174 
MR. A. C. DIXON ON SIMULTANEOUS 
the minor of -y ~y in the determinant 
(10). PTence the constituents in the second column of the transformed array are as 
stated, and those of the third are found in like manner. It is not, of course, neces¬ 
sary that these columns should be the same as would be found by actual substitution 
of the values of p x , p 2 , q { , q 2 in the columns of the original array ; a linear trans¬ 
formation is allowable, with constant or variable coefficients. 
The above process gives fifteen independent rows of the array ; the others are 
deduced from the consideration that y, z are known in terms of x v x. 2 , u l , u 2 , u 3 , u i 
from the equations <j> = 0, i/j = 0. 
the coefficient of the Jacobian written being 
Examples. 
§ 25. I. As a first example of the method of solution, take the equations 
a i = a -2, Ai = As, 
where a ]5 denote known functions of x v p 1} q x and a,, /3. : known functions of 
X Z> P-2’ 9.2' 
In the array (5) multiply the seventh row by « 
3(«iAi) 
3(A> Pi)’ 
the fifteenth by 
0 («] 
,&) 
( i\ Y 
the 
twenty-fourth by ^j , and add. The result in the first column is d(a x , J3 X ), in the 
second, by virtue of the particular forms of f x and f 2 , 
{ x i, P\] fax, q 2 ] + i x i> 9i] Wz’Pi] + hh’ 9\) i x i, <1-:} or 0, 
and in the third, 
{ x i> Pi) {P- 2 ’ 9\) + i x 1 , 9u {Pi,P 2 } ~ {Pi, 9i) Pz) or °- 
Hence eq, are two functions satisfying the auxiliary equations, and a solution is 
given by finding p x , q v p 2 , q 2 from the equations 
a 1 = a 2 — a, 
Ai = Aa = 
and integrating. Two constants will he introduced by integration, so that the result 
is a complete primitive. 
§ 26. II. Take, secondly, the equations 
V = Vi x 1 + F ( x 2, Pi, 9i, P-2, 9-2), 
z = qpc x + G (x 2 , p x , q x , p 2 , qd). 
Here the twenty-fourth row is 
d (Pi, 9i), °> °> 
