Calendar: 1880-1881 Page 631

GENERAL LITERATURE AND SCIENCE 631 What is 1partial differentiation If χ and be both functions of third variable and xy write down the value of du dV explaining carefully the meanings of the differential co- efficients If then xL yi χ ax ay If ν e&mu where £ is infinitesimal express in simple formula tan- in terms of tan What is meant by 4changing the independent variable' If ux cos 6י uy sin0 transform dx dhj into an expression where and θ are the variables Define the maxima and minima values of function and investigate the conditions that φ χ may have maximum or minimum value when χ and discriminate between the two cases Inscribe the greatest rectangle in given semicircle and shew that it is half the square inscribed in the whole circle 10 In any curve investigate an expression for the cosine of the angle between the radius vector and the tangent at any point The co-ordinates of any point in the cycloid satisfy the equations χ α θ sin θ cos Prove that the angle which the tangent at any point makes with the axis of is 11 Explain the connection between integration and sum- mation of series When η is infinite prove that -i- -f --- -τ- ϋ£ η η 42 tut