Joint distrib of gamma distribs

Let X_i, with i \epsilon {1,2,3} be independant r.v. and suppose that X_i \epsilon Gamma(r_i,1).

Now set:

Y_1= X_1/(X_1+X_2)

Y_2= (X_1+X_2)/(X_1+X_2+X_3)

Y_3= X_1+X_2+X_3

Determine the joint distrib of Y_1, Y_2, Y_3.

What is the best strategy to tackle this problem? rewrite X_i in Y_i and try to find a jacobian? I tried that but i got some divided by 0 in it.

(answer: f_\vec{y}(\vec{y})=1/(gamma(r_1)*gamma(r_2)*gamma(r_3))*y_1^{r_1-1}*y_2^{r_1+r_2-1}*y_3^{r_1+r_2+r_3-1}*(1-y_1)^{r_2-1}*(1-y_2)^{r_3-1}*exp(-y_3) with 0<y_1<1, 0<y_2<1 and y_3>0 )

im sorry about the notation but i did not get the mathmode to work...