302 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 


Ex. 9. AU,+ aA? U,+6U,=X 
1 z =i 1 ? =a 
Sha 0+X) — — = 
tes gS Re OO ee B)~ 0+) 
a 1 4 -—1l aff 4 “oat 
= 2\x 2 ae: hs x, Velie a 
=AGs Ope eR Agee @auak aap Ee 
= AC + a%)"+ BQ + 6)°4—, (a? +a) (1 + a2)? 3 
a—6 (1+a?)e+1 
ae carad eae | 2 aa 
ae TEE ae) * 4 Br 
Ex. 10. Wy tax d® 1c—2. © 
: : : A i 
This equation gives “= 7G 
__l-art_y 
~ 1l-a? wat ax at 
=(1—awa*)?,, where v, is determined by the equation 
0, -@ 2d? wo dtv,=X 
ak 
Now AP @ A? v,=@ AV, +4 Pr 41 (Prop. 3.) 
2 92 a” x 
0, —@ 22Av,——20 = 
ax a 2 a+l 
Aa - a 1+a?*2? | 2x 
a+1 a? 44+22° a” @ (#+227) 
which being solved by the ordinary method, v, and therefore u, (provided A? Uy can 
be found) is known. 
Equations of Differences with two independent variables are not capable of 
solution to any great extent. An example or two will suffice to illustrate our 
process. 
Ex. Ll. Ay Uy, y—Ay Ue, y= 0. 
The solution is ub, = aw 

Treat 4, as a constant c, then the solution of (4.—¢) #2, y=6 is 
a b 
Uy, y= (c+1)7JA oar 
ty, y= (Ay +1)? zy —4y—1 6 
d 
—— Nia 
= (a) Uy—by 
=VUyp~—5y 
vy+« being an arbitrary function of y+z2. 

