Newtonian Form of the Thin-Lens Formula

The formula \frac{1}{p}+\frac{1}{i}=\frac{1}{f} is called the Gaussian form of the thin-lens formula. another form of this formula, the Newtonian form, is obtained by considering the distance x from the object to the first focal point and the distance x' from the second focal point to the image. I am going to show that xx'=f^{2} is the Newtonian form of the thin-lens formula.

In this situation, my diagram is going to look as follows below:

 

Starting with the thin lens formula, \frac{1}{p}+\frac{1}{i}=\frac{1}{f} , I note that the object distance is x+f , and the image distance is x'+f , where the focal distances for the two lenses are equivalent. After making these substitutions into the thin lens formula, and solving for f I obtain the equation f=\frac{xx'+fx+fx'+f^{2}}{x'+x+2f} . Now I can multiply each side of the equation by x'+x+2f and solve for f^{2}  to obtain the final solution f^{2}=xx' , which is the Newtonian form of the thin lens equation.

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