294 
PKOFESSOR A. K. FORSYTH OX THE 
and therefore 
a. = V, /S = 2Q, y = 2R. 
In order that the three equations (IL) may be satisfied, we must have 
« (28 + ^) + /^ {2k + 0 + <7^ + ^) = 0, 
h (2S + 13)+ h (2k +. 0 +f{\ + «) = 0, 
,7(28 + 13) +/(2k -f ^) + k (X + (3) = 0, 
and tlierefore 
2 S +;8 = 0 , 2 ; c +4 = 0 , \-^0 = O . 
Tlie tl)rep equations (Ilg) similarly give 
2e + y = 0, 2/x + t = 0, X + 77 = 0 : 
the three equations (II^) give 
p = 0, C=0; 
the three equations (II5) give 
2-77 + /5 = 0, 2r + X ~ d + 77 = 0 ; 
and the three equations (Ilg) give 
V = 0, y = 0, t = 0. 
When these equations are solved, it apj)ears that the onh" coefficients (other than 
a, /3, y) which do not vanish are 8 and e; their values are 
8 = - Q, e = - R. 
Inserting these values, we obtain the quantity which has been denoted by a. The 
other five solutions can be constructed in the same way. 
15. It may be remarked that there exists a certain symmetry in the quantities 
already obtained (and in corresponding quantities of higher orders) which mav be 
used for comparative verification of results obtained or for avoidance of long stretches 
of algebra by deducing new results from results obtained. The symmetrv arises 
through the effects caused by interchanging w and v and ic, u and 7c, as follows:— 
a 
h 
c 
f 
ff 
h 
a 
b 
c 
1 1 
f g h 
P 
Q R 
u and r 
1 
a 
c 
f 
h 
b 
a 
c 
g , f : t 
■ 
Q 
r R 
j 
V and w 
a 
c 
h 
f 
h 
9 
a 
c 
b 
f b g 
p 
R Q 
u and w 
c 
h 
a 
h 
[j 
f 
c 
b 
a 
b g f 
1 1 
R 
Q i r 
1 
