AND LINES IN THE INTERSECTIONS OP A SYSTEM OF SURFACES. 
143 
Now denoting differentiation when x, y, z are the only independent variables by 0, 
0A _ 0Q ^ , n /HA , HA d*i 
dx dx ^ \Dx Dcq dx 
Hence, since a — a x satisfies (2) 
0A 
dx 
0 M 
dx + 4 Dx 
(?)> 
assuming that D/JDa 1 dajdx vanishes when DA/Doq vanishes. 
Now when x — y = y, z = £, jA = 0, therefore 
0A 
dx 
Q 
Dx 
X= f 
y = v 
z = { 
■ W- 
Hence when x = tj,y = r),z=L, 
•0A/HA 
0X / Dx 
0A /HA _ 0A 
0y / % — 03 / D. 
(9). 
Now the tangent plane to the surface 
/(x, y, z, a) = 0 
at the point a? = £ y = y, z = £ is 
(X-f)^ + (Y-,)| + (Z-i)|(=0. 
■ (10) 
( 11 ). 
Now D//D£ stands for D/(x, y, z, a)/Da?, when x — y — y, z = £. And the 
value of DA/Da?, be., DA(ay y, z, aq)/Da?, when x = £, y — y, z — £, and, therefore, 
oq — a is the same as the value of D f{x, y, z, a)/Dx, when x = y = y, z — £. 
This may be expressed thus 
0A fDf _ 0A /Df _ 0A /H/ 
0£/ D£ — 0t? / Dt? ” 0f / Df' 
Hence the tangent planes to the surfaces A = 0, /(a?, y, 2 , a) = 0 at the point 
A y, £ coincide. 
This proves the envelope property in general for the locus of ultimate inter¬ 
sections. 
Hence A vanishes, if in it x, y, z be made respectively equal to £•, y , £, the coordi¬ 
nates of any point on the envelope-locus. 
Therefore A contains E as a factor. 
