32 
ON EVOLUTES AND PARALLEL CURVES. 
[493 
At any point of a curve we have a normal, viz. this is a line at right angles to 
the tangent; or, what is the same thing, it is the line joining the point with the pole 
of the tangent. The locus of the pole of the tangent is the reciprocal curve, and for 
any point of a given curve, the pole of the tangent at that point is the corresponding 
point of the reciprocal curve. Hence, also the normal is the line from the point to 
the corresponding point of the reciprocal curve. And the curve and its reciprocal have 
at corresponding points the same normal. 
The envelope of the normals is the evolute; any curve having with the given 
curve the same normals (and therefore the same evolute) is a parallel curve; in other 
words, the parallel curve is any orthogonal trajectory of the normals of the given curve. 
The parallel curve is also the envelope of a circle of constant radius having its 
centre on the given curve; or, again, it is the envelope of a circle of constant 
radius touching the given curve. 
The theory in the above form is directly applicable to spherical, or rather conical, 
geometry ; but in ordinary plane geometry the absolute degenerates into a point-pair, 
the two circular points at infinity, or say the points I, J; and this is a case that 
requires to be separately treated. The theory in the general case, the absolute a conic, 
is the more symmetrical and elegant, and it might appear advantageous to commence 
with this; but upon the whole I prefer the opposite course, and will commence with 
the case of plane geometry, the absolute a point-pair. 
The subject connects itself with that of foci: I call to mind that a common 
tangent of the curve and the absolute is a focal tangent, and the intersection of two 
focal tangents a focus. In the case where the absolute is a point-pair, the focal 
tangents are the tangents from / to the curve, and the tangents from / to the curve, 
or say these are the /-tangents and the /-tangents; a focus is the intersection of an 
/-tangent and a /-tangent ; the line //, when it touches the curve, and (when the 
curve passes through / and / or either of them) the tangents at / or / to the curve 
are usually not reckoned as focal tangents; and other singular tangents, for instance a 
double or stationary tangent through / or /, are also excluded from the focal tangents; 
and the number of foci is of course reckoned accordingly, viz. it is the product of the 
number of the /-tangents into that of the /-tangents. So when the absolute is a 
conic; if this is touched by the curve, the common tangent at the point of contact 
is not reckoned as a focal tangent; and we may also exclude any singular tangents 
which touch the absolute; and the number of foci is reckoned accordingly, viz. it is 
equal to the number of pairs of focal tangents. 
Let the Pliickerian numbers for the given curve be (m, n, 8, k, t, l), viz. m the 
order, n the class, 8 the number of nodes, k of cusps, t of bitangents, l of inflexions ; 
and suppose moreover that D is the deficiency, and a the statitude ; viz. 
a = 3m + t, = 3 n + k] 
2D — (m — 1) (m — 2) — 28 — 2k, = (n — 1) (n — 2) — 2t — 2i, = — 2m — 2n + 2 + a 
= n — 2m + 2 + k, = m — 2n + 2 + i,