L 6 8 
ME. A. C. DIXON ON SIMULTANEOUS 
just appeared is not that in which complete primitives were discussed in § 14, since 
the equations are not here supposed to be solved for the arbitrary constants. 
§ 19. In addition to the six pairs (u;, Uj) of functions satisfying the auxiliary 
equations, we have also the six pairs (x ; , Xj) ; of these twelve, eleve i are indepen¬ 
dent, the other being a bifunction of them. If we can find : bifunction of the 
eleven pairs which is not a bifunction of either set of six it will give a new complete 
primitive ; whether every, or indeed any, other primitive is thus given is a matter 
for further inquiry. 
Suppose Vi = bi (i = 1 , 2 , 3, 4) to be a new complete primitive, then it gives six 
more pairs of functions satisfying the auxiliary equations, and thus we have in all 
eighteen pairs. The bidifferentials of these must be connected by (18 —13) five linear 
relations, one of which has been written (8) ; by means of the other four, an 
expression of either of the following forms— 
A d(v lt v 2 ) + B d(v lt v 3 ) + Cd(v 1} %), 
A d(v 2 , v 3 ) + B d(v 3 , rq) + Cd(v : , v 3 ), 
can be found which will be equal to a linear combination of the twelve bidifferentials 
d (xj, xj) and d (u ,, Uj). It is natural to ask whether, conversely, any linear combina¬ 
tion of these twelve which can be written in one of the above forms will lead to a 
complete primitive ? In the first case this is not so, for if we take any function 
whatever, y, of six independent variables, ^q, . . . £ c , we may choose the coefficients 
oq, . . . a G , so that 
6 
- «; d(v), £•) 
i=l 
shall be a linear combination of eleven^ given bidifferentials; the expression S a, d£, 
may then be reduced to three terms, /3 1 d£ } + /3. : d£ 2 + {3 : . d£ 3 , so that for an 
arbitrary function (y) a combination of the eleven given bidifferentials can be found 
of the form /3 1 d(y, £j) (3 2 d(y, £ 2 ) + /3 3 d(y 7 ,^ 3 ), which is the same as A d(v x , v. 2 ) + 
B d{y { , v-) -f C d(v 1} rq). This argument does not apply to the second form 
A d(v 2 , v 3 ) + B d(v 3 , + C d(v lt v 2 ), 
and further investigation may show that any combination of the eleven that can be 
reduced to this formf will lead to a primitive. 
* Not of any lower number in general, since the most general bidifferential expression in this number 
of variables contains fifteen terms, while the expression just written vanishes identically if 
U; = 3 rj/dgi, 
so that there are virtually only five coefficients, of which one must be left arbitrary. 
t The conditions necessary that a bidifferential expression may be reducible to this form include 
algebraic ones which are the same as for a complete bidifferential, since 
Ad(v 2 ,v 3 ) + Bd(v 3 ,Vi) + 
C dvx 
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