360 
IklE. A. CATLET ON THE DOUBLE TANGENTS 
or substituting for □ and □ ' their values, we have 
(a, h, g, hjx, y, b', c', p', g', h'X«, b, c, 2/, 2g, 2h) 
— (a', b', c', f', g', R'Xax+hy+gz, ha;-\-by-]rfz, gx+fy-\-czf 
={a', b', c\f, g\ h!Xx, y, z)^(A, b, c, f, g, hJ^', b\ c', 2/', 2g', 2b!) 
—(a, b, c, f, g, iiX«'^+%+A 1}!x+b’y-\-fz, g'x-\-fy+G'z)\ 
which is Hesse’s theorem. 
If in particular («, c, /, g, h) are the second differential coefficients of a function 
y-, z)^-> and second differential coefficients of a function 
y-, zy\ then the equation becomes 
p{'p — l)zi .(a', b', c', f', g', h'X^^, 'by, —(2? -1)^5 b', c', p', g', byU, b,u'f 
=y(y— 1 )w'.(a, b , c, f, g, hX^,, byfu'—{p'—l)\A, B, c, P, G, hX^X byv!, b,uj 
and if for ii, u' we take the quartic function U and the sextic function H, its Hessian, 
we have 
12U.(a', b', c', f', g', h'X^^, by , 9 (a', b', c', p', g', h'X^^U, B^U)* 
= 30H.(a, b, c, f, g, hX^., ^3,, B^)"H-25(a, b, c, p, g, hX^^H, B^H, B^H)'; 
and if in this identical equation we write U=0, then from the resulting equation and 
the equation 
n, = — 3 H(a, b, c, f, g, hX^., BJH+(a, b, c, p, g, hXB,H, Bj,H, B^H)* 
we may eliminate any one of the three terms 
(A', B', c', f', g', h'XB,U, 
H.(a, B, C, F, G, hX^, 
(a, b, c, p, g, hX^*H, ByH, B^H)"; 
and in particular if the second term be eliminated, we obtain the equation 
n,=5(A, B, c, F, G, hXB,H, B,H, B,H)^-3(a', b', d, d, g', h'X^.U, B,U)^ 
and the equivalence of the two forms n, = 0 and n 2=0 is thus established. 
But Hesse’s theorem leads also to the demonstration of the equivalence of the third 
form n3=0. To use it for this purpose, I remark that if (a”, b”, d’, f’, g”, h") are the 
second differential coefficients of H — SH,, where after the differentiations x^, y^, 2 , are to 
be replaced by {x, y, z), then the theorem gives 
12U.(a", b", c", p", g", h"XB., BJU-9(a", b", c", f", g", ff'X^X, B,U, B,U/ 
=(«", b", d’,f, g", li<Xx, y, z)\{A, B, c, p, G, hXB„ B„ B,f(H-3H.) 
—(a, b, c, f, g, Rja!'x-\-h!'y-\-g"z, h!'x-{-b''y-\-f'z, g"x-\-f"y-^d’zf. 
But on putting {x, y, z) for (x^, y^, z,) we have (since H is a homogeneous function of 
the order 6, and H, before the change is a homogeneous function of the order 3 in 
{x,y, z)) a".r+A"y+^''2=5B^H— 3.2B,H,=5B,H — 3B,H (since, -on making the substi- 
