186 
MR. A. C. DIXON ON SIMULTANEOUS 
(f>z = 0, (fro = 0 will give suitable values for u { , u. z . This array will have four columns 
and forty-five rows, ten such as 
cl (pc i, Xj), 0, 0, 0, 
ten such as d (u h Uj), 0, 0, 0, 
and twenty-five of the following type. In the first column there is d(xu/), in the 
(r -J- l)th the minor of d'j) r /dx t duj, in the determinant 
d ~cf) r 
d 2 p 
d 2 cf) r 
d~P 
d 2 cf) r 
dp 
dcf> o dcp 3 
dap dn/ 
dx % du/ 
dx 3 du/ 
dap du/ 
dap dip 
dip 
dip dip 
dp 
dp dp 
* * * 
du 2 
du , dip 
dp 
dcf/ dp 
dip 
dip du 3 
dp 
dp dcb 3 
dip 
dip dip 
d 2 P 
d~cf) r 
dp 
dp d4 >, 
0q du. 
dap du- 
du- 
du- du- 
d<b l 
dp 
djh 
dp 
dp 
0 
0 0 
0q 
dap 
dr s 
0q 
dap 
dcf) o 
dcf)* 
dp 
00o 
dp 
0 
0 0 
0q 
dap 
dap 
0q 
dap 
dp 
dcf>. 
dp 
dp 
0 
0 0 
0q 
0q 
dr a 
dx i 
dap 
r 
= 1, 2, 3. 
This interchange of variables and parameters may take place whenever their 
numbers are equal, the differential equations being of the first degree. 
Examples. 
§ 38. I. As an example of the method of solution take the equation a = /3, where 
a is a function of r, s, p — sy, x and /3 a function of s, t, q — sx, y. 
In the array (§ 32) multiply the first row by da/dr, the fifth by da/dp, the fourteenth 
by da/dx, the seventeenth by — s da/dp, and add; the resulting row is 
d(a, s), 0, 0. 
Hence we take a = fi = a, s = b, 
z = bxy + X -fi Y, 
X being a function of x only and Y a function of y only. Then a = a is a relation 
connecting x, clX/dx, drX/dx 1 , and (3 = a is a relation connecting y, clY/dy, d~Y/dy-, 
and by solving these for X, Y respectively we shall have the complete primitive. 
