TRANSACTIONS OF SECTION A. 409 
The solution of (4) resolves itself into one of three forms, according as (first) 
I? —27J* is positive, in which case all the roots are real; (secondly) 7*—27J* is 
negative, but J is still positive; or (thirdly) J*—27J? is negative, and J is negative 
—in each of which latter cases there is only one real root. 
In the first case we have the well-known solution of the cubic by comparing it 
with the trigonometrical identity 
4 cos* a—3 cos a—cos 3a=0. 
In the second case it can be solved by comparison with the identity 
4 cos h® a—3 cosh a—cosh 3a=0; 
and in the third by comparison with 
4 sin h$ a+3 sinh a—sinh 3a=0. 
[These identities are apparent at once on using their exponeniial values. } 
First then, when £° —27J? is positive, we have 
27,72 
& =N COS a, where cos 3a= ae : 
secondly, when J*—27J? is negative, but Z is positive, 
27,J2 
a2=n cosh a, where cosh 3a= ae : 
thirdly, when J is negative, 
v=n sinh a, where sin h Sa= 4/24 : 
— 3’ 
in each case n being numerically equal to a/ 5 and of opposite sign to J. 
It seems worthy of notice that, since the two quadratic factors into which the 
biquadratic can be split [see equation (2)] differ only in the sign of q, the biquad- 
ratic is equivalent to a single quadratic equation— 
y+ 2qy + 29°+ 8H—F=0 at 2 
in which g may have any one of six values—namely, any one of the roots of the 
cubic in g®. The six different forms of (5) are, of course, the six different combi- 
nations-in-pairs of the linear factors of the biquadratic. It is easy to see from 
this the intimate association of equal and zero roots in the cubic with equal 
roots in the biquadratic; for if two of the roots of the biquadratic are equal, we 
ean only form four different quadratic expressions (5), and hence two roots of the 
cubic must be equal. Again, if the biquadratic has two pairs of equal roots, the 
two equal roots of the cubic must be zero, since only three combinations can be 
made: and so on. 
8. On Symmetric Functions, and in particular on certain Inverse Operators 
in connection therewith.| By Captain P. A. MacManoy, R.A. 
This paper is more particularly concerned with non-unitary symmetric functions, 
on account of the author’s recent discovery that they are in fact seminvariants of 
an allied quantic. Some recent theorems in seminvariants are herein applied to the 
calculation of symmetric functions. 
§ 1. The equation considered is in every case 
SLOT EAR dor r-oreest/ Piypelast le by alley ehrany c—=0 
and if partitions in ( ) refer to this equation, and those in (( )) to the 
equation 
TNL e) Tere — OE" O oan. Nee =O) 
? Published in extenso in the Proceedings of the London Mathematical Society. 
