580
PROBLEMS AND SOLUTIONS.
[705
Cunningham in a paper in the last number of the Quarterly Journal of Science*;
and the question having been proposed to me by Mr Glaisher, I have also solved
it in a paper [580] printed in the April Number of the Monthly Notices of the Royal
Astronomical Society. I there obtain
pkx+lx" 1
Nk = 1.2 ... k coeff. x k in j- rr,
(1 — xp
viz. writing
rp rr&
u = N 0 + N 1 j + N 2 y~2 + • • •,
I show that u satisfies the differential equation
_ du 1..
2 s H1 + æ +
1 — X
U,
giving when the constant is determined
u =
glx+izt
(1 - xf ' ■
Writing the differential equation in the form
2(l-.)£-(S-<K
we at once obtain for Nk the equation of differences
N k - JcN k _! + \{k — l){k — T) N k -s = 0,
which is in fact a particular first integral of Mr Roberts’s equation; viz. from the
above equation we have
N k -x - (k -1) N k - 2 + i (k - 2) (k - 3) N k -, = 0,
and multiplying this last by k — 1 and adding, we have
iV* - N k -, - (* -1) 2 N k _ 2 + i (k - 1) (k - 2) {N k - 3 + (k - 3) N k -,} = 0,
which is the equation obtained by Mr Roberts. It thence appears that the general
first integral of his equation is
The equation
N k - kNje-i + i (k - 1) (k - 2) N k - 3 = (-)* Cl. 2 ... (k -1).
N k = kNk-x — I* (k — 1) (k — 2) N k - 3
gives very readily the numerical values, viz.
1 = 1.1-0
2 = 2.1 -0
5 = 3.2-1.1
17 = 4.5 - 3.1
73 = 5.17- 6.2
388 = 6.73-10.5
2461 = 7.388 -15.17
18155 = 8.2461-21.73.
* I have not the volume at hand to refer to, but he obtains an equation of differences, and gives the
numbers 1, 2, 5, 73, 398 (should be 388),...