Explanation of the Problem:
The problem states that two planets revolve around the sun with frequencies N1 and N2 revolutions per year, and their average orbital radii are R1 and R2, respectively. The task is to find the ratio of R1 by R2.

Formula for Orbital Frequency:
The frequency of a planet's revolution around the sun is given by the formula:
N = 1/T
Where N is the frequency in revolutions per year, and T is the time taken by the planet to complete one revolution around the sun, in years.

Formula for Average Orbital Radius:
The average orbital radius of a planet is given by the formula:
R = (2πa)/T
Where R is the average orbital radius, a is the semi-major axis of the planet's elliptical orbit, and T is the time taken by the planet to complete one revolution around the sun, in years.

Solving the Problem:
We are given that the two planets have frequencies N1 and N2 and average orbital radii R1 and R2, respectively. We can use the above formulas to find the ratio of R1 by R2.

Step 1:
Using the formula for frequency, we get:
N1 = 1/T1
N2 = 1/T2

Step 2:
Taking the ratio of the above equations, we get:
N1/N2 = T2/T1

Step 3:
Using the formula for average orbital radius, we get:
R1 = (2πa1)/T1
R2 = (2πa2)/T2

Step 4:
Taking the ratio of the above equations, we get:
R1/R2 = (a1/a2) x (T2/T1) x (N1/N2)

Step 5:
Substituting the value of N1/N2 from Step 2, we get:
R1/R2 = (a1/a2) x (T2/T1)^2

Conclusion:
Thus, we have derived the formula for the ratio of R1 by R2 in terms of the semi-major axes and periods of the two planets. We can use this formula to find the required ratio.