414 Proceedings of Royal Society of Edinburgh. [sess-. 
If we investigate the general case independently, we shall get 
as the condition that the line lx + my -j- n = 0 may be a tangent 
A l 2 Bm 2 + Cn 2 + 2F mn + 2G nl + 2fIZra = 0 
with a known notation ; and replacing Z, m, n by the above ratios- 
we have the required equation to the pair of tangents in the most 
reduced form. 
We may observe that on substituting the values of x", y" in the 
first form of equation, the quantity ab - h 2 will be a factor 
throughout, which on being removed gives the more reduced form 
last obtained. 
Since the above results hold alike for oblique and rectangular 
axes, we may deduce corresponding equations in the areal and 
trilinear system by changing to oblique Cartesians with two sides 
of the triangle of reference as axes. 
In the case of the general equation 
ux 2 + vy 2 + ivz 2 + 2 uyz + 2 vzx + 2 id xy = 0 
in areals, we shall find the result corresponding to the first form 
of equation to be 
K 2 
+ u — v - w )(x - x) 2 4- — • (xy z") 2 = 0 
with a known notation; while if it be treated independently of 
Cartesians, we shall get as the condition that the line Ix + my + nz 
= 0 may touch the conic 
UZ 2 + Y m 2 + W n 2 + 2U W + 2Y nl + 2 W'zl L 0 
where U, Y, W, etc., have their usual meanings ; and replacing 
Z, m, n by yz - yz, zx — zx, xy ' — xy respectively, we have the- 
required equation to the pair of tangents in the most reduced form.. 
A like observation to that made above also applies to these two- 
forms of equation. 
( Issued separately June 5 , 1903 .) 
