126 THE ROYAL SOCIETY OF CANADA 
Whence 
N'(u) + p'(u)o(v) = 0, 
u’(v) + p(u)o’ (v) = 0. 
Thus either p(w) or a(v) is a constant. But o(v)=V and o’(v)=V 
which does not vanish; since if V’ vanished ¢ would vanish. Hence 
a(v) is not a dir therefore p(w) must be a constant, 1.e., 
pu) =a U3; + ——> Tr = (U’ U", +4 U"U':) = © (a constant). 
Now (uz) can be written in the form 
Il 
Au) = une DÉPART VD) au u, NIUE 
U' Uni + c Ur V Tv] , 
which becomes (from the previous equation) vee 
À (uw) = @ U — © (UU3 — U2) — Uy NU! 
Equation (11) therefore becomes 
G1) GU= UN Lae UU) E TAN La TEE 
From the form of this equation, viz., 
a(u) + B(v) = 0, 

we conclude that es 
(12) a(u) = ce U— On JU’ + © (Ur — U U3) = — 6s, 
BO) = VE VAN Va (VENT) = Le 
where c3 is a further constant. 
Equation (12) can now be solved for U’; and V’;; their values are 
ue an [a (ue - U Us) + Co U+c | ; 
VU 
Vi= = [ar V V3) av a |. 
Hence 


i= | [aw UU) +aU +a] saad 
v= | (HEP | Ce 
A Ce YO ee => 
1 1 2 3 Ex 
Substituting these values for U; and V; in (8) we find as the general 
integral of the Normal System 
[+ Lee -vu ta U + cs + SF 
[Ce aig à eae = 
Therefore, on evaluating the double integral, we obtain the following 
fundamental set of solutions for the Normal System: 
6; = Us V3 + Us Vo + (ue Ui Uy) etre A 
NU! = 

