$10\mathrm{.}22.$ For the two$-$cylinder $90V$ engine shown in Fig. $10\mathrm{.}42,$ derive the following equa. tions of unbalance as a function of ${}_1$ : resultant primary force $F_p,$ resultant secondary force $F_s,$ resultant shaking force $S,$ direction of shaking force, and the distance $a_R$ from the plane of the first cylinder to the line of action of $S.$ Using the equations, determine $S$ and $a_R$ for ${}_1=60.MR{}^2=1,\frac{R}{L}=\frac{1}{4},a=100mm.$

FIGURE $10\mathrm{.}42$