WHICH SATISFY GIVEN CONDITIONS. 
87 
viz. the single condition X has the five characteristics (yJ", . . . r"), . . .; the four conditions 
4X, the characteristics (+ v) as in the original theory ; and the five conditions 5X a 
single characteristic yj 0 . 
34. We thus see the origin of the notion of the representatives (a, (3) of a single con- 
dition X ; for considering the arbitrary four conditions 4Z, the characteristics whereof 
are (y, v), and assuming that the single characteristic, or number of the conics (X, 4Z), 
is (3v, and taking for (4Z) successively the conditions 
(.-./), (://), (•///), (////), 
having respectively the characteristics 
(1,2), (2,4), (4,4) (4,2) (2,1), 
we have 
^/"= la +2/3, 
»/"=2a+4/3, 
£'"=4«+4/3, 
o-"'= 4a + 2/3, 
r"' = 2a + l/3, 
that is, the characteristics (yJ", g"', d". d") of a single condition X are not independent, 
but are representable as above by means of two independent quantities (a, /3) ; or, what 
is the same thing, we have 
i/"=2yJ\ <r'"=2r", f=% (/+®' ff ), 
which being satisfied, the representatives (a, /3) are given by 
«=i( 2d"-yJ n ), (3=jt(2yj'"—7 J "). 
35. I find that a like property exists as to the characteristics (yJ', v", g", o') of the two 
conditions 2X, viz. these are not independent but are connected by a single linear relation, 
&-¥+&-*=& 
This may be proved in the case where the conditions 2X are two separate conditions 
(X, X') ; viz. let the representatives of these be (a, /3), (a', /3') respectively, then com- 
bining with them the three arbitrary conditions X", X'", X'" having respectively the 
representatives (a", /3"), ( a' ", /3"'), (a"", (3 we have the general equation 
(X, X', X", X'", X"")=(l, 2, 4, 4, 2, l£a, ,3)(«', /3')(a", /3"')(«"", fT); 
taking herein 
(X", X'", X w ) =(.*.), (:/),(• //), (///) 
successively, and observing that the representatives of ( • ) are (1, 0) and those of (/) are 
(0, 1), we thus obtain for ( yJ v", g", o"), characteristics of (X, X), the values 
l“"=(l, 2, 4£a, /3)(a', (3% 
'"=(2, 4, 4£a, /3)(a', (3). 
(={ 4, 4,21a, £)(«', /3'), 
<f'=(4, 2, 1X«, /3)(«', /3'), 
