G76 
PROFESSOR W. K. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
[ Theory of Derived Points on an Elliptic (or Bicursal ) Curve. 
Sylvester’s theory of derived points on a plane cubic is as follows : — Starting from 
any given point on the curve, we may construct its tangential, or point where the 
tangent at the original point meets the curve again ; similarly we may construct the 
tangential of the tangential, or second tangential, and so on. By joining any two 
non-consecutive points on this series, we can find their residual, the point where the 
joining line meets the curve again. In this way we obtain an infinite group of points 
derived from (and including) the original point, such that, the line joining any two of 
them is either tangent at one of these or passes through a third point of the group. 
It is to be observed that all points on the curve uniquely derived from the given 
point by any geometrical process (e.g., the point where the conic of five-pointic contact 
meets the curve again, the point where cubics of eight-pointic contact meet the curve 
again, &c.) are included in the group. 
The coordinates of any derived point may be expressed rationally in terms of the 
coordinates of the original point, and the order of the functions to which they are 
proportional is always a square number. Thus the three coordinates of the tangential 
are proportional to quartic functions of the coordinates of the original. If the square 
root of the order of these functions be called the order of the derived point, then we 
have the theorem that when three derived points are in a straight line, the order ol 
one of them is equal to the sum of the orders of the other two. It is observed that 
there is no derived point whose order is divisible by 3. By help of this observation 
it is easy to make out a scheme of the orders ; for when we join two points, the order 
of their residual must be the sum or the difference of the orders of the points, and one 
or the other of these is always divisible by 3. 
This theory is really a geometric representation of the multiplication of elliptic 
functions. The coordinates x 1} x 2 , x 3 of any point on the cubic curve may by proper 
choice of axes be made proportional to elliptic functions of a parameter u, so that 
x x : x 2 : x 3 = 1 : sn 3 (u-\-iK') : sn (u -j- i'K') cn (u -j- i K ') d n (u + IK') . 
This being so, if u, v, w are parameters of three points in a straight line, we shall 
have u -\-v+ w— 0. If v be the tangential of u, the three points u, u, v are in a 
straight line, and 2u-\-v=0, or v— — 2 u. Hence the series of tangentials has for 
parameters 
u, —2m, -j-4M, —8 m, &c. : 
and in general the parameter of any derived point is of the form nu, where n is a 
positive or negative integer. The number n, taken positively, coincides with what 
was called the order of the derived point. For the elliptic functions of nu are of the 
order rd in the elliptic functions of u. 
In this way all points of the theory are explained, excepting the fact that no 
derived point has its order divisible by 3. 
