The collected mathematical papers of Arthur Cayley.

591]

SOLUTIONS AND REMARKS.

231

viz. since X + Y=-P, and XY= Q, this is
(p2 4Q) X'' = 0,
and writing also
X = - {P + V(P2 - 4Q)}, Y= - { - V(P2 4Q)},
we find
X'r' = P'2 (PP'2W)')
A-1/= 4 {pt2-( pP 4Q }*
the differential equation thus is
(p2-4Q) {p'2 (PP'- 2Q')} = 0.
The application to the theory of singular solutions is that, in the case where the
function (1, P, Q...)(c, l)n breaks up into rational factors c-X, c-Y,..., the factor
=(X- Y)2(X -Z)2... divides out and should be rejected from the differential equation,
which in its true form is X'Y'Z'... = 0; viz. this is what we obtain immediately,
considering the given integral equation as meaning the system of curves c - X = 0,
c- Y= 0,..., and there is not really any singular solution; whereas in the case where
the factors are not rational, the factor in question, when the product X'Y'Z'... is
expressed in terms of the coefficients P, Q,..., and their derived coefficients does not
divide out from the equation; and in this case, equated to zero, it gives a proper
singular solution of the equation.
11. In the theory of elliptic motion, v denoting the mean anomaly and e the eccenl+e
tricity, if m' be an angle such that tan v= etan m', find in terms of e, m' the
mnean anomaly m.
Taking as usual u for the eccentric anomaly, to commence the solution write down
tan -v = -e) tan u
l+e
= tan im',
that is,
tan u =/( + e)tan a,
and u being given hereby as a function of m', we have by substitution in the equation
m=u-esin u, to find m as a function of n'.
A creditable approximate solution would be m =m'+ 0. e, viz. this would be to
show that neglecting terms in e2, &c., we have m = mn'. In fact, taking e small, we have
tan 2u = (1 + e) tan m',
and thence if Ai = m' + x, we have
tan n m' + -x sec2 n' = (1 + e) tan 'm,
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