388 Dr Dixon, An example of complex double integration. 
By A. C. Dixon, 
An example of complex double integration. 
Sc.D., F.R.S., Trinity College. 
[. Received 15 December 1907.] 
Let U, V, W be rational integral algebraic functions of x, y of 
degrees m, n, m + n — 3 respectively ; then it is a known result, 
given by Jacobi, that £ W ^ = 0, where the summation 
/ o{x, y) 
extends to all pairs of values of x, y, such that U = V — 0 : it is 
assumed that these pairs are all distinct. If homogeneous co- 
ordinates x, y, z are used the corresponding theorem is that 
XzW /YY’ - - 0 and in this form it is unaffected by a change 
/ b(x, y) 
of the triangle of reference, which may therefore be supposed to 
have a quite general position. The object of the present note is to 
prove this theorem by double contour integration. 
Suppose that the line z= 0 does not contain any intersection of 
the curves U — 0, V — 0 or any singular point on either, and that 
the values x = 0, z = 0 do not satisfy either equation. Put z= 1. 
When x has any given value the equation U— 0 gives m values 
for y, say y u y,, ... y m and V— 0 gives n, say ...y n - When 
x is made great the ratios y^x, y 2 /x ..., rj^/x, y 2 jx... tend to 
definite limits which are all different, and we may therefore take a 
radius r, such that when | x j > r, every difference such as y s — y t is 
numerically greater than e\x\, where e is a definite constant, 
depending to some extent on the coefficients in U, V. 
Let the complex variable x describe a circle of radius R (> r) 
about the origin in its plane. Then in the y plane the points 
y ]t y 2 ..., 7 ]. . 7/ 2 ... will also travel about but will not approach each 
other, and in fact if a circle of radius leR is described about each 
of them these circles will always be separate. Take for y a path 
enclosing the circles about tj^, rj 2 ... rj n and excluding those about 
7/j, y 2 ... y m . The length of this path may be supposed <IcR 
where k is a finite quantity. 
ff w 
Let us estimate the value of I \^jydydx over the region thus 
defined, which is closed, since the initial and final positions of the 
point x and of the y-path, which is closed, are the same. 
We have U = a(y — y x ) (y — y. 2 ) ... (y — y m ), where a is the co- 
efficient of y m , and thus on the path | U \ <£ | a \ ( leR ) m , since each 
difference, such as y — y r , never falls below ^ eR . Similarly, on the 
path | V | <(: ] b j eR) n where b is the coefficient of y 11 in V. Hence, 
IF being of degree m + n — 3, we have, on the path, 
| W/UV c/R 3 where c is a constant. 
