23. In any curve referred to an axis, the ordinate is a maximum or a minimum, when in the equation y = fx, an odd number of differential coefficients becoming = o the differential coefficient of the succeeding order is negative or positive. And there is a point of contrary fexure when an even number of differential coefficients becoming = 0, the differential coefficient of the succeeding order is real and finite.

24. A body urged by a constant force in an uniform resisting medium is projected in a direction contrary to the action of the force with a certain velocity; it is required to determine the velocity at any point of the ascent, the resistance being supposed proportional to the square of the velocity. Find also the greatest height to which the body will ascend.



1. In a plane triangle the vertical angle, the perpendicular and the rectangle under the segments of the base being given, it is re quired to construct the triangle.

4.2.2 17% 2. Solve the equation 2

+ 2 = 0, two roots 3


being of the form a and

And find the number of all the pos


sible values in integer numbers of x, y and z in the equation

5x + 7y + llz = 224. 3. What are the dimensions of the strongest rectangular beam, that can be made out of a given cylinder, when placed to the most advantage ? and what is its strength, compared with that of the greatest square beam cut out of the same cylinder ?

4. In the wheel and axle (the inertia of which may be neglected) required the ratio between the radii, when a weight () acting at the circumference of the wheel generates in a given time the greatest momentum in a weight (W) attached to the circumference of the axle.

5. Tangent of half the spherical excess =

tan. }b x tan. fc x sin. A

1 + tan. 6 x tan. fc x cos. A where b, c, and A are the two sides and included angles of a spherical triangle.

6. The excess of the Sun's longitude above its right ascension may be found by the equation

tan.4T X sin. 2L. tan. (L – R.A.)

1+ tan. 17 x cos. 2L 7. Find an expression from which the effect of parallax upon the horary angle may be accurately calculated; the horizontal parallax, the polar distance of the heavenly body and the time before or after transit, being the only given elements.

8. If an orifice were opened half way to the centre of the earth, what would be the altitude of the mercury in a barometer at the bottom of it, when the altitude at the surface is 30 inches?

9. A vessel formed by the revolution of a parabola round its axis is placed with its vertex downwards, in which there is an orifice one inch in diameter. A stream of water runs into the vessel through a pipe of two inches diameter at the uniform rate of eight feet per second. What will be the greatest quantity of water in the vessel, supposing the latus rectum to be six feet?

10. Find the present worth of the reversion of a freehold estate after the death of a person now sixty years of age, the rate of interest being given ?

11. When a ray of light passes out of one medium into another, as the angle of incidence increases, the angle of deviation also increases.

12. To find the least velocity, with which a body projected at a given angle of elevation will not return to the earth's surface.—To find also the latus rectum of the orbit described, and the position of the axis.

13. Supposing the Moon's orbit at present to be circular, what would be the excentricity of it and the periodic time, if the attraction of the earth were diminished




14. Find the sum of the following series:

(1.) sin. (A)+sin. (A + B)+sin. (A+2B) &c. ad infin.
(2.) cos. A + cos. 3A + cos. 5A ad infin.



&c. to n terms,
by the method of increments.
15. Find the following fluents:

xi (1.)

(2.) 2(a + bx')


[ocr errors]

= a.

(3.) J Vas — x* xxi, between the values z=0, and r=a; and solve the fluctional equation (ie+y')}

ху 16. A and B put down equal stakes -A has m chances of success, and B n chances. There are, moreover, p chances, which entitle both parties to withdraw their stakes,-to find the gain of A.

17. Two equal weights are placed at a given distance from each other on a straight rod supposed to be without weight. Find the point of suspension, so that the pendulum may vibrate seconds.

18. Construct the curve whose equation is (a – X)* . (a + 1) = x’y®, and shew whether there are any cusps.

19. Invenire incrementum horarium areæ, quam luna, radio ad terram ducto, in orbe circulari describit.

20. Let the force to a centre vary as the distance, to find all the various curves, along which a body may move, so that its oscillations may be isochronous.

21. Given the diameters of two planets, and the periodic times and distances of their respective satellites; compare their densities and quantities of matter.


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