152 
MAJOR P. A. MAGMA HON ON A CERTAIN CLASS OF 
ill the manner 
.Ti 
^12/ \*yi3^12^23y 
X a multiple of x^\ 
.r. 
etc,-. 
23, 
fa —nii—iii 
where, of the three portions, the first accounts w^holly for the second wholly 
for x^^, and so on; and not in any other manner. 
Put 
(^ 12 , '^23) ~ (^n ^2)) 
Xi, x„, Xg, and consider the 
X 4*i^-2’ 
^1 ■ 
, ao. 
4>.-)Xg 
X^ x^ 
/y» 
^1^2 ^2 
Xg 
in which unnecessary quantities are omitted. 
Further, omitting a multiplier, independent of x-^, a’g, a’g, on the right-hand sides, the 
following six inequalities must be satisfied, 
putting 
^ — ’ 
^2^3 
4^2 ^ 2 ” 
XjX.^ 
2 ^ 
tX/O 
x„ 
a?. 
t- 
X., 
I I ^’2 
= </'!“’ 
Xo 
X^ 
% = h ’ 
these conditions are representable by the single inequality 
^ cp^sq, _|--L ^ _L Jh , ^ 2<p 3 I — I — 
^ 2 1 -r ^^ 3 ^^ ^ ci>2 ^ epy ^ 2 1 -t- 
As regards functions of ccj, x^, x^, this inequality being satisfied, and may be 
put equal to any functions that may be desired. Like inequalities may he obtained 
in respect of the fourth and higher orders. 
