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. 261
By integrating the simultaneous system
dxl dxI,, dz' dpn- = d. (25),
I -Ip/.... -I,- — i-pi, —1 —[ — 0 - '" -- -0.....................(25),
H,' Y-pi ï Hp i n- l H O -O
we have the corresponding one-parameter group of contact transformations
X = x + rnptP, ' = z + (=PiUpi - t, =...... (i = 1, -..., n).........(26).
Let t have for the moment a fixed value; the corresponding contact transformation
of the group changes the point (x,..., x,,, z) into a surface whose equation in current
coordinates (xl',..., x,', z') is obtained by eliminating the pi from the first n + 1
equations; this elimination yields the equation
(t ' t ' t " t ' t )......... (27).
The form of this equation enables us to find the characteristic property of these
transformations as the following considerations will make evident.
1~. In the first place it is clear that contact transformations in n + 1-dimensional
space may be determined by a system of r equations
1 (X... -, Zn, Z 1,..., Xn)= 0, 2 = 0,..., O,.= 0...................(28),
where r may have all values from 1 to n + 1; in the last case the transformations if
existent will be point transformations, since the n + relations will give the n+ 1
quantities x', z', as functions of the n + 1 quantities xi, z alone.
In fact the problem of determining all finite contact transformations of a space
of n + 1 dimensions is that of resolving the total differential equation
n n
dz' - pidx' - p (dz - pidxs) = O () 0,...... (,,n)...............(29)
i 1
where the z', x', pi' are functions of the 2n +1 variables z, xi, pi to be determined.
This equation shows that there ought to exist at least one relation between the variables
z', x', z, xi containing z' and z*. Taking the general case of r different relations expressed by (28), the equation (29) ought to be a consequence of
do1 = O, do = O,...,. =.................................(30);
that is, it ought to be possible to find r coefficients ÀX,..., X,. such that the identity
n n r
dz' - pi'dxi'- p (dz - lpidxi) = 2Xidci
1 1 1
exists. This demands the following equations:
1 = l...,,z Pi i au),...........(3 );
p = — x a. =i — ii;* See Goursat, Leçons sur les équlations aux dérivées partielles du premier ordre, Paris, Hermann, 1891, p. 258,