PARTIAL DIFFERENTIAL EQUATIONS. 
171 
In the case of two equations of Clairaut’s form 
y = V\ X \ + 1A X 2 + P-2, ? 1 , <? 3 )> 
z = q x x x + q 2 x % + xjj(Pi, p. 2 , q x , qft), 
which will be more fully considered later, the number of differential relations among 
the parameters is two, so that one parameter may be taken as an arbitrary function 
of a second, and the other two found in terms of the second by solving two ordinary 
differential equations. 
If the primitive^ is 
y = aa + bfi + cy + eS 
z — Aa -f- B/3 -f- Cy -f ES, 
where a, b, c, e are the parameters, A, B, C, E known functions of a, b, c, e, and 
«, ft, y, § known functions of x L , x. 2 , then the variations of the parameters must 
satisfy the relations 
ad a + j3db + ydc + Sde = 0, 
ad A ~b /3dB -f- ydC + = 0, 
and thus, in general, if a, b, c, e are all functions of one variable they are connected 
by three relations 
dA/'da = dB/db = dC/dc = dEjde. 
The integral equivalent of these equations consists of three relations connecting 
a, b, c, e with three arbitrary constants, and by eliminating a, b, c, e we find a new 
solution of the original differential equations which is not a complete primitive, 
since it only contains three arbitrary constants. 
These examples show that the number of conditions to be fulfilled by the para¬ 
meters when all four are taken to be functions of one of them, may be one, two, or 
three ; this number is to be made up to three by assuming arbitrary relations (two, 
one, or none, as the case may be). 
§ 22. Usually the parameters will not be functions of one variable only, and we 
may suppose two of them, w 3 , tq, to be functions of the other two, u x , u 2 . 
The partial differential coefficients 
are then given by the 
for instance, gives 
du z du z clu 4 dv A 
du x ’ du 2 ’ du x ’ du 2 
equations (9), each of which is equivalent to two. 
0q bduz 00 du± _ 
?)u x dn s du x du 4 du x 
30 90 du 3 00 du 4 _ q 
0« 2 du z du 2 0iq du 2 
The first, 
The derivatives are thus given in terms of ?q, u 2 , u z , ^q, x Y , x 2 , x 3 , x±, and the last 
* It is unnecessary to give the differential equations. 
z 2 
