13] 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
85 
so that all the terms on the second side of the equation vanish. If, however, ¡3 = a, 
whence, on the second side, we have 
c.QR ... P -\- cPR ... Q &c. —p . cPQR ... + &c. + &c. 
= pU, 
or the theorem in question is proved. 
In the case of an incomplete hyperdeterminant, the corresponding systems of 
equations are of course to be omitted. In every case it is from these equations that 
the form of the function u is to be investigated; they entirely replace the system (A). 
A very important case of the general theory is, when we suppose the coefficients 
rst... to have the property r's't' ...= r"s"t" ..., whenever r's't'... and r"s"t" ... denote 
the same combination of letters ; and also that the coefficients A are equal to the 
coefficients ¡x, v ... , each to each. In this case the coefficients rst... have likewise 
the same property, viz. that r"s"t" ... = r's't'..., whenever r's't' ... and r"s"t" ... denote 
the same combination of letters. 
The equations (B, 1), (B, 2), become in this case 
u = L np . u 
(B, 3), 
where only different combinations of values are to be taken for r, s, t, ... and 
a, /3,... express how often the same number occurs in the series. In the equation 
(C), /x, v must be replaced by A, the equations (D) are no longer satisfied; the 
equations (A) reduce themselves to a single one, (so that there can be no question 
here of incomplete hyperdeterminants) : but this is no longer sufficient to determine 
the function sought after. For this reason, the particular case, treated separately, 
would be far more difficult than the general one ; but the formulae of the general case 
being first established, these apply immediately to the particular one \ The case in 
question may be defined as that of symmetrical hyperdeterminants, (a denomination 
already adopted for ordinary determinants). It would be easily seen what on the same 
principle would be meant by partially symmetrical hyperdeterminants. 
I have not yet succeeded in obtaining the general expression of a hyperdeterminant ; 
the only cases in which I can do so are the following : I. p = 1, n even, (if n be odd, 
there only exist incomplete hyperdeterminants). II. p = 2, m— 2, n even. III. p = 3, 
m = 2, n = 4. 
I. The first case is, in fact, that of the functions considered at the termination of 
a paper in the Cambridge Philosophical Transactions, voi. vili, part i. [12] ; though at 
that time I was quite unacquainted with the general theory. 
1 See concluding paragraph of this paper.