184 
ME, A. C. DIXON ON SIMULTANEOUS 
three linear relations can be formed connecting the twenty bidifferentials ; one is 
formed from each pair of equations as at § 17 ( 8 ). Hence seventeen bifunctionally 
independent solutions of the auxiliary equations are known when we have one com¬ 
plete primitive. The full number is nineteen — 2 j, and in order to know all 
we must have one, or possibly two (see § 41, p. 190), more complete primitives. 
§ 34. New solutions found by varying the parameters may be divided into two 
classes, according as the parameters are or are not all functions of one variable ; 
solutions of the former class only occur in exceptional cases, and the principles of £ 21 
apply to them with slight modification. 
Let the three equations connecting 
be 
V, P, q, u 2 , u a , u 4 , u 5 
M x > y> z > P > q> U 1, u z> Mg, w 4 , U 5 ) = 0 (i— 1 , 2 , 3); 
(the forms 0 i5 0 .,, 0 3 are not unrestricted, but must be such that the following rela¬ 
tions hold identically 
0(01, 0g) | 03> 0 . 3 ) _ |) 
3 ip,V,q) ** d(z,p, q) ~ 
3(01, 03, 0 ,'.) ■ 3(0U 02, 0s) _ Q . 
3 (y,p,<i) ^ 3 (z,p,q) ~~ ’ 
or we may take 0 , as not involving p, q and 0 2 as p d^Jdz -f- d(f>Jdx 
03 as q d<f)Jdz f 30031 /), 
then the variations of the parameters must satisfy the three equations 
iu, = 0 (i = 1, 2, 3), 
r=l 'JUi r 
in order that the same relations may subsist among x, y, z , p, q, r, s, t and the para¬ 
meters, as held when the parameters were constant. 
If the parameters are functions of one variable, their forms must be so chosen that 
the three equations last written reduce to one only, otherwise we shall have five 
relations connecting x, y , 2 , p, q with this single variable. 
§ 35. If the parameters are not functions of one variable, only the equations 
v 5 
a*'. 
du r — 0 
are equivalent to six, and determine the partial derivatives of w 3 , v A , u 5 with respect 
to u x , u 2 in terms of the five parameters and x, y, z, p>, q. By help of the relations 
0 ; =r 0 we may suppose x, y, z, p, q eliminated and thus arrive at a system of four 
partial differential equations connecting u Y , u 2 , w 3 , u 4 , u- a . 
