196 
ME. A. CATLEY ON THE DOUBLE TAA^GEYTS OE A PLAXE CTETE. 
that H . . . are the Hessians of U, U„ U„ . . Hz. H is the deteiminant formed 
with the second derived functions of U with respect to {x,y, z), H, the like deteimmant 
with the second derived functions of U, mth respect to the same quantities (.d y i) ; 
and so on. Moreover let D«-H, =(Xd.+Yd,+Z^.r^H, denote the e/z-2)thic 
emanant of H with respefct to the current coordinates (X, Y, Z) as facients of emanation ; 
and similarly let . . . denote the (7i-2)thic emanants of H^, H, ... 
in respect to the same facients of emanation— it being understood that in aU these 
functions, {w, y, zj are after the differentiations to be replaced by (x, y, z). It is to 
be observed that U, is of the degree {n-r) in {x, y, z), and consequently H of t^ e 
degree 3(w~2-r); hence is of the degree 3(w=-2--r)-(w-2), ~-2(?z -) or, 
which implies that r>|(?i~2), for otherwise would be identically equal to zero 
Upon replacing (zr, by (^, y , .), (r satisfying the above condition) becomes o 
the degree 2{n-2) in {x, y, z), and it is obviously of the degree 3 in the coefficient, o 
U, and of the degree (w-2) in the current coordinates (X, Y, Z). 
3. This being premised, we have 
n=(tIx,Y,z)-“ 
= D»-^H + &c. = 0 
for the equation of the curve of the order (n-2), which by its intersection with the 
taiment gives the tangentials of the given point; the numerical coefficients are the 
binomial coefficients of the order (^-1) taken ivith the signs + and - alternately-, and 
the series is continued as long as the terms do not vanish, that is, if as before r denote 
the suffix of H, for so long as r>|(.z-2) ; but of course the value mil not be altered 
by continuing the series to In particular, for the quartic we have 
a=D"H-3D^Hn 
for the quintic 
n=D*H™4D*H,+6D^H„ 
and so on. The function fl, like the several component terms, is of coimse of the degiee 
3 ill the coefficients of U, and of the degree 2(n~2) in (x, y, z). ^ 
4. It is to be remarked that the formula applies to a cubic ; we have here simp > 
which agrees with a result already mentioned. It may be noticed also that ui 
the general case the formula gives at once the condition for the points ol inffexion; m 
fact if the point (^, y. z) be a point of inflexion, then one of the tangentials must coin- 
cide with this point, or the equation 0 = 0 will be satisfled by writing therein {x, y, * 
for (X Y, Z) ; but when this is done D^^H, &c. reduce themsffives (to numerical 
factors j?m) to H, and the equation becomes simply H = 0, which is the well-knomi 
condition for the points of inflexion. _ , Tr_n 
5. If two of the tangentials coincide, or what is the same thing, if the tangent 
touches the curve 0=0, then the point {x, y, z) ivill be the point of contact of a doub e 
tangent. The equation which expresses the condition in question, treatmg tierein 
