1931　京都帝国大学　理学部

【１】　Find the number of combinations of $n$ things $r$ at a time.

　Prove by the mathematical induction.

【２】　$a$,$b$ are such that $2x+3y=5$ always touches the curve ${\displaystyle \frac{y^2}{a^2}}-{\displaystyle \frac{x^2}{b^2}}=1\text{.}$

　Represent graphically the condition between $a$ and $b\text{.}$

編注　どの原典でも双曲線の式中，$x$と$y$の位置が不明確．逆になっている可能性あり．

【３】　The curve ${\displaystyle \frac{x^2}{{\alpha }^2}}+{\displaystyle \frac{y^2}{{\beta }^2}}=1$ revolves around one of its tangents whose distance from the centre of the figure is $p\text{.}$

　Prove that the volume of revolution is $\pi \alpha \beta \cdot 2\pi p\text{.}$

【４】　In the preceding problem, find the maximum or minimum of the volume of revolution when the tangent varies.