is applied to find the areas of curves and the volumes of solids. 7. Trace the curve ra (1+cos): find its area, and the angle at which it cuts the axis of y. 8. Integrate the following equations: 9. What is meant by the singular solution of a differential equation? Find the general integral and the singular solution of the equation 10. Enunciate and prove Newton's seventh Lemma. 11. A body moves in an ellipse so as to describe areas proportional to the times of describing them round one of the foci; find the direction and law of the force acting on it. 1. If f(x)=0 is an equation under a rational form, and two quantities a and b when substituted for x in f(x) give results with different signs, show that an odd number of the roots of f(x) = 0 lie between a and b. Show that the equation x+x2-8x-15=0 has two, and only two real roots of contrary signs lying between 3 and -2. 2. Transform the equation a3+qx+r=0 into one whose roots are the squares of the differences of its roots; hence show that the proposed equation has three real roots or only one, as +: is negative or positive. 4 27 Ex. Find the nature of the roots of 3-6x+6=0. 3. Show that an odd number of the roots of the equation f'(x)=0 lie between each adjacent two of the roots of the equation f(x)=0. 4. Show that the equation f(x)=0 cannot have more positive roots than f(x) has changes of sign, or more negative roots than f(x) has changes of sign. 5. Solve the following equations: x2-7x3+13x2+3x-18=0 which has equal roots, x2 - 8x3-16x2+128x=0 whose roots are in arith- 6. Define accurately a spherical triangle, and explain what angles are represented by its sides and angles respectively. Show that the angles of a spherical triangle together cannot be less than two or greater than six right angles. 7. Prove the formula cosa cosb cosc+sinb sinc cos A for all cases; and write down a formula connecting a, b, A and C. 8. State and explain Napier's rules for the solution of right-angled spherical triangles; and apply them to deduce the requisite formula of solution when the hypothenuse and one side are known. 9. If the arcs which form the sides of a spherical triangle be very small relatively to the radius of the sphere, show that each of its angles will exceed the corresponding angle of the plane triangle, whose sides are the same length as the arcs forming the sides of the spherical triangle by one third of the spherical excess. 10. Find the relations between the constants in the equations to a straight line and a plane that they may be perpendicular to one another. Find the equation to a line perpendicular to the plane α 1 11. Find the angles which the line of intersection of the planes lx+my+nz=d and l'x+my+n'z'd makes with the coordinate axes. 12. Explain how two equations represent a curve in space. If u=0 v=0 are the equations to a curve, show that du du du (E-x) + (n-y) +(5-2) =0 (20-3) dq dv dq dy dv dz +(n−y) + (b −z) dv dy =0 dz are the equations to a tangent to it at the point x, y, z. Ex. Find the tangent to the curve and show that it is inclined to the axis of z at a constant angle. CHEMISTRY.-DR. ANDREWS. 1. Give a description of the thermo-multiplier, and mention the advantages it possesses over the differential thermometer in experimental inquiries on radient heat. 2. What is the true relation according to recent investigations between the latent heat of steam generated at 212° F. and at lower temperatures ? 3. What is the cause of the cloud which generally appears in the receiver of an air pump soon after the process of exhaustion is begun ? 4. Explain the meaning of the term isomorphous, and enumerate the principal groups of isomorphous bodies. 5. Give an account of what is usually termed "catalytic action." 6. State the law of gaseous combination commonly known as the theory of volumes, and illustrate it by giving the composition by volume of the compounds of oxygen and nitrogen, of those of carbon and oxygen, of marsh gas, of olefiant gas, and of the vapours of alcohol and ether. 7. Explain the chemical changes which occur during the solution of gold or platinum in nitro-muriatic acid. 8. What are the chief sources of bromine, and how is it prepared? 9. How is the identity of the different forms of carbon proved? 10. Give a general account of the properties and composition of the principal ores of iron. 11. State the composition, mode of preparation, and principal properties of the oxalic and formic acids. 12. How is oxamide prepared, and what is its composition? 13. Describe the acetous fermentation, and explain the chemical changes which occur during the process. 14. Give an account of the theory of substitution, and illustrate it by expressing in symbols the relation between the acetic and chloracetic acids, and between the benzoic and nitro-benzoic acids. 15. Describe the chemical action which takes place in a closed voltaic circuit composed of platinum, zinc, and dilute sulphuric acid. 16. If 32.5 grains of zinc be dissolved per minute in each cell of the battery, how many cubic inches of oxygen and hydrogen will be obtained in the same time in the decomposing cell? 17. With copper poles and a solution of sulphate of copper in the decomposing cell, what is the final result of the process of electrolysis? NATURAL HISTORY.-DR. DICKIE. ZOOLOGY. 1. State the general influence of heat on animal and vegetable organisms, and illustrate by examples. 2. Describe the tissues which enter into the formation of the teeth, and give examples illustrating the relative development of these tissues in different Vertebrata. 3. State the principal differences between man and the higher Quadrumana, more especially those having reference to the skeleton and teeth, and the order of development of the latter. 4. State the peculiarities of the teeth and alimentary canal in the frugivorous Cheiroptera and in certain Phyllostomidæ of the same order. 5. State the general structnre of the teeth and nature of the dentition in the ruminating Artiodactyla, and mention any relation between the development of certain teeth and of frontal appendages. 6. State the general characters of the Articulata. 7. Describe the nature of the changes through which certain insects pass before reaching maturity; state also the leading characters of the Coleoptera. 8. Describe the peculiarities of the Cirropoda in reference to development and habits. 9. Describe the modifications in form, position, &c. of the respiratory organs in the Annelida. 10. State the general characters of the Nudibranchiate and Gasteropodous Mollusca. BOTANY. 1. Define and explain by plans or reference to examples the following terms: pari-pinnate, impari-pinnate, umbel, capitulum, orthotropous. 2. Describe the nature of the hard tissue in certain stone fruits, and illustrate by a plan. 3. Explain the nature of those organs called Phyllodia and Pitchers. 4. Describe the different ways in which dehiscence occurs in fruits. 5. State the general structure of the mature pollen, and describe its mode of development. 6. State the leading organic constituents of plants, and mention their sources. 7. Silica is abundant in the tissues of certain plants; whence do they derive it, and in what form do they receive it? 8. Enumerate the proofs having reference to the functions of Stamens and Pistils. 9. Describe the general structure of the seed in Exogens, and state any modifications in the relative development of its different parts. 10. Mention the different organic arrangements usually associated with the Endogenous structure of stem. |