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PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
or, to avoid fractions, a numerical multiple of the congregate, and on the right hand 
side a linear function, with numerical coefficients, of the segregates. 
371. The syzygy is irreducible or reducible; and in the latter case it is, or is not, 
simply divisible : viz., if the congregate on the left hand side contains any congregate 
factor (the other factor being literal) then the syzygy is reducible : it is, in fact, 
obtainable from the syzygy (of a lower cleg-order) which gives the value of such 
congregate factor. But there are here two cases ; multiplying the lower syzygy by 
the proper factor, the right hand side may still contain segregates only, and then no 
further step is required : the original syzygy is nothing else than this lower syzygy, 
each side multiplied by the factor in question, and it is accordingly said to be simply 
divisible (S.D.). But contrariwise, the right hand side, as multiplied, may contain 
congregates which have to be replaced by their values in terms of the segregates 
of the same cleg-order : the resulting expression is then no longer explicitly divisible 
by the introduced factor : and the original syzygy, although arising as above from a 
lower syzygy, is not this lower syzygy each side multiplied by a factor : viz., it is in 
this case not simply divisible. 
For example (see the subsequent Table No. 96, under the indicated deg-orders) 
(6.6), from the syzygy 9 d 2 = aj — b 5 -\-2bh — eg, we deduce (7.11) the syzygy 
{ )ad~ = a~j — ab 3 -\-2abh — aeg, which (all the terms on the right hand being segregates) 
requires no further reduction : it is a reducible and simply divisible syzygy. But we 
have (6.8) a syzygy giving de, and also (6.10) a syzygy giving e 3 ; multiplying the 
former of these by e or the latter of them by d, we obtain a value of de 2 , but in each 
case the right hand side contain terms which are not segregates, and have thus to be 
further reduced; the final formula (9.13) is 
3 de 2 = — - 4 a 2 bj-\- 3a~dg-\- 4a& 4 — 8ab 2 h-\- Aaheg — 1 2b~cd,, 
which is not divisible by any factor : the syzygy is thus reducible, but not simply 
divisible. 
A syzygy which is not in the sense explained reducible, is said to be irreducible. 
372. The number of irreducible syzygies is obviously finite : it has, however, the 
large value 179 as appears from the annexed diagram, showing the congregates 
determined by these several syzygies, and the deg-orders of the syzygies : — 
