114 
PROCEEDINGS OF SOCIETIES. 
t, and to replace the two equations which contain z, by the equation 
given by the elimination of this quantity, and in like manner to reject 
the equation containing t\ and to replace the two equations containing *', 
by the equation given by the elimination of this quantity, and the system 
will be reduced to the required form. 
The reduction of the equation which gives n' is effected as follows, 
we have— 
(2b + 3 c) (n - 2) in - 3) = U + 18 h + a(2b + 3 c) + 12 be - 36/3 - 24 7 
- 12$’ — 4/o — 9 <r, 
3 b(n — 2) — 6/3 + 9<y + 9 1 - 1 * 3p ; 
and thence— 
(2b + 3c) (n - 2) (n - 3) + 3 b(n - 2) 
= a(2b + 3c) + 12 be + 8 ^ + 18A - 12$’ + 9t - p - 9o- - 30/3 - 15 7 
= a(2b + 3c) + I2bc + 8k +1 8h - 12* + 9t -18/3 -12 7 - p - 9<r - 3(4)3 + 7 ); 
and consequently— 
~8Jc- 18 h + 18/3 + 12 7 + 12* - 9 1 
= [a-(n- 2) (»-3)} (2b + 3c) - 3b(n - 2) + \2bc 
- p -9(7 - 3(4/3 + 7 ), 
which (observing that the left-hand side is precisely the combination of 
terms which enters into the equation for ri) shows that the reduction 
is possible; to complete it, putting for a its value n(n - 1) - 2b - 3c, 
we have— 
-8&-18A + 18/3+ 12 7 + 12*-9* 
& b(5n - 6 ) + c(\2n - 18) - 4 b 2 - 9c 2 - p - 9<r - 3(4/3 + 7 ); 
and substituting this value in the equation for n', we obtain 
n - n(n - l ) 2 - b(2n - 2) - 3c - p - 9o- - 3(4 (3 + 7 ). 
Some of the other equations admit of simplification: the equation 
a(n - 2) (n - 3) = 2d + a(2b + 3c) - 4p - 9c, 
if we put for a its value n(n - 1 ) - 2b - 3c, becomes— 
(4 n - 8 - 2b - 3c) (n - 2) (n - 3) = 25 - 4p - 9 <r, 
and the prescribed combination 
(2b - 3c) (n - 2) (n - 3) = 8k - 18 h + a(2b - 3c) -4 p + 9 <r, 
gives in like manner, putting for a its value 
(- n 2 + n + 4b) (n - 2) (n - 3) = ( 8 1c -ISh) - 4p + 9 <r. 
