226 
we have, for the reduct equation, 
A?. (au, + 0, + 10,) =F (au, + wv, + Wz); 
for the solution of this equation 
AUz+ ev, +w,=C(1+k+C' (1-4); 
while & is determined by the same equation as m the last example. 
As this equation is of the sixth degree, it might be supposed that 
the complete solution of the problem should consist of six equations, 
each involving two arbitrary constants. It will be observed, however, 
that since the roots of the equation in & are of the form 
+h, +hp, +k,, 
and a, », » depend only on 4”, these six equations, each of which is of 
the shape, in the former case 
- ¥ 
Au, + wv, +0,= Ck + C' (-k)*, 
and in the latter case, 
Mz + pV, + W,= C(1+kP+C' (1 -ky, 
are reducible to three, and there are not virtually more than six arbitrary 
“constants. These constants are, in general, to be determined by given 
values of ¥,, Vz; Wz, Ura, Vray Mra, Corresponding to a given value of z. 
6. If the system of simultaneous equations, proposed for solution, 
were given in the form, 
AUzin + Opn tent +--+ =Uz | 
AUren + OVrin + CMrnt -- + =Vz 
Agtbzin + Orin + CWimt--- =, {’ 
&e. 
or, in the correlative form, 
a,A°u, + OA", + OA", +... =Uz ) 
AnA"U, + b,.A"0,+ GA, +...=0, | 
a,A"u, + 6,0", + 6,0", +...=Uz [{’ 
- &e. 
the first equation being multiplied by 2, the second by «, the third by 
y, &e., and all being added together, subject to the conditions 
