598 
NOTES AND REFERENCES. 
Tó complete Boole’s solution : the equations easily give 
and 
s'tx'y st'xy 
u — ay) u — ßq 
st' 
1 — u 
s't'x' 
s't'y' 
stxy 
1 — ap' 
1 — ßq' — u ap + ßq 
= V; 
and multiplying together the first three values, and also the second three values, we 
have in each case the same numerator ss' 2 tt' 2 xx'yy', and we thus obtain the equation 
(u - ap) {u — ¡3q) (1 — u) — (1 — ap — u) (1 — ftp' — u) {ap + fiq — u) = 0, 
which, the term in u 2 disappearing, is a quadric equation; it is in fact 
u 2 (— 1 + op + /3q') + u {1 + a (p —p') + /3 ( q — q') — a 2 pp — ¡3 2 qq + a/3 (— 1 + 2p'q')} 
+ {— ap — ¡3q + a 2 pp' + ¡3 2 qq + aj3 (1 —p'q) — {op + [3q) a(3p'q'} = 0 ; 
or, what is more simple, if we write with Boole ap = a, ¡3q = b, 1 — ap = a', 1 — ¡3q' = b', 
ap + (3y = c, then the equation is {u — a) {u —b){l—u) — {a — u){b' — u) {c — u) = 0, that is 
(1 — a' — b') u 2 — [ab — a'b' - 1 - (1 — a' — b') c'} u + {ab — a'b'c') = 0, 
giving 
_ab — a'b' + (1 — a — b') c' + Q 
U ~ 2(1 -a'-b') ’ 
where 
Q 2 = {ab - a'b' + {l- a '~ b') c'} 2 -4(1 -a'- b') {ab - a'b'c'). 
We have as conditions which must be satisfied by the data, that each of the 
quantities a', b', c is greater than each of the quantities a, b; or say, each of the 
quantities 1 — ap', 1 — (3q, ap + (3q greater than each of the quantities ap, f3q : Q 2 is 
then real, and taking Q positive, we have u equal to or greater than each of the three 
quantities and greater than each of the two quantities. The difficulties which I find 
in regard to this solution have been already referred to. 
139. See volume I. Notes and References 13, 14, 15, 16 and 100. I have in the 
last of these noticed that the terms covariant and invariant were due to Sylvester: 
and I have referred to papers by Boole, Eisenstein, Hesse, Schlafli and Sylvester. 
Anterior to the present memoir 139 we have other papers by Boole and Sylvester, 
one by Hermite (with other papers not directly affecting the theory), a paper by 
Salmon, and a very important memoir by Aronhold: it will be convenient to give a 
list as follows: 
Boole. 
1. Researches on the theory of analytical transformations with a special application 
to the reduction of the general equation of the second order, Camb. Math. Jour. t. n. 
1841, pp. 64—73.