230 
and the arbitrary portion is 
C,(1 +h)" + 0 (A-h)? + C2 (1 + he)® + C'n (1 - fe)? 
C3 (1 +43)? + C’, (1 — h5)* : C,(1 + ky)? + OC", (1 - hy)’. 
9. Let the system proposed for solution be 
Unig = Uz + bv, + 6, 
Vgig = Ugly, + bVz + Co)” 
Proceeding as above, we obtain 
OP (Aly + a.) =? (AU, + Vz) FAC, + CQ, 
whence, at once, result the two equations for the determination of u, 
and v,, namely, 
AC, + KE, 
Miz +02 = Chk + C1 (-hy + otal M 
1-k? | 
! Veate'e 
Nite pg = Calg + Cry (ye ¢ MOSES | 
1—k? ss} 
the equation in & being, as before, 
(2 — a,) (# — 6,) = bya. 
Hence, 
0 (1—6,)+ re 
G22: 
pp Gh ae ee {Dik + Di, (-hy)*} 
1-a.1-6, -d,a, b, : 
ee + D', (~k,)* + D,k,* + D', (- kz)’, 
U,= 
kh? — a 
+ i 
10. As the values of the constants a, b,, ¢,, &c., are supposed to be 
given by observation, and are therefore liable to certain small errors, it 
may be worth while to consider what corrections should be introduced, 
if these constants should become, respectively, a+ 0m, 6,+0b,, e+e, 
&e. 
-{ Dyk? +-D', (— ky)*}. 
It is obvious that, as the roots of the equation in & depend on the. 
values of the constants stated, these roots will receive certain increments, 
which may be, for convenience, respectively denominated by the ex- 
pressions 0h, dk,, dk3, &e. 
Thus in the second example, quoted under the first article, the value 
of uv, will become 
Uz = CO; (ky + dk,)* + C, (hy + 0h,)* + Cy (hes + dhs)*, 
