222 
these quantities, corresponding to v= 0, are respectively, «, 8, y; then 
we shall have, for the determination required, 
a=C,+ C,+ C, >) 
B=D,+ D+ Ds > 
y=L,+ #,+ EF; j 
writing, for simplicity, the constants D,, #,, &c., in their primitive 
form. 
The following is the geometrical problem, of which the given system 
of simultaneous equations of finite differences is the analytical state- 
ment. 
To find three curves AL, BM, CN, such that, 
taking OD=. and Oo as the unit-increment of the abscissa, 
OP= Uny Op = Uxir5 
OQ=,, Whence 09 =Vx, 
OR=vw,, OF = Wr) 
we may have 
op=b,. OQ+¢,. OR) 
09 =. OP +0. OR +, 
or = 03. OP +b,.0Q J 
2. In the preceding article, the solution of the system 
Uns = UU, bVz+CWet.. +) 
Vou = Aqlt, + Oz + CoWzt ... \ 
Wri Ali + DDe t+ CsWnt » » » | ‘ 
&e. J 
was found by the aid of indeterminate constants. The following is, per- 
haps, a simpler and more elegant method. 
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