84] 
TWO SURFACES OF THE SECOND ORDER 
495 
The substitution of these values gives after all reductions the result 
f 2 g 2 h 2 w« + 4 (pf- qg)fgh 2 zw 5 
+ 4 (r 2 h 2 — Qpqgf) h 2 z 2 w* + 2 (cfg 2 + 2prfh) flixhv 4 + 2 (pf 2 + 2qrgh) ghy 2 w 4 
— 16 {pf — qg) z 3 w 3 — 4 (q 2 g 2 — 4ipf 2 — Qpqfg) qhx 2 zw 3 — 4 (pf 2 — 4<q 2 g 2 — 6pqfg)phy 2 zw 3 
+ 16p 2 q 2 h 2 z 4 w 2 - 8 (pf+ 4 qg) q 2 phx 2 z 2 w 2 — 8 (qg + 4pf)pq 2 hy 2 z 2 w 2 
+ (q 2 g 2 + 8prfK) q 2 x i w 2 + (pf 2 + 8 qrgh) p 2 y 4 w 2 + 2 (10r 2 /i 2 — pqfg) pqx 2 y 2 w 2 
— lQp 2 q 3 hx 2 z 3 w 4-1 6p s q 2 hy 2 z s w 
+ 4 (4p/+ 5qg) pcfoftzw — 4 (4 qg + 5pf) p 3 qy i zw — 4 (pf— qg) p 2 cfx 2 y 2 zw 
+ 4p 2 q*x i z 2 + 4p A q 2 y 4 z 2 + 8p 3 q 3 x 2 y 2 z 2 + 4<pq 4 rx s + 4>p 4 qry 3 + 12p 2 q 3 rx i y 2 + 12p 3 q 2 rx 2 y A = 0 ; 
which is therefore the equation of the envelope for this case. This equation may be 
presented under the form 
+ 4pq (qx 2 -f py 2 ) 2 (qrx 2 + rpy 2 + pqz 2 ) = 0, 
and there are probably other forms which I am not yet acquainted with. 
C. The reciprocals of the two surfaces made use of in determining the Intersect- 
Developable, although in reality a system of the same nature with the surfaces of 
which they are reciprocals, are represented by equations of a somewhat different form. 
There is no real loss of generality in replacing the two surfaces by the reciprocals of 
the cones x 2 — 2yz, y 2 = 2zw; or we may take the two conics 
(x 2 — 2yz = 0, w — 0) and (y 2 — Zzw — 0,x = 0), 
for the surfaces of which the envelope has to be found, these conics being, it is 
evident, the sections by the planes w = 0 and x = 0 respectively of the cones the 
Intersect-Developable of which was before determined. The process of determining the 
envelope is however essentially different: supposing the plane %x + r/y + £z -f cow = 0 to be 
the equation of a tangent plane to the two conics (that is, of a plane passing through 
a tangent of each of the conics) then the condition of touching the first conic gives 
If 2 — 2rj£=0, and that of touching the second conic gives f — 2£Vi> = 0. We have therefore 
to find the envelope (in the ordinary sense of the word) of the plane f-x + yy -1- "Qz + cow = 0, 
in which the coefficients If, y, f, co are variable quantities subject to the conditions 
If 2 -2^= 0, ?7 2 -2£« = 0. 
The result which is obtained without difficulty by the method of indeterminate 
multipliers, [or more easily by writing If : y : £ : w = 26 3 : 26 2 : 6 4 : 2] is 
8 y 4 z — S2y 2 z 2 w + 32 z 3 w 2 — 27x A w + 27 x 2 yzv) — 4 x 2 y 3 = 0, 
which may also be written under the form 
8^ (y 2 — 2zw) 2 — x 2 {4y s + 9 (3&' 2 — 8yz) w } = 0. 
[Another form, containing the factor w, is 4 (y 2 + 2ziu) 3 — (2y 3 + 27x 2 w — SQyzw) 2 = ().]