■ROYAL IRISH ACADEMY. 
115 
The system of seventeen equations then is— 
d ~ ct 
a'= %(% - 1) - 2 5 - 3 c, 
k— 3 %(% — 2) — 65 — 8 c, 
B'= \n{n - 2) (% 2 - 9) - (% 2 - n - 6 ) (25 + 3c) + 25(5 - 1) + 65c + |c(c -1), 
a{n - 2 ) = k + p + 2a, 
c(% - 2) = 2<r + (4/3 + 7 ), 
(4% - 6-25-3c) (w- 2 ) (%-3)=2S-4/>-9<r» 
(- % 2 + % + 45) (%- 2) (% - 3) = ( 8 /; - 18A) - 4p + 9<r, 
%'=%(% - l) 2 - 5(2% - 2) -3c - p - 9a - 3(4/3 + 7 ), 
a = %'(%' - 1 ) - 25'- 3c', 
/c = 3%'(%'- 2 ) - 65'- 8 c', 
^^=|^(w'-2)(^ 2 -9)-(%' 2 -^-6) (25'+3c , ) + 25'(5'-l) + 65'c / +|c'(c'-l). 
« / (»'- 2 )=V •+/»' + 2 ^, 
c'(^'-2) = 2<r' + (4/3' + 7 '), 
(4%' - 6 - 25' - 3c0 (%' - 2 ) (%' - 3) = 25' - 4/7 - 9<7, 
(- %' 2 + %' +• 450 O' “ 2 ) O' ~3) = (87c' - 18 ti) - 4/a' + 9a', 
% = %'(%'-1) 2 - 5'(2%'- 2) - 3c'- />' - 9<r'- 3(4/3' + 7 '). 
We may here take as data %, 5, c, 4/3 + 7 , 8/;-18A, the equations 
exclusively of the two marked (*), then give a, B, k, p, a, %', a', B', k', 
5', c', p , er', 4/3' + 7 ', 8 //- 187'; and then, since all the quantities enter¬ 
ing into the two excepted equations are expressed in terms of the data, 
these equations are satisfied identically, and it is easy to see that this 
proves what was before assumed, viz., that in the system of twenty-one 
equations, the second of the equations marked (*) is satisfied identi¬ 
cally. 
Several of the other quantities may be expressed without difficulty 
in terms of the data %, 5, c, 4/3 + 7 , 8^-18 li : we in fact have (besides 
a, a', k, which are originally so expressed)— 
2a = (% - 2) c - (4/3 + 7 ), 
8 P = (16% - 24) 5 - (15% - 18) c - 2(45 2 - 9c 2 ) + 2(8/; -18 h)- 9(4/8 + 7 ), 
Sic = 8 %(% - 1) (n - 2) - (32% - 56) 5 - (17% - 46) c 
+ 2(45 2 - 9c 2 ) -2(8/;- ISh) + 17(4/3+ 7 ), 
2B = %(% - 1) (% - 2) (% - 3) - (4 % 2 - 20% + 24)5 - ( 6% 2 -1 bn +18)c 
+ 125c + 18c 2 + ( 8 /; - 18/0-9(4/3+ 7 ), 
8 %' = 8 %(% -l) 2 - (32% - 40) 5 - (21% - 30) c 
+ 2(45 2 - 9c 2 ) - 2(8 1 - 18 k) + 21(4/3 + 7 ), 
= 4%(%- 1) (%- 2) - (16% - 28) 5- (10%-26) c 
+ (45 2 - 9c 2 ) - ( 8 /: - 18A) +10(4/3 + 7 ), 
