[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 125 
where U; and Vj, are arbitrary functions of # and v respectively. 
To find the general solution of the Normal System it is necessary to 
choose the functions VU, and V; so that the value of @ given by (8) 
will also satisfy the proposed system of equations. 
We shall now determine the forms of the functions U; and V4. 
The equation 
2 10S pe 
Ou dv a9 
can readily be transformed into one of the Liouville type from which 
the general solution of the above equation is found to be (Sullivan C.T. 
Trans. Am. Math: Soc., Vol. XV, Series III, p. 176) 
VU'V' 
(9) p =} ms 
(O+V) 
where U and V are arbitrary functions of # and v respectively and the 
primes indicate differentiation of the functions with respect to their 
arguments. 
Let us introduce the notation 
U du 
vam EE Us = | Te 
V dv 
We => —— dv, V => 5 
: | JV : | VV" 
The direct differentiation of equations (8) and (9) gives (in terms of 
the notation just defined) 


08 ej C1 ! 
00 = Ci ! 
av Tae Nez (VU3+ Us) + V', 

If we substitute these expressions in (A) there results an equation 
which can be reduced to 
AD — = [or a UU" UV UL +e, Us | i 
Cv: NV’ V'+a VV) | + v[aus+ 
DA U"!, + Le DES a. 
Oe 
U’ = U’ 
1.e., to the form 
A(u) + w(v) + p(u)o(v) = 0. 
Sec., III Sig. 4 
