AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
153 
Section II. (Arts. 7-9).— Consideration of the Cases reserved in which two 
ROOTS OF THE EQUATION DfjDct = 0 BECOME EQUAL AT ANY POINT ON THE 
Locus of Ultimate Intersections. 
Art. 7 .—Consideration of the exceptional case of the Envelope Locus , in which two 
consecutive characteristics coincide. 
(A.) It will be shown that this is the case reserved in Art. 1 , viz., where 
D^/Dcq 2 = 0 . The geometrical meaning of the condition will first of all be 
determined. 
The surface 
/ (x, y, z, a) = 0 
intersects the surface 
f(x, y,z,a + 8a) = 0 , 
where 8a is indefinitely small in the curve whose equations are 
/ (x, y,z, a) = 0 , 
D/ (x, y, z, a) _ 
Da 
This curve is called a characteristic. The equations of the next characteristic are 
obtained by changing a into a -\- 8 a in the above. Hence they are 
f(x,y,z, a) + (8a) 
D/ (x. y. 3, a) _ 
Da 
0 , 
D /fc y,% “) D~y Qg, y, z, «) _ 0 
Da 
Da 3 
Now, if the two consecutive characteristics coincide, 
f(x, y, z, a) = 0 , 
D/(.r, y,z, a) _ 
D« 
v*f(x,y,z, et) _ 
Da 2 
at every point of the coinciding characteristics. 
Hence, the characteristic counts three times over as an intersection of the envelope 
and the surface, instead of twice as in the ordinary case. 
(B.) It is now necessary to repeat the investigation in the case in which equation 
( 2 ) has equal roots when x — £, y = rj, z = (, the co-ordinates of a point on the locus 
of ultimate intersections. 
MDCCCXCII.—A. 
X 
