ROYAL IRISH ACADEMY. 
113 
n f =n(n-iy-n(7b + l2c) + 4b i + 9c‘ i +8b + 15c-8&-18A + 18/3 + 12 7 
+ I2i-9t, 
a = n'{ri - 1 ) - 2 b' - 3c', 
K = 8ri{ri-2)-8b'-8c f , 
* z=i n '( n '-2) (»*-9) - (n'>- ri-6 ) (2 b'+ 3c') + 2b\b'-l) + 6 £'c'+ fc'(c'-l), 
«'(»' - 2 ) = k + P + 2a, 
b'(ri - 2) = p + 2ft + 87 ' + 3t', 
c'(ri - 2) = 2a - 1 - 4/3' + 7', 
fl'O'-2) O'-3) = 2S' + 2«'&' + 3 a'c' - 4p' - 9a, 
b'(ri - 2) (ri - 3) = 4ri + a'b' + 3b'ri -9/3' -67'- 3*' - 2/>', 
c'(w' - 2) (w' - 3) = 6A' + a'ri + 2b'c' - 6(3' - 47 ' - 2*' - 3«r, 
*n = ri(ri - l ) 2 - ri(7V + 12c') + 4b' 2 + 9c ' 2 + 8b' + 15c' -8ri-l 8h! + 18>3' 
+ 127 '+ 12$"'- 9t'; 
to which may be joined— 
q = b % - b - 27c - 37 - 6t, 
r = c 2 - c - 2h - 3/3, 
/= 5' 2 -V — 2k' - 37 ' - 8t', 
ri- c' 2 - c' - 2A' - 3/3'. 
Considering the twenty-one equations, and taking as data w, 5, c, 
(3, 7 , h, 7c, then, by means of the several equations, other than the two 
equations marked (*'), we may express in terms of the above data a, S, 
k, t, i, p, a, ri, a!, S', k, V , c', p, a, 2/3' + 37 ' + 3f, 4/3' + 7 ', 4 7c' - 3 i' , 
6ri- 2i'; the quantities which enter into the first of the marked equa¬ 
tions are then all given in terms of the above data, and it is clear that 
the equation must be satisfied identically : the quantities which enter 
into the second of the marked equations are given in terms of the data and 
of V, i', and it is not clear, a priori, but that the equation might lead to 
a relation between the data and t', i'; it will, however, appear in the 
sequel that the equation must be satisfied identically, independently of 
any particular values of t', i'. Thus, Mr. Salmon’s theory does not de¬ 
termine the values of these two quantities, nor, consequently, the values 
of /3', 7 ', Ji, ri ; it does, however, determine the values of the combina¬ 
tions 4/3' + 7 ', 8 ri - 18A'. But the twenty-one equations between the 
twenty-eight quantities may be replaced by seventeen equations be¬ 
tween the twenty quantities— 
n, a, S, k, b, c, p, a, 4/3 + 7 , 8 7c - 18 7i, 
ri, a', S', k, V , c', p, a, 4/3' + 7 ', 8ri — 18 7i, 
this will clearly be the case if it is only shown that the equation which 
gives ri can by the other equations be transformed into one of the form 
in question ; for a similar transformation will, of course, apply to the 
equation for n, and then we have only to reject the equation containing 
