220 
consequently, in the above solution, £ may be supposed to have any one 
of x known values. 
Hence, writing down the series of solutions corresponding to the 
several roots /,, ks, &3, &c., it is obvious that the general solution of the 
given system of simultaneous equations in finite differences is exponible 
in the form— 
Ue = Onley? + Onhs? + Cphget « . . + Cyha? ) 
0, = Dyk? + Dok? + Deke + . . » + Dyley# | 
W,= i,k? + Eke + Eks* sP ve disp Ei kent | 2 
&e. J 
where, of the constants C,, D,, £,, &c., » only are arbitrary. 
When some of the roots ky, h2, £3, &c., are equal, or when there are 
pairs of imaginary roots, modifications sufficiently evident must be in- 
troduced in the general form of solution. 
Thus, in the case of 7 equal roots, whose common value is /,, the ge- 
neral form of solution becomes 
Uy = hy” (On 24+ Ciga’?+...4+C\vt+Cy)+...+ Ck," ) 
Ve = hy? (D2) + D, ga? +... +Di¢+ Dy) +... + Dalek 
w= hy? (Lv) + Bea? +... thet h)+...+ fh (7 
&e. J 
Lastly, in the case of a pair of imaginary roots, the general form of 
solution becomes— 
t= Ci, (Fey + hn yf -1)° + Ch (ly — a -1)* + Oslige ts . + Ola? 1 
0, =D, (hy + hey f -1)* + D, (hy — rg -1)* + Doig? +. « » + Dylin® 
w,= Fy, (hy + fgof-1)° + Ey (hy oa Tiga/ -1)" + Fishes? + eo 8 + Ek, ; 
&e. J 
which may obviously be reduced to the simpler form— 
Uz=(ke+h?)? Cos {x tan? (2) + Cn} + Cohgtt+ . 2. + Cyhn? } 
| ey 
0, =(k2+h,7)? D’,cos {x tan (2) +D',} + Dslis+ 2. . + Dylin® 
1 
? 
w,=(h? +h?) EB’, cos {x tan (7) +f") +Bkset. .. +#h,? 
1 
&e. 
EXAMPLES. 
(1.) Let it be proposed to solve the system of two simultaneous 
equations— 
