536 
[959 
959. 
NOTE ON PLUCKEE’S EQUATIONS. 
[From the Messenger of Mathematics, vol. xxiv. (1895), pp. 23, 24.] 
It is well known that if 
then the equations 
A, B, C, D = 2, 3, 6, 8, 
n = m-— m — AS — Bk, 
l = 3m 2 — 6m — C8 — Dk, 
m= n 2 — n — Ar—Bc, 
k = 3?i 2 — 6?i — Gt — Dl, 
are equivalent to three independent equations giving n, t, l in terms of m, 8, k. 
It is easy to show that the necessary conditions in order that this may be so, are 
that is, 
C = SA, and D=SB-1, 
A, B, C, D = A, B, 3A, SB -1, 
where A and B are arbitrary. 
In fact, from the last two equations eliminating r, and for n, i substituting their 
values, we have 
Cm — Ak = (G — 3A) (m- — m — A3 — Bk) 2 , 
— (C — 6H) (m 2 — m — H3 — Bk), 
+ (AD — BG) (Son 2 — 6m — G8 — Dk), 
which must therefore be an identity. In order that the term in m 4 may vanish we 
must have C = 3A ; and then substituting this value for C, we must have 
3Am — Ak = SA (m 2 — m — H3 — Bk) + (AD — SAD) (3m 2 — 6m — 3AS — Dk). 
Here the coefficient of m 2 must vanish, that is, 
0 = SA + SAD - 9AB, or D = SB-1, 
and, substituting this value, the equation is 
SAon — Ak = 3A (— m — — Bk) 
that is, 
- A {— 6m -SAS- (SB - 1) «}, 
3 m — k = — Son — 3HS — 3 Bk 
an identity. 
+ 6m + SA8 + (SB — 1)k,