227 
re) 
AA + ope + gy +. paca 
# 
bAtbpt+byt+.. aga be 
CA + Oye +vt... =f 
&e. 
we get for the reduct equation, in the former case, 
OD (Aig + ez tz +.. = RM (Au, + evr t+ ..), 
and, in the latter case, 
A”. (Ate +e, +r +.. =k (au, + ev, ++.» +); 
the corresponding solutions being, in the former case, 
Aly + Vz +0, +...=C (ek) + OC! (Wh) +C” (ah) +... 
and, in the latter case, 
Ap + Kz t+ pt... =C (ak +1)° +O" (wWh+1)24+0" (E+ 1)7 +... 
It should be noticed that the values of the constants and of the several 
roots of the equation in & are, of course, wholly different from those 
occurring in the previous article, in which a notation similar to that 
just used was employed. 
7. If the system of equations to be solved were of the form 
A.U, = Uz t+ bv, + ew,+...+fi(2) 
A.V, = Ally + by, + Cr+... +f2() 
A. W,= Ag; + bv, + C3 +... +f3(x) f’ 
&e. J 
where fi, fo, fs, &c., are given algebraic functions,—proceeding as be- 
fore, and with the same system of conditions, we obtain the equation 
A (Atle + pe + 80.) = Te (tle + pe + &0.) + (afi + wfot &e.), 
(A-&) . (Au, + 20, +0, + &e.)=Afi + wfet fg + Ke. = F(a), 
the solution of which is, in its primary symbolic form, 
. Au, + ev, + vw, + &. =(A-k)'. F(z) +(A-A£)".0, 
or, in its semi-evaluated form, supposing, for simplicity, that / only 
contains positive integer values of z, 
