What is the correct way to apply Kirchhoff's rules to this circuit?

[kliker]

[PLAIN]http://img17.imageshack.us/img17/201/73448266.jpg

let's suppose that we know all resistors and voltages

by applying kirchhof rules we have

V1 - i*R1 - i*R2 -i2*R3 = 0
V1 - i*R1 - i*R2 - i1*R4 - V2 = 0
-V2 + i2*R3 - i1*R4 = 0

we know V1,V2,V3 R1,R2,R3 and we can find the currents

now we have this image

[PLAIN]http://img220.imageshack.us/img220/4225/53762410.jpg

it's the same circuit and we suppose that there are two currents ia and ib

now here we have this, is this right?

V1 = ia*R1 + ia*R2 - ib*R2
-V2 = ib*R4 + ib*R3 - ia*R3

is this correct? but if we find ia and ib, we know just two currents, I mean in the previous image we have 3 currents, i i1 and i2

here in the second image if these equations are correct we get the current ia which is the current i and ib which is the current i1

but we don't have only these 2 currents we have a third current too called i2, how can I find it here?

thanks in advance

 

[Zorba]

I don't know what the second image is about but the first one looks correct, except you didn't apply the junction rule, so you won't be able to solve it until you do.

It's worth noting that you should always try to apply the junction rule before you start writing any equations as it greatly simplifies things.