[m 
123] 
113 
ies to any 
aere be a 
first side 
of them 
the side 
i property 
then will 
, any two 
s of con- 
$e only a 
arder into 
ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN 
INTEGRAL. 
[From the Philosophical Magazine, vol. vi. (1853), pp. 414—418.] 
The equation of a surface passing through the curve of intersection of the surfaces 
x 2 + y 2 + z 2 + w 2 = 0, 
ax 2 4- by 2 + cz 2 -f dw 2 = 0, 
is of the form 
8 (x 2 + y 2 + z' 2 + w 2 ) + ax 2 + by 2 + cz 2 + dw 2 = 0, 
where 8 is an arbitrary parameter. Suppose that the surface touches a given plane, 
we have for the determination of 8 a cubic equation the roots of which may be 
considered as parameters defining the plane in question. Let one of the values of 8 
be considered equal to a given quantity k, the plane touches the surface 
k (x 2 + y 2 + + w 2 ) + ax 2 + by 2 + cz 2 + dw 2 = 0, 
and the other two values of 8 may be considered as parameters defining the particular 
tangent plane, or what is the same thing, determining its point of contact with the 
surface. 
Or more clearly, thus:—in order to determine the position of a point on the 
surface 
k {x 2 + y 2 + z 2 + w 2 ) + acc 2 + by 2 + cz 2 + dw 2 = 0 ; 
the tangent plane at the point in question is touched by two other surfaces 
ies to any 
lere be a 
first side 
of them 
the side 
i property 
then will 
, any two 
s of con- 
$e only a 
Drder into 
p (x 2 + y 2 + z 2 + w 2 ) + ax 2 + by 2 + cz 2 + dw 2 = 0, 
q (x 2 + y 2 + z 2 + w 2 ) + ax 2 + by 2 + cz 2 + dw 2 = 0; 
C. II. 
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