=” 
117 
As an illustration, consider the integral : 
if F (2, y) de. 
y= \Le—9 + Va (2—a) | (t—2a). 
Here we have: 
(1.) fry? —2y, (x —a)(2—2a) + (xa) (02a) 5 =o, 
and this curve has a triple 
point at (2a, 0). Taking A at 
this point, and 
(2.) y=o0,7—2a=o0 
as the equations of lines 
through A, we are to solve for 
the intersections of f,—o, 
and the line: 
(3.) y+4(x— 2a) =o. 
Since three solutions are 
known, we readily find: 
(7=) X (x, A) =a (A? —1) 2? —a(2A4— 2142+ 1) =o. 
(8.) typed (20° = 2A LD, 
. Nephi (A —1) ? 
Syne 2 a ie De fee (2h 
(9.) “y¥=—A(2—2a) = ae 
If the curve ) fn =o, instead of having a multiple point of order (n—1), has 
z (n—-1)(n—2) double points, that is, if its deficiency is zero, then it is a uni- 
cursal curve, and hence x and y can be expressed rationally in terms of a single 
parameter, and hence the reduction can be performed. 
ALTERNATE Processes. By Proressor ARTHUR S. HATHAWAY. 
I, INTRODUCTION. 
1. The alternate (and symmetric) procesess that we develop seem valuable 
from their simplicity and power, and their general applicability in all depart- 
ments of mathematics. They may be employed in any algebra in which addition 
is associative and commutative without regard to the laws of multiplication. 
