Here is a problem I have struggled on for hours. Hopefully it shouldn't be too difficult for someone more skilled at math.
Here I have my first function:
$$R_{oM} = \frac{p R_t^2 + 2 R_t R_n^2}{2 R_n^2 + p (1+ 3 c_M r_m) R_t}$$
All of the variables have positive values. This function is to be calculated individually on multiple objects, which each have their own separate values for $p$, $R_t$, & $R_n$.
Now I want to make a similar function that uses this first function as input. This formula applies when there is only one object:
$$R_{oR} = \frac{p R_{oM}^2 + 2 R_{oM} R_n^2}{D} = \frac{p R_t^2 + 2 R_t R_n^2}{2 R_n^2 + p (1+ 3 c_R r_m) R_t}$$
The formula on the right is like the first, $R_{oM}$, but with $c_M$ replaced with $c_R$. Now I must find the value of the divisor, $D$, for the above equation to be correct. $c_R$ will never be less than $c_M$.
$$0 \leq c_M \leq c_R$$
I also want $R_{oR}$ to work on a group of objects. Here is the complete formula that should work on a group of objects:
$$R_{oR} = \frac{p (\sum{R_{oM}})^2 + 2 (\sum{R_{oM}}) (\sum{R_n})^2}{D} \leq \sum{R_{oM}}$$
What must be the divisor, $D$, for the last two equations to be correct?