[324 
324] 
NOTE ON A THEOREM RELATING TO SURFACES. 
99 
FACES. 
something to the evidence of the theorem asserted, but I should be glad if a more 
simple one could be found. Analytically, the theorem is—“ If 
(x, y, z, ax -f Ay + 7z) m+n , 
where (a, /3, 7) are arbitrary, break up into factors (x, y, z) m , (x, y, z) n , rational in 
regard to (x, y, z), then (x, y, z, w) m+n breaks up into factors (x, y, z, w) m , (x, y, z, w) n , 
rational in regard to (x, y, z, w).” It would at first sight appear that (a, /3, 7) being 
arbitrary, these quantities can only enter into the factors of (x, y, z, ax + ¡3y + yz) m+n 
through the quantity ax + /3y + yz; that is, that the factors in question, considered as 
functions of (x, y, z, a, /3, 7), are of the form 
(x, y, z, ax+ {3y + yz) m , (x, y, z, ax + fiy + yz) n ; 
and then replacing the arbitrary quantity ax + /3y + 72 by w, the factors of (x, y, z, iv) m+n 
will be (x, y, z, w) m , (x, y, z, w) n . But the objection proves too much; for in a 
similar way it would follow that if (x, y, ax + ¡3y) m+n , where a, ¡3 are arbitrary, breaks 
up into the factors (x, y) m , (x, y) n , rational in regard to (x, y) (and qua homogeneous 
function of two variables it always does so break up), then (x, y, z) m+n would in like 
L, 62.] 
manner break up into the factors (x, y, z) m , (x, y, z) n , rational in regard to (x, y, z): 
and a simple example will show that it is not true that the factors of (x, y, ax + (3y) m+n 
ires, I think, a 
the order m + n 
be surface breaks 
only contain (a, /3) through ax + /3y ; in fact, if the function be = # 2 + y 1 + (ax + ¡3yf, 
then the factor is 
1 . 
Va^+ 1 ^ a2+ 1 )« + ( a /3 + ^Va-’ + ^ 2 + l)y}, 
surface in m + 2 
the m + 2 points, 
e line A A', the 
wo points A, A'; 
i surface of the 
rsection with an 
en surface with 
r plane through 
then the branch 
lving round BB' 
; passing through 
.at is, the conic 
or, what is the 
rbitrary plane in 
iven surface thus 
two surfaces of 
to add at least 
which cannot be exhibited as a function of a, /3, ax + ¡3y. 
I am not acquainted with any analytical demonstration; the geometrical one cannot 
easily be exhibited in an analytical form. 
2, Stone Buildings, W.G., November 26, 1862. 
3s of an ellipsoid, or 
equator or any other 
13—2