267 
1909-10.] Dr Muir on the Theory of Orthogonants. 
where L, M, N are, as in his paper of 1827, the roots whose reality is to be 
established, he obtains at once 
a = Lal + Mal+Nal, 
& = L# + Mj8j + N$, 
■ C = Lyl + Myl+Xyl, 
and thence derives 
f — L/? 1 y 1 + M^ 2 y 2 + N/3 3 y s , 
g = Ly^j + My 2 tt 2 + Ny 8 a 8 , 
h = L a-ifii + Ma 2 /5 2 + Na 3 /3jj , 
A = MNa? + NLai + LMag , F = MN/3 iyi + NL/3 2 y 2 + LM/l B y 3 , 
B = MN^-+ NLj8| + LM/8J, G = MNy^ + NLy 2 a 2 + LMy 3 a 3 , 
C = MNy 2 + NLy 2 + LMy! , H = MNa^ + NLa 2 ft + LM a 3 /? 3 . 
From these it can be shown with more or less trouble * that 
#H — JlGr = II • a 1 a 2 a 3 , 
^>F — ,/B + cF — fC — 2aF + 2/A = II • (cq/? 2 /^3 + a 2 ^ 3 /?i + a 3 /h/3 2 ~ a i7273 — a 2737i — a 37i72) > 
6C - cB + cA - aC + aB - 6 A = n • (cq&yg + a 2 /5 3 y 1 + a^yg + oq^yg + a 2 /5 1 y 3 + a^yj , 
where II stands for (L — M) (M — N) (N — L). Kummer’s sum of squares is 
thus made to take the form 
h 2 • [ 15 2( a i a 2 a s) 2 + 2,H(A& _ Wi) + a 2 (AA - y S Yi)+ “«(£A - YiY»)} 2 
+ j“i(Ays + Ay 2 ) + a 2< AsTi + A ys) + “s(Ay 2 + A Yi)| > 
where we use S to indicate the sum of a set of terms produced by the 
cyclical substitution a—^/3, /3—>y, y—>a. After this the cofactor of II 2 is 
shown with seeming ease to be 1, and the desired result is reached. 
A knowledge of the relationships existing between the elements of the 
orthogonant | a t /3 2 y 3 1 is, of course, a constant requirement throughout the 
demonstration ; and to two of these relationships special attention is drawn 
by Jacobi himself. The first is 
HaWA+Pim + yhhl} 
— a l/?273 ' a 2^37l a 2^2,7l ’ a 3^l72 a 3^l72 ' a l/^273 
+ ai /? 3 y 2 . a 2^lYs + a 2^l7s * a All + a 3&7l • «1^ 3 72 » 
* The modern reader would do well to use Binet’s theorem regarding the determinant 
which is viewable as the product of two rectangular arrays. Thus 
A = 
\ h f\ 
ll L,8 > 
M^2 
Nft, l| 
j £1 /k ^3 || 
\fc\ 
II 1*71 
m 72 
N 73 II ■ 
1 7i 72 73 ' 
LM | ) 8 1 7 2 | 2 + MN | /3 2 7 3 | 2 + NL | ^y 3 \ 2 = LMa 2 + MN a\ + NL«| : 
1 9 
h 1 _ I 
1 L 
M 
N II 
, I 7ltXl 
72«2 73«3 | 
\ G 
H 
MN 
NL 
LM 
1 «i/h 
a 2^2 a 3^3 1 
and so forth. 
N(L 2 - M 2 ) . 0l a 2 | 7 1 /8 2 | + . . . . 
