158 
MR. A. C. DIXON ON SIMULTANEOUS 
Taking the values thus given for X,,, X,-,, X,„- we have 
, .. 0Xr, 0X.V , SXst 
( rsi )= U + ~ + 
0r, 
XL, 
x 13 ‘ 
/ 3 X„ 
l e®, 
+ 
1 ") + 
dr, / 
x. v/ 
/0X, 
+ 
0X 1; \ , 
xj 
v 0*i 
d) + 
X rs 
0 X 12 
X ; , 0X 12 
x 13 
0r, 
X 12 0.r s 
0 X 
7*2 
+ 
dx r 
0 x 2; \ X;,/ax* 0x^ 
cte, / ^ X 12 \ 0a* " h 0> 
0X 2) .\ , x 2; /0X ls , 0X„ 
x ls \ a«v 
+ 
00 
+ ^ + ^) + ^ + ^) + 
_ X.j 3x„ 
^Xy 
= ^ (lri) + fa (r2i) + I 5 -" (2*0 + (sl«) + f 2 (s2 r) + f 2 (lsr) 
-A. i2 -A -Ain -A 12 -A-jo A^ 
+ Y v (Xj> X,. -f- X,i X 2 , + X„ X 2i ) + u -- ur (X )7 - X ls -f- Xu X lr -f- X, r X i; ) 
Ajo Ai2 C/-X-2 
+ f 2 (21s) + |s ( 2 lr) + ^ (210. 
Afo -A-12 A]2 
Tims the conditions (2) are not independent, but all follow from those in which at 
least one of the suffixes 1, 2 enters. If they are satisfied then the equations 
X X],, dx r = 0, 2 X 2r dx r — 0 
r= 1 
can be satisfied by two integrals of the form u = a, v — b ; that is, these last 
equations will give aq, x 2 as functions of the rest, such that 
0^ _ X*. 
0o. - x 13 » dx r - x,; 
For the conditions necessary and sufficient* for this are the vanishing of such 
expressions as 
0 + XL, _0_ X,, X s , 0 X,i _ 0_ X„ _ XL, _0_ Xrt _ X« _0_ X* 
0i, Xjo Xi 2 0<q X 12 Xj 2 ^12 0*O‘ X^ 2 Xj 2 dp X_j 2 Aq, 0a 2 X.j 2 
in which 1, 2 may be interchanged and r, s are any two of the other suffixes. This 
expression may be written 
(rlS) - If (12s) + L 1 (12r) + — (X» X„ + X,„ X„ + X 2 , X„), 
A_i2 A_i2 A_i2 A_i2 CX-^ 
so that it vanishes and the conditions of integrability of the equations SXj ,dx r = 0, 
7 * 
%X 2 r dx r = 0 are satisfied. If u — a, v = b are the integrals, then, since 2 X ]r dx r 
r 
does not contain the differential of aq, we must have 
* For proof of this statement see Forsyth, ‘Theory,’ part I., pp. 43-51. 
