50 
[718 
718. 
ADDITION TO MR GENESE’S NOTE ON THE THEORY 
OF ENVELOPES. 
[From the Messenger of Mathematics, vol. vn. (1878), pp. 62, 63.] 
The example, although simple, is an instructive one. Introducing z, g for 
homogeneity, the equation is 
X 2 y (y — hz) + 2X/ixy + y?x (x — az) = 0, 
giving the envelope 
xy [(x - az) (y - hz) - xy] = 0; 
that is, 
xy (hx 4- ay — abz) z = 0 ; 
viz. we have thus the four lines 
x — 0, y = 0, - + j- — z = 0, z — 0. 
CL 0 
Writing these values successively in the equation of the curve, we find respectively 
X*y (y - hz) = 0, 
/i 2 x (x — az) = 0, 
(b\-a^f b = 0, 
(Xy + gxf = 0; 
viz. in each case the equation in X, ¡i has (as it should have) two equal roots ^ but 
in the first three cases the values are constant; viz. we find X — 0, g = 0, hX — a/i = 0, 
respectively ; and the curves x = 0, y — 0, j +1 - z = 0, are for this reason not proper 
envelopes.