ON THE TRIPLE ^-FUNCTIONS. 
435 
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the equations, which define the ^-functions A, B,...,H, ABC,...,FGH, and the 
co-function 12, are 
A = CL fa (8 equations) 
ABC = il Vabc (56 equations) 
and one other relation which I have not as yet investigated. 
As regards the algebraical relations between the 64 ^--functions, it is to be 
remarked that, selecting in a proper manner 8 of the functions, the square of any 
one of the other functions can be expressed as a linear function of the squares of 
the 8 selected functions. To explain this somewhat further, observe that, taking any 
5 squares such as (ABC) 2 , we can with these 5 squares form a linear combination 
which is rational in x, у, z. We have for instance, writing down the irrational part 
only, 
(ABC) 2 = | 2 {abc (z — x)(x-y)f YZ + aJbjCj (x — y)(y — z)\/ZX + a. 2 b 2 c 2 (y — z) (z — x) fXY), 
and forming in all five such equations, then inasmuch as the coefficients abc,... of 
(z — x) (x — у) V YZ are each of them a cubic function containing terms in tXfi у ОС у ОС у 
oc?, we have a determinate set of constant factors such that the resulting term in 
(z — x)(x — y)fYZ will be =0; but the coefficients a^bjCj,... of (x — y) (y — z) fZX only 
differ from the first set of coefficients by containing у instead of x, and the same 
set of constant factors will thus make the resulting term in (x- y)(y — z) f ZX to 
be = 0 ; and similarly the same set of constant factors will make the resulting term in 
(y — z) (z — x) VXY to be = 0 ; viz. we have thus a set of constant factors, such that 
the whole irrational part will disappear. It seems to he in general true that the same 
set of constant factors will make the rational part integral ; viz. the rational part 
is a function of the form -3- multiplied by a rational and integral function of x, y, z, 
C/" 
and if this rational and integral function divide by в 2 , then the final result will be a 
rational and integral function, which, being symmetrical in x, y, z, is at once seen to be 
a linear function of the symmetrical combinations 1, x + у + z, yz + zx + xy, xyz. Such 
a function is obviously a linear function of any four squares А 2 , В 2 , C 2 , D 2 ; or the 
form is, linear function of five squares (ABC) 2 = linear function of four squares A 2 , 
that is, any one of the five squares is a linear function of 8 squares. 
As an instance, consider the three squares (ABC) 2 , (ABD) 2 , (ABE) 2 , which are 
such that we have a linear combination which is rational : in fact, we have here in 
each function the pair of factors ab, which unites itself with (z — x) (x — y) fXY, 
viz. it is only the coefficient of ab (z — x)(x — y)fXY which has to be made =0; 
the required combination is obviously 
(d - e) (ABC) 2 + (e-c) (ABD) 2 + (c-d) (ABE) 2 . 
55—2