Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 265
exists between their symbols, we can verify by this principle that the transformations
of the above group are commutative* 1~ with all dilatations, 2~ with rotations about
the origin, 3~ with all spiral transformations starting from the origin, 4~ with all pedal
transformations, 5~ with all point and contact transformations commutative with all
rotations about the origin, 6~ with all transformations of the infinite group whose
characteristic function is
_1 _2 X +1 ), 2)(44)
) X <> ) *............................... (44),
where
1, X= i $a +.................... (45).
1 axi aZ ) \W axJ
The first case of commutation is especially interesting because of reasons given in
~ 6. The second may be shown even more simply by introducing polar coordinates.
The aequationes directrices (39) themselves exhibit certain geometrical properties
of the transformations. For example they show that every point (z, xi,..., xn) is
changed into a circle whose points are at the same distance from the origin as the point
(z, xi,..., xn) itself. Further the radii vectores of (z, xL,..., Xn) and (z', x,..., xn')
make an angle with each other whose cosine is k.
11. The particular transformation of the above group, namely that corresponding
to k = and accordingly defined by the two equations
Z2- z2 +- (i/2 - x i2) = O, + Z iXi' 0........................(46),
1 1
was first studied as a contact transformation by Goursat, in three dimensions t.
If in equations (46) z', x',..., xn' be regarded as constants and z, xz,..., x, as
current coordinates, these equations define a certain circle C in n + 1-dimensional space,
the locus of (z,,..., xn). That is the equations make a circle C correspond to
every point (z', x',..., Xn'), and similarly, since the equations are symmetrical in both
sets of variables, to every point (z, xz,..., xn) there corresponds a circle C' in the current
coordinates (z', xz,..., xn'). When the point (z, xz,..., x,) describes a surface S, the
circles C' relative to the several points of 2 form a congruence. The focal surface of
this congruence is the surface,' into which Z is transformed. -' is also the locus of
the points (z', xz',..., X,') such that the corresponding circles C' are tangent to S.
The focal surface of the congruence of circles C' is a plane passing through the
radius vector OP and the normal PN to the surface at P. Thus to construct the
point P' corresponding to P it is only necessary to draw, in the plane passing through
OP and the normal PN, the perpendicular OP' to OP, cutting off a distance OP' equal
to OP.
* In the last loc. cit. Lie shows indirectly that the enumerated commutative properties appertain to these
transformations in three dimensions.
+ See loc. cit. p. 267.
VOL. XVIII. 34