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.

262 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS.
the 2n 2 +r equations (30) and (31) in general determine the 2n 2 + r functions
z', Xi', pi', Xj, p as functions of z, xi, pi.
Eliminating p we can write the following n + r+ equations for z', x,', Xj,
S^^+^2X^ii_.. L-~=0, j=l,..., n,
i=l o,....a=
Si=i~, cl=0., coi )..=. 0
resolving these for z', xi', Xj, the remaining functions pi', p are found by substituting
the values of the former in the remaining equations of the system (30) and (31).
2~. In the second place two transformations S and T are commutative when the
symbolic equation
ST= TS
obtains. Consider the contact transformation S and the point transformation T. That
the point P is changed into the point Pl by the transformation T is expressed by the
symbolic equation
(P) T= (P1).
In the same manner, that S transforms P into the surface Z is expressed by the
equation
(P) S= ().
Then if (P) ST= (P) TS,
we have also (Pl) S () T.
That is, if S transforms the point P into the surface E, and T changes the point
P into the point P1, the latter is changed by S into the surface into which the
surface 2 is changed by T.
3~. In the third place let S be a contact transformation of an n + 1-dimensional
space commutative with all translations T of that space. If S changes a definite point
P into the surface 2, the surfaces into which all other points are changed by S may
be determined, for there always exists a translation which carries the point P to any
other arbitrary position Pi; then by the second paragraph above, the point P1 is
changed by S into the surface Y, into which 2 is changed by the last-named translation; hence all points are changed by S into congruent surfaces similarly situated.
Accordingly the contact transformations that are commutative with all translations of a
space of any number of dimensions are determined by a single function of the form
V((x1-x X, X=-X21,..., Xn,-xn, Z - ') =........................(33);
it is not to our purpose to construct the explicit forms of these transformations here;
the most general one in the plane has been given by Lie in his geometry of contact
transformations to which reference has been made.