EQUATIONS POSSESSING 
498 
[318. 
effecting the quadrature, and dividing out by one of the constants 
a 1} a 2 , a 3> a 4 , we have 
u = <p (a’j , ..., x 7 , a, h, c) + a', 
where a, b, c are three arbitrary constants, and a is an additive 
constant. As u = 0 is the intermediate integral, we can take the 
latter in the form 
which is an equation of the first order. 
319. To obtain the primitive of the differential equation 
constructed with (f> + a = 0 as an intermediate integral, we might 
proceed to construct the primitive of the equation of the first 
order: but the theory of § 284 can be generalised, so as to allow the 
primitive to be constructed merely by operations of elimination. 
When we substitute 
u = (p + a' 
in the differential equations L = 0, M = 0, N = 0, these are satisfied 
identically: hence 
dv, dv<> d\ dp 5 t d\ dp 6 ^ d\ dp 7 _ ^ 
dp 5 da dp 6 da dp 7 da 
dn. dn> du dv* da dv« da dv-? 
du 
Now let the Poisson-Jacobi combination for u and ¡r- be con- 
da 
structed: it is 
(&P* 
\3a