which puts in evidence the tangent 77 — /3'z. It is easy to see that the equation may 
be written in any one of the four forms 
VZ (f- az) + Vra (£- 0z) + (m - l)(77 - o'«) = 0, 
Vm (£ - a*) + V Z (£ - £2) + *, (m - Z)(t7 - /3 r 2)= 0, 
V Z (77 — <*'2)+ Vm(r)— (3'z) + /y/— ( m ~ Z) (£ - a2 ) ••= 0, 
Vm (77 — Zz) + V Z (77 - /3'2) + ,y/- 4^ * ( m -Z)(f- 782) = 0, 
viz., in forms containing any three of the four radicals \/¡~ — az, — /32, \Zrj — afz, 
V 77 — fi'z. The conic is thus expressed as a trizomal curve, the zomals being each a 
line, viz., they are any three out of the four focal tangents; the order of the curve, 
as deduced from the general expression 2" -2 r, is = 2 ; so that there is here no depression 
of order. 
115. But the ordinary form of the focal equation is a more interesting one; viz., 
SI, 33 being as usual the squared distances of the current point from the two given 
foci respectively, say 
21 = (£ ~ a z) (77 - az), 
23 =(£—£*) (v-Pz), 
then 2a being an arbitrary parameter, the equation is 
2az + V2l + V23 = 0, 
viz., the equation is here that of a trizomal curve, the zomals being curves of the 
second order, that is, the zomals are (z 2 = 0) the line infinity twice, and the line-pairs 
AI, AJ and BI, BJ respectively: the general expression 2 v ~ 2 r gives therefore the order 
= 4 ; but in the present case there are two branches, viz., the branches 
2 az + V21 — V23 = 0, 2az — V21 + V23 = 0, 
each ideally containing (2 = 0) the line infinity; the curve contains therefore (z 2 = 0) 
the line infinity twice, and omitting this factor the order is = 2, as it should be. 
116. To express the equation by means of the other two foci A lf B u writing the 
equation under the form 
21 + 23 + 2 V2i23 — 4a 2 2 2 = 0, 
and then if 2fi, 23i are the squared distances of the current point from A l} B L 
respectively, we have (ante, No. 65), 
2133 = 2l 1 33 1 , 
21 + 33 — 2l x — 33 a = kz 2 ,