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),...