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.

10 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION
unknown functions, the centre of the stars being the initial value of the time. In fact,
it is sufficient to employ M. Mittag-Leffier's expansions to obtain the unknown functions
for any value of the time. The coefficients in the expansions will be determined by the
initial conditions of motion.
1~. A very extensive class of dynamical equations can be reduced to the integration
of differential equations of the type
= SK PPrK,
1 1
where a(r + a_)= 0. Since in this case a finite strip enclosing the real axis is contained
in the stars of the functions Ps, the centre being t= O, new forms of the integrals of
these equations can be derivable by M. Mittag-Leffler's expansions*.
2~. Passing to the problems of attraction, it may be remarked that the problem of
the motion of a point attracted by fixed points placed in a straight line, the force being
according to Newton's law, has not been resolved when the number of attracting points
is greater than two. Let us consider the general case and suppose that the moment of
the initial velocity of the moving point m, with reference to the axis x of fixed points,
is not zero. Then e being the angle which the plane mx makes with a fixed plane
through x, and r being the distance of m from the axis x, we have the areal integral
r2% = C = constant,
and the integral of vis viva T- P=h= constant, where
T= 'm( 2r2 + 2 x2), = Mim
T being the vis viva and P the potential: in the latter expression the masses of the
fixed point are denoted by Mi and their distances from m by ri. It is at once obvious
that r cannot vanish. In effect, if for t =t0, r can become indefinitely small, let us
take this quantity as an infinitesimal of the first order. On account of the areal
integral, à would be infinitely great of the second order, and consequently r232 (= C)
would also be of the second order: T therefore would be infinitely great of the second
order. But P if it become infinitely great, can be so only to the first order because the
quantities ri are greater than r; hence if r could become infinitely small, the integral of
vis viva would not be verified. It therefore is to be inferred that the real axis of the
time is contained in the stars of the unknown elements: and consequently these elements
are expressible by Mittag-Leffler's series.
3~. Given n points repelling one another according to the Newtonian law of force,
the integral of vis viva may be written
ei is,+ i + s) + m
* I have studied this class of equations in three Notes class can be still further extended so as to include many
published by the Academy of Turin in 1898 and 1899. The of the classical problems in dynamics.