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A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
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a linear function of 1, x + y, xy. This is the case as regards the difference of any
two of the squares (Vab) 2 , (Vac) 2 , &c.; hence selecting any one of these squares, for
instance (ydde) 2 , any other of the squares is of the form
\ + fi (x + y) + vxy + p (Vde) 2 , (p = 1);
and obviously, the other squares ('da) 2 , &c., are of the like form, the last coefficient p
being =0. We hence have the theorem that each square can be expressed as a linear
function of any four (properly selected) squares.
127. But we have also the theorem of the 16 Kummer hexads.
Obviously the six squares
(Va) 2 , (\ r bf, (V c) 2 , (Ve) 2 , (dff
are a hexad, viz. each of these is a linear function of 1, x + y, xy: and therefore
selecting any three of them, each of the remaining three can be expressed as a linear
function of these.
But further the squares (Va) 2 , (V6) 2 , ( f dab) 2 , (Vcd) 2 , ('dee) 2 , ('ddef form a hexad.
For reverting to the expression obtained for ('dab) 2 — (\/cd) 2 , the determinant contained
therein is a linear function of aa / and bb y , that is, of ('da) 2 and (Vb) 2 ; we, in fact, have
(a-b)
= (b — c) (b — d) (a — x) {a — y) - (a — c) (a — d) (b — x) (b — y).
1, x + y, xy
1, a + b, ab
1, c + d, cd
Hence (Va6) 2 — (Vcd) 2 is a linear function of (Vu) 2 , (VTi) 2 ; and by a mere inter
change of letters (Vab) 2 — (Vcef, (Vab) 2 — (Vde) 2 , are each of them also a linear function
of (Va) 2 and (V6) 2 ; whence the theorem. And we have thus all the remaining 15
hexads.
128. We have a like theory as regards the products of pairs of functions. A
tetrad of pairs is of one of the two forms
V a 'db, 'dac'd be, VadV bd, V ae 'dbe, and V/ \i ab, Vc V'de, 'd d'd ce, Ved'cd;
in the first case the terms are
V aa / bb / ,
^ |(ab / + a b) Vcdefc / d e / f / + (cfd^-f c^de) Vaa/bbJ,
+ (dft^e, + d / f / ce) „ },
+ (efc/i, + e / f / cd) „ },
and as regards the last three terms the difference of any two of them is a mere
constant multiple of Vaa^b,; for instance, the second term — the third term is
= ^ (cd, — c,d) (fe, — f,e) Vaa^b,, = (c — d) (/— e) Vaa^b,;
If
d 21
If
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