AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
183 
These give 
[*, «.] + K. «,] ^ + [a„ 6,] 
3a; 
[*. M + K, 4 J I 1 + [«-!, &i]f£ 
= 0 
= 0 
(52). 
Similar equations exist for finding 
3a, 36, , da, db, 
^ ~ cUI Cl 5 • 
dy dy dz d: 
These, however, are not jet required, it being seen at once that, in general, 
3A_3R „ p Df 
dx ~ dfJ + U \ h: 
(S3). 
[For a case of exception, in which yp- p 1 -fi 1 ~ ^ does not vanish, see Art. 19, 
Ex. LI, E.] 
Differentiating again with regard to x, 
a^_3[R 
he 2 ~ d.j? J + 
0r iy 
dr, Dx 
+ R 
!>/ , D 3 / da, D 2 / 86, 
Dor I)./; Da , 0a; D.i D6, dx _ 
9 2 A _ 3R D/ 0R D/ T Dy_ D;/ 8a, 
3./; 3y dxdyJ dx Dy Dy Dx [_D.c Dy Da; Da, 3y 
D 2 / 36 
Da: D6, 3// _ 
Hence, bj (52), 
3 2 A 3 2 R 
3R Df 
D 
Bx> - a- sjf+ 2 a. n- + l! 
3.i Do: 
py iy D/n 
; d 
p/ 
D fl 
Dx ’ Da, ’ D6, 
/ 
/ 
Da, 
D6, 
D [r, a„ 6,] 
D [a, 
(54), 
. (55). 
(56), 
D 
3 2 a _ sry 3 r iy 3 r d/ 
0.i dxdy' 3./; Dy 3y Do; ^ D [«, a„ 6,] 
iy 
D/1 
/ D 
p/ 
iy~ 
Da, 
D6, 
./ 
Da, 
D6, 
D [a„ 6,] 
■ (57). 
Art. 8 .—Proof of the Envelope Property. 
Let 7], £ be a point on the locus of ultimate intersections, and let the values of 
ci, b satisfying the equations (3), (4), (5) when x = £, y — rj, z = £. be a, yd. [It 
will be supposed first of all, that only one value of a exists, viz., a, and only one value 
of b exists, viz., /3. But the following cases will afterwards be noticed, viz., (1) 
