[931 
932] 
265 
those of 
932. 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
[From the American Journal of Mathematics, vol. xv. (1893), pp. 1—74.] 
The principal object of the present memoir is to develop further the theory of 
seminvariants, but in connexion therewith I was led to some investigations on 
symmetric functions, and I have consequently included this subject in the title. The 
two theories, if we adopt the MacMahon form of equation, 
may be regarded as identical; but there are still two branches of the theory, viz. 
we may seek to obtain for the symmetric functions of the roots expressions in terms 
of the coefficients (which expressions, in the case of non-unitary symmetric functions, 
are in fact seminvariants), or we may attend to the properties of the functions of 
the coefficients thus obtained and which we call seminvariants. But I do not in 
the first instance use the MacMahon form, but retain the ordinary form of equation 
0 = 1 + bcc + c% 2 + dot? +..., and we have thus only a parallelism of the two theories, 
and in place of seminvariants we have functions which I call non-unitariants. In 
regard as well to these as to unitariant functions, I consider certain operators 
®<r, A, P — 8b, and Q — 2mb, which under altered forms present themselves also in 
the theory of seminvariants. 
As regards seminvariants, I consider what I call the blunt and sharp forms 
respectively: the great problem is, it appears to me, that of sharp seminvariants, 
valuable 
otherwise the 7-and-P problem—viz. fpr any given weight we have to determine the 
correspondence between the initial and final terms in such wise as to obtain a 
system of sharp seminvariants. I obtain a “ square diagram ” solution, which is so 
far theoretically complete that for any given weight I can, without any tentative 
operation, determine by a laborious process the correspondence in question: but I 
am not thereby enabled to establish or enunciate for successive weights any general 
rule of correspondence; and my process is in fact, as regards practicability, far 
inferior to that which I call the MacMahon linkage, but of the validity of this I 
have not succeeded in obtaining any satisfactory proof. 
c. XIII. 
34