ 3 we shall have x—a k raised to the third power; and 
therefore if we expand N in ascending powers of x — a k the first term is of the order 
(x—a k )\ 
has A(x — a K ) as its first term and so vanishes when x=a K . But this 
is the additional condition that x=a k should satisfy the third equation, and being 
satisfied it proves that a K may occur as a triple root provided 
M=0 N=0 
when x=a k . That is to say, two conditions are necessary for each root, or 2 p in all; 
but as there are 2 p disposable constants these can all be satisfied and so the truth of 
the proposition is established. But as N is only of the degree p — 1 in x, while it has 
to vanish for p values, it must be identically zero; and we choose M=y 3 , so that the 
equation is M 2 y 3 =0 which is obvious beforehand. 
14. Applying now the general theorem from § 6 we have 
% P \ P (x)dx | fb P (x)dx | pA P {x)dx | constant 
A =ilJ (x — a^y/F^x)' J — (x—a^s/ h(®)J 
Hence 
rfN\ 
dx ) 
f 
