Law of Ratios? derving a differential Fick Equation

[sadsol]

Hello,

I'm trying to make sense how a mathematical derivation (equation 4) from equations 2 and 3. Basically it defies any algebraic law I know. The paper is cited below and the the equations are attached as picture.

Do you think this is a correct derivation?
Is there such a thing as "Law of Ratios"? If so can someone tell me what it is?

Thanks! Paper: Jaffe, M. B. "Partial Co2 Rebreathing Cardiac Output--Operating Principles of the Nico System." J Clin Monit Comput 15 6 (1999): 387-401. Print.

 

[JThompson]

I've never heard of the Law of Ratios, but the result is mathematically sound.

I'm going to simplify this by saying that $C=\frac{A_{1}}{B_{1}}=\frac{A_{2}}{B_{2}}$. Then $CB_{1}=A_{1}$ and $CB_{2}=A_{2}$. Subtracting the second equation from the first,
$CB_{1}-CB_{2}=C(B_{1}-B_{2})=A_{1}-A_{2}$, which means $C=\frac{A_{1}-A_{2}}{B_{1}-B_{2}}$.

 

[JThompson]

Of course, we must have $B_{1}\neq B_{2}$...