MR. A. C. DIXON OX SIMULTANEOUS 
176 
§ 28. The solution may be verified. We have taken b x = F(a 1 ), a known hut 
arbitrary function of a x , and « 2 , b 2 other functions of ci x , such that 
da x 
00 00 pw , 00 dec 2 00 db 2 
da l db x da 2 da x db 2 da x _ 
d\ _ T71 // \ 
r - B 
= F( ai ) 
00 , 00 x , 00 d «2 , dj> db 2 ~ 
5-1 ^7 T (Cfj) + 'X 7 + 
dor, co, v /7/ » 
La l 
0« 2 da x db 2 da x 
Then we have the further relations 
Act, ,00 , 00 T-,,, x , 00 da 0 00 AA 
: i + %^-' + %) + 
X, 
Aaq 0 a' x A 6 : 
_|_ rz. _ q ? 
0a 3 da x 0& 3 Acq 
-r,// \ . AA 0-0 0-0 . , 0"0 da. 2 00 AA 
!%) + + 0^ + 0T F ( a j) + o- + IT— = °> 
CO, 
0rt. 2 Arq (/A da x 
which are of course not distinct. Also 
so that 
Pi = a i + 
da x 
dT x 
y — a x x x + cl 2 x 2 + 0 (a x , a 2 , b x , b. 2 ), 
da 2 00 00 da. 2 00 db x 00 db 2 
' l i X ' 2 da x da x 0« 2 da x do x da x db 2 da x _ 
= «i, 
and in like manner p 2 = a 2 . 
Again z — b x x x -\- b 2 x 2 0 (a 1} a 2 , b x , b 2 ), and 
da x 
9.1 — &i + 
and similarly q. 2 = A 
c7A, 
x. 
+ % W + 
00 00 A« 2 00 A6 : 00 A A 
+ 
+ 
+ 
Aq ' 1 Acq 1 Act], 1 0cq 1 0cq da x 1 0Zq da x ' AA da x _ 
= K 
Hence the original differential equations are actually satisfied. If the arbitrary 
relation assumed—which may if convenient involve more than two of the parameters— 
contains two arbitrary constants, the new solution will generally be a complete 
primitive, since two more constants are introduced by integration. # f 
* The ordinary equations to be integrated may have a singular solution with one arbitrary constant, 
or with none: if the arbitrary function has been chosen so as to involve three or four arbitrary constants, 
the whole number being thus raised to four, the solution so given may quite well be a complete primitive, 
and, in general, will be so. 
f The above investigation in a modified form shows how to find integrals of a system of three 
equations 
fi («, % pi, p o, (ji, q%) = Om 
ft (A % Pu P- 2 , ?i, 2 a) = °, >.( 12 ) 
fi ( U > V i Ph P 2 , tjl, p>) = 0) J 
where u = p x x x + p 2 x 2 - y, v = q x x x + q- 2 x-> - z. 
One solution is to take u, v,p u p 2 , q h q 2 as constants connected by the three relations (12); if they are 
not constants we have 
du = X\ dpi + %2 dp-2, 
dv = X\ dq x + x 2 dq- 2 . 
