84] 
TWO SURFACES OF THE SECOND ORDER. 
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equation If + mg + nh = 0) in the coefficients f g, h, l, m, n ( 1 ). But re-establishing the 
actual values of the coefficients A, B, &c., A', B', &c. (by which means the function 
□ becomes a function of the sixteenth order in x, y, z, w) the quantities f g, h, l, m, n 
ought, it is clear, to disappear of themselves; and the way this happens is that the 
function □ resolves itself into the product of two factors M and ''P, the latter of 
which is independent of f g, h, l, m, n. The factor M is a function of the fourth 
order in these quantities, and it is also of the eighth order in the variables x, y, z, w : 
the factor ' V P is consequently of the eighth order in x, y, z, w. And the result of 
the elimination being represented by the equation A P = 0, the Intersect-Developable 
in the general case, or (what is the same thing) for systems of the class (A), is of 
the eighth order. In the case of a system of the class (.B) the equation obtained as 
above contains as a factor the square, and in the case of a system of the class (G) 
the cube, of the linear function which equated to zero is the equation of the plane 
of contact. The Intersect-Developable of a system of the class (B) is therefore a 
Developable of the sixth order, and that of a system of the class (G) a Developable 
of the fifth order. The elimination is in every case most simply effected by supposing 
two of the quantities X, y, v, p to vanish (e.g. v and p): the equations between which 
the elimination has to be effected then are 
A X* + B y- + 2 H Xy = 0, 
A'X~ + B'y 1 + 2H'Xy — 0 ; 
and the result may be presented under the equivalent forms 
(AB' + A'B - 2HHJ -4 (AB-H*) (A'B' - H" 2 ) = 0, 
and {AB' - A 'By +4 (AH'- A'H) {BH' - EH) = 0, 
the latter of which is the most convenient. These two forms still contain an extraneous 
factor of the eighth order in x, y, z, w, of which they can only be divested by sub 
stituting the actual values of A, B, H, A', B', H'. 
A. Two surfaces forming a system belonging to this class may be represented bv 
equations of the form 
a x 2 + b y 2 + c z 2 + d vP — 0, 
a'x 2 + b'y 2 + c'z- + d!w- = 0, 
1 I believe the result of the elimination is 
□ = 4 (PB - Q 2 ) = 0, 
where, if we write uA + u'A' = A, &c., the quantities P, Q, 11 are given by the equation (identical with respect to u, u') 
Pu- + 2Quu' + Ru' 2 =(Aa 2 + ...) (Aa' 2 + ...) - (Aaa'+ ...) 2 
=?t 2 {(BC - F 2 ) f 2 +...} + uu' {{BC’ + B'C- ‘IFF’) f 2 +...}+ u' 2 {(B'C’ - F' 2 ) f- + ...} 
a theorem connected with that given in the second part of my memoir ‘On Linear Transformations’ (Journal, 
vol. i. p. 109) [see 14, p. 100]. I am not in possession of any verification a posteriori of what is subsequently 
stated as to the resolution into factors of the function □ and the forms of these factors. 
C. 
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