3.37 The magnitudes of two forces acting on a material point at a right angle, and the magnitude of their resultant, form an arithmetic progression. Determine the ratio of the magnitudes of the forces.
3.37 According to the condition,

$a^{2}+(a+d)^{2}=(a+2 d)^{2}$, i.e., $3 d^{2}+2 a d-a^{2}=0$.

Let $d=a k(k>0)$; then

$3 a^{2} k^{2}+2 a^{2} k-a^{2}=0$, from which $3 k^{2}+2 k-1=0$.

Only the root $k=\frac{1}{3}$ is suitable. We find the required ratios: $\frac{a+d}{a}=1+\frac{d}{a}=\frac{4}{3} ; \quad \frac{a+2 d}{a}=1+\frac{2 d}{a}=\frac{5}{3}$.

Answer: $3: 4: 5$.