CEETAIN APPLICATIONS TO THE THEOET OF DEFINITE INTEOEALS. 773 
be exhibited in the form 
Po{u—vy,){u—vy^),.{u—vy^)=0, (5.) 
being the different values of as determined by giving different signs to the 
radicals in its expression. Hence we have 
^^<p(w)ydw= dx=^e [^(^)^] ^ \og^,{u—vy,){u-vy ^) . . {u-vy^) 
8 m — y^lv 
J [_T'V M — 
( 6 .) 
the summation in the second member extending from r=l to T—m. 
Here, transposing the symbol 2, the function upon which the operation 0, whose 
VL 
int&rp'etation is derived from <p{oc'fy> is to be performed, is 
which may be resolved into 
Hence 
, ,M 8m — Ifr^V 
wiST’ 
8 m — yr^v 
U — yrV 
^^(p{x)ydx=^lQ(p{x)y, (7.) 
We are especially to remark, that while 2 in the first member has reference to the n 
different values of x furnished by the equation (3.) or (4.), 2 in the second member has 
reference to the different values of y furnished by the equation (2,), its numerical range 
being from r=\ to r-=m. 
Now considering the term 
all that part of the operation 0 which depends upon v produces no effect. For none of 
the factors of v can enter in any way into y„ since those factors contain the variables 
a,, «i • • from which y^ is wholly fr’ee. Again, they cannot enter into the denominator 
u—y;o^ for then they would enter into w, and the fraction - would not be in its lowest 
terms. Hence the first term in the second member of (7) becomes 
We can now transpose the symbols J and 0 and integrate. The result is 
Q\(p{x)'\^y^og{u—y;o) (8.) 
The last term of (7.) will, on account of the interpretation of 0, be properly written in 
the form 
20 
which may be resolved into 
' j 
20 
r 
.in 
<p{x)- <p(x)- yM 
