AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 185 
Therefore 
0A 
dx 
mE 
m -1 
0E 
^ + E 
Hence when x = g, y — 17 , 2 = £, 9 A/S.r = 0. 
Consequently, A contains E once, and once only in general as a factor. 
(B). It is necessary to examine the cases of exception. 
(i). If equations (11), (25), (26) are satisfied by more than one set of distinct values 
of a, b ; take, for example, the case where there are two sets of solutions, oq, {3^ ; a 2 , (3. 2 . 
Let cq, 6 1? a. 2 , b 2 become cq, « 3 , (3 2 respectively, when x = f, y — 77 , z = £. 
Putting 
A = R'/(®, y, 2 , oq, Zq) /(a, y, z, a 2 , b 2 ). . . 
/ P/ (*> &i) 
^ = y> z ’ ai ’ 2A 2:5 a2> ^ + R 
+ R/(*, y, z, a ls Zq) 
Da? 
D/ (.z, ?/, 2 , cq, &„) 
. . (59). 
/(*, y, z, « 2 , 6 2 ) 
. . . (GO). 
Now, when £C = £, y = 17 , z = £, 
/(a?, y, z, oq, Zq) becomes/(£ 17 , £, a 1? /3 X ) and vanishes, 
/(#, y, 2 , cq, Zq) becomes /(f, 77, £, a 2 , (3 2 ) and vanishes. 
Hence dA/dx vanishes. Similarly oA /dy, dA/dz vanish. Therefore A contains E 3 
as a factor. 
Similarly if there be p distinct sets of values of cq b satisfying ( 11 ), (25), (26), 
it can be shown that all the partial differential coefficients of A up to the ( p — l) th 
order vanish. Hence A will contain E^ as.a factor. (See examples 4 (C.), 5 (C. ii.), 
6 (C.) in Arts. 10 , 11 , 12 respectively.) 
(ii.) The case in which two of the systems of values of the parameters satisfying 
equations ( 11 ), (25), (26) coincide, is dealt with in Arts. (12)-(25). The case, in 
which more than two systems of values of the parameters satisfying equations ( 11 ), 
(25), (26) coincide, may be treated in a similar manner. 
Example 2 .— Ordinary Envelope. 
Let the surfaces be 
2 + (* — a) (y — b) = 0 . 
(A.) The Disci 'iminant. 
A — 2. 
(B.) The Envelope Locus is 2 = 0 . 
Every point on 2 = 0 is the point of contact of one only of the surfaces. Hence 
2 occurs as a factor once only on the discriminant, 
MDCCCXCII.—A. 2 B 
