18 
ON A DIFFERENTIAL EQUATION. 
[710 
where N is now of the order n — 2 at most, is expressible as the logarithm of a 
quasi-algebraical function, that is, a function containing powers the exponents of which 
are incommensurable (for instance, x^ 2 is a quasi-algebraical function): in fact, the integral 
is of the form 
/(■ 
,, A B \ dx 
M H 1 h ...) —¡= 
x—p x—q JxX 
where each term is separately integrable, 
^=log [ax + b + Va.Vjf}, 
1 
dx 
(x—p)\^X VP 
log 
(cip + b)x + (bp + c) + VP. VX' 
x — p 
where P is written to denote ap 2 + 2bp + c: the integral is thus = log Cl, where il 
is a product of factors 
ax + 
b + Va.Vx, («P + *)« + (»P + c) + ^-yx, etc> 
x—p 
M -A 
raised to powers —¡=, —— , etc.: hence, if we have a differential equation 
xa VP 
N'dz Ndx iV' V Zdz N\/Xdx 
^ = —r-, or 
n*Jz dVx’ 
U 
D 
where Z (= a'z 2 + 2b'z + o'), and N', D' are functions of 0 such as X, X, D are of 
x; then, taking log C for the constant of integration, the general integral is 
log Cl' = log C + log Cl: 
viz. we have the quasi-algebraical integral 11' — CCl = 0. 
The constants a, b, c, p, q, ... etc. may be such that the exponents are rational, 
and the integral is then algebraical: in particular, for the differential equation 
Vz 2 4- 14>z + 1 dz ’dx 2 + 14# -\-ldx 
z(z- 1) 
x (x — 1) 
the general integral is in the first instance obtained in the form 
(z+l + */Z)(z-l) 2 = c (#+l + VX) (.x-l) 2 
V'z (2z + 2 + V Z) 2 \lx{2x + 2 + VX) 2 
which, observing that (2x+ 2) 2 — X = 3 (x — l) 2 , may also be written 
(z + 1) (z 2 — 34z + 1) + ZX r Z _ q (x+ 1)(# 2 — 34a; + 1) + XVX 
V0 (z — l) 2 Vx (x — l) 2