In a genetic mapping experiment, the crossing over frequencies...

Crossing over frequencies are directly proportional to the distance between genes on a chromosome, where $1\%$ crossing over equals $1$ map unit.

The maximum recombination frequency is between P and S ($35\%$), which means P and S are located at the two extreme ends of the gene map.

The distance between P and R is $25\%$, and between R and S is $10\%$. Since $25 + 10 = 35$, R must be located between P and S.

The distance between P and Q is $10\%$, and between Q and R is $15\%$. Since $10 + 15 = 25$, Q must be located between P and R.

Checking the remaining distance, Q and S is $25\%$. From the map, the distance from Q to S is the sum of Q to R ($15\%$) and R to S ($10\%$), which equals $25\%$. This confirms the arrangement.

Thus, the correct linear order of the genes on the chromosome is P, Q, R, S.

Answer: P, Q, R, S