OEDEE WITH TIIEEE YAEIABLES. 
But these are the equations (10) (11), Art. 8, by which the 
system of solutions founded upon the complete primitive is 
constructed. 
The argument then is briefly this. If z — % (x, y) is a 
solution of the given partial differential equation, it is possible 
to determine a and b in the given complete primitive so as 
to satisfy the equations (23); therefore so as to satisfy the 
equations (25); therefore so as to indicate a necessary in 
clusion of z — % {x, y) in the system which is founded upon 
the given complete primitive. 
Coe. 1. Hence the connexion of a given solution with a 
given complete primitive may be determined in the following 
manner. Adopting the foregoing notation, determine the 
values of a and b which satisfy the system (23). If those 
values are constant, the solution is a particular case of the 
complete primitive; if they are variable, but so that the one 
is a function of the other, the solution is a particular case of 
the general primitive; if they are variable and unconnected it 
is a singular solution. 
Coe. 2. Hence also any two systems of solutions founded 
upon distinct complete primitives are equivalent. For each 
is virtually composed of all possible particular solutions. 
Ex. The equation z =pq, has for its complete primitive 
• (oS) 
z = (x + a) {y + b), and for a particular solution z = — 
What is the connexion of this solution with the complete 
primitive ? 
We have by (23), 
(x + a) {y + b)