20 
ON A DIFFERENTIAL EQUATION. 
[710 
and the rationalised equation is 
(1, -12, 807, 2504, 807, -12, 1\z, l) 6 
z(z- 1) 
(1, -12, 807, 2504, 807, -12, l%r, 1) 8 = 0. 
x(x — l) 4 
This is a sextic equation in z, of the form 
^ + ii +x ( z,+ ?) +i ‘(^ + i) +! ' =0, 
/x, v = - 12 — il, 807 + 412, 2504-612, 
where 
if 12 denote the function of x which enters into the equation; and writing z + - = 8, this 
z 
becomes 
8 3 — 38 + \ (8‘ 2 — 2) + ¡jl6 + v = 0. 
But the equation in z is satisfied by the value z = x, and therefore the equation in 8 by 
the value 8 = x + ^ = a suppose, we have therefore 
a 3 — 3a + X (a 2 — 2) + /¿a + v = 0, 
and thence subtracting, and throwing out the factor 8 — a, 
8 2 + 8a + a 3 - 3 + A (8 + a) + fi = 0 ; 
viz. writing for A, fi, a their values, this is 
02 + q L + 1 _ 12 - a) + X 2 - 1 + - 2 - lx + -) (12 +12) + 807 + 412 = 0, 
or, what is the same thing, 
8 2 + 8 (x - 12 + - - Si) + a? - 12« + 806 - — + \ - (x - 4 - 12 = 0, 
V x J x x 2 \ x) 
where 
12 = 
(1, -12, 807, 2504, 807, -12, 1$«, l) 6 . 
x{x— l) 4 
Hence in the quadric equation, the coefficients, each multiplied by (x — l) 4 , are 
and 
(®-lW®-12+-1-^(1, -12, 807, 2504, 807, -12, l\x, l) 6 , 
X! x 
12 1 
(x - l) 4 (x 2 - 12x + 806 - — + 
\ XX 2 / 
- 1 (¿»-4 + -W - 12, 807, 2504, 807, -12, 1%», l) 8 , 
which are respectively rational and integral quartic functions of x; and, writing for 8 its 
value, the equation finally is 
/ iy A (_ 1^(1, 188, 646, 188, l%ar, l) 4 | ^(1, -644, 3334, -644, l\x, I) 4 A