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. 259
The left-hand member of this equation is a series of ascending integral positive
powers of St; thus the function p must be an ascending series in integral positive
powers of ît; as the term of zero degree in the left-hand series is dz - pidxj, p must
therefore have the form
p = 1+ + t........................................... (13).
Inserting this value of p and equating the coefficients of corresponding powers of
St we have
d - p idi - rxidxi = -(dz - Cpidxi)...........................(14),
or d ( - piti) + i dp - ri dxii = o (dZ - p.idxi)..................... (15).
This linear and homogeneous condition in dz, dxi, dpi must be true for all values
of these differentials; hence, writing
Ç-spi p =- (, X,... æ P,.. p......................(16)
for convenience, we have
n i - 7ri = pi, z —, p- i =..........................(17).
Eliminating a and solving (16) for Ç we find
i= P, r=2jifpi-f, 7rri= —fl-pAi..................... (18).
The infinitesimal transformation is therefore completely determined, g, ei, 7ri being
given by an arbitrary function f2.
5. Let the preceding results be now applied to the infinitesimal contact transformation defined by the characteristic function
fi = v/1 + Pl2 + p 22 +... + pn2
The formulae (18) show that the coordinates of a surface element, by which we
mean the ensemble of a point and a plane through it, receive the infinitesimal increments
i =' aEit, 8z = - 2 St, p~ = 0.(19).
8x^ pi î-t, îZ ît, îPi- 2~0.....................(19).
Vl+ Sp.i2 Vl+2/1 + -Pi
This infinitesimal transformation generates a one-parameter group of contact transformations, namely the group of dilatations, whose finite equations are found by
integrating the simultaneous system
_ _ _ _ _ _ _ _ _ _ _ >V 1 ~ 3pi d p1 _ dp n
1 - =... --- P — dxn- _ dz = p=.....(20)
p/ p~' - 1 0 0
the integration effects itself, without any difficulty, and yieids the integral equations
piti t
xi = xi + \ z z-, i'= pi,...... (i = 1,..., n)......(21)
v1 + Epr 41 + E pi2
where t is an arbitrary constant.
33-2