170 
ME. A. C. DIXON ON SIMULTANEOUS 
number of conditions, which will be of the nature of ordinary differential equations, 
thus imposed on the four parameters must not be greater than three ; for if they 
are subjected to four conditions they are made invariable ; it may be, however, 
less than three. For instance, a complete primitive of the equations Pi = p 2 , q x = q 2 
is given by 
y — ff - x.f) ff- b, z — c{x± ff- xf) -(- e ; 
the equations given by varying the parameters are 
(aq + x. 2 )da + db = 0, 
(aq -f- x 2 )dc -f- de — 0, 
which give the single differential equation connecting the parameters 
da de = db dc. 
We may then assume arbitrary forms for two parameters in terms of a third, and 
find the fourth by integration. Say, for instance, 
b = 4>(a), c = xfj(a), 
then e = l(f>'(a)x}j\a)da, 
Xl + x. 2 = - ; 
thus we arrive at the known general solution 
V = x( x i + x i)> z = °>{ x i + x z)- 
whence, by elimination of X\, 
(x 2 2 db + dc ) 2 (xylb + dc) + x^de^da = 0 . 
This equation must fail to define x 2 , so that b, c, and a or e must he constant, thence it follows that all 
four parameters must be constant. 
I lay stress on this, because it is not in agreement with the results of Professor Konigsberger (‘ Crelle,’ 
vol. 109, p. 318), and appears in fact to show that his method there given is faulty. Professor Koxigs- 
berger assumes (p. 313; I take m = 2) that the most general integral of the equations 
/i(g'i, y> -> Ph P2> 2n 9.2) ^ 
flip 1 , * 2 , y, Z, Pi, P- 2 , 2l, (pi) = 0 
has the form 
y = Wipn, * 2 , <ki[fi(x h a- 2 )], *»)]) 
Z = u>. 2 (xi, X-2, f>l[fl{x h * 2 )], f-ilfdxi, !&>)])> 
where <£ 1 , </> 2 denote arbitrary and ^ 1 , y 2 definite functions. But suppose these equations solved for 
<f>i, </> 2 in the form 
2>i[yi(.tii, xff] = Xid'b ^ 2 , y, Z) 
folMvu * 2 )] = Xa(sb x- 2 , y, ~) 
and the arbitrary functions eliminated by differentiation. The differential equations thus formed are of 
the first degree in p 1} p- 2 , 2n 2‘» and are not by any means of the general form assumed. The differential 
equations in the examples given by Professor Konigsberger are, in fact, linear (see pp. 319, 328). The 
method appears to be founded on an interpretation of the last clause of § 2 (p. 290), which is not justified. 
