TRANSACTIONS OF SECTION A. 653 
1st. Consider the non-integrable linear total differential equation 
(2 — ay — y)dx + (x? —aryz — x)dy + ds = 031 
the equation 
P(Q, — R,) + Q(R, — P.) + RP, — Q,) = 0 
gives 
(2 — zy) (@ - zy — y) = 0, 
and 
s—ay =0 
is a singular solution of the given equation. 
2nd. The Pfaff equations 
(y + xy? — yz)dx + (x + a’y — 2x)dy — dz 
(y — vy? + yz)dx + (xv + v?y — zx)dy — da 
ll 
= 
both admit of the infinitesimal point transformation 
0 
@— api 
as is readily verified ; both admit of the singular integral 
S= xy. 
The first of the two equations is integrable, hence the construction of the 
aes example is not applicable here ; but by Lie’s general theory the function 
z — zy) ‘is an integrating factor of the equation, by which we find the general 
integral to be 
ay + log (xy — 2). 
ord. The non-integrable Monge equation 
Qx(2 — «x — y) 22—-x-y) (2 +1)@-w-y) 
e du* + e dy? — dz* + 2e dudy = 0 
admits of the infinitesimal point transformation — 
Of _ of. 
oz oy” 
the equation (27) becomes 
Q2(e — a — y) 2(e-—- x —y) (2 + 1)\(2-2-y) 
e + 
e — Qe = 0, 
which gives the singular solution 
Z=U+ Y. 
11. On Fermat's Numbers. By Lieut.-Col. Antawn Cunnineuan, R.£., 
Fellow of King’s College, London. 
These are numbers of form N, =2°"+1. Until about 1729 they were sup- 
posed to be all prime, although only the first five (N,, N,, N,, Nj, N,) had been 
proved prime. But, about December, 1729, N, was completely factorised by Euler: 
and between 1876-86 four more were determined composite by various mathemati- 
cians, viz., N, completely factorised; and N,,, N,,, Ny) one factor of each found. 
1 This form is taken from Guldberg’s note in the Comptes Rendus, to which refer- 
ence has been made. ; 
