OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 559 
Then as c, and c, are not both constant, we have N = 0 where 



This is a further condition connecting b,, b3, ¢), ¢:, uw. From it we deduce 
oN oN oN oN oN 
ano o ae, 02 + my = ay lear an adj = 0. 
A 
Cc 
From this and the former equations we can again eliminate the differentials. 
Suppose the resultant equation to be P= 0. Then the equations N = 0, P = 0 afford 
an integral of the differential equations. We will verify this. 
§ 47. N = 0 may be replaced by 



oM oM 
0b, Cr T Ob C, = 0, 
OM OM 
ne LS Gna 
where C, and C, do not both vanish. 
Also P = 0 may be replaced by 
oN ON\ ,, , /ON oN Ne 
ae Ay, | Chee ty) Gta K= (10), 
oM \ 8M OM 
(uo) 1 +(e 2) © + Rew oes (Up 
We have also M = 0, 0M/ou = 0, o?M/op? = 0. 
The last three equations give 

aM fie ONE 
Ob, (Hey bY) a ab, (ue's an v’,) = 
if we write c’, for de,/dax, &c. 
We may then put 
poy + by = uC, pe, + bn = uC, 
They also give 
oM OM , OM. pou 
A Ow ee Oe Cle Ce 
