(3.) 18

5. Solve the following equations :

(1.) 28 – 6.3* + 6x + 8 = 0; one root being 1 + 3.
(2.) 2.8 3.x2 – 46x — 72 = 0, by the method of divisors.

13.2° +54x - 72=0, the roots being in harmonical

progression. (4.) x' – 10.2" + 35.x"_502+24 = 0, arithmetical

progression. (5.) 24

5.r 5x4 + 45.x 36 = 0, two roots being of

the form + a and 6. The roots of every equation of the form a** + px&*-' + +px+1=0 may be found by the solution of an equation of half the number of dimensions. 7. Solve the equation x + 1 = 0,

(1.) By means of the preceding question.

(2.) By trigonometrical formulæ. 8. The equation 26 – 2.25 + 6.8* 8.20 + 12.02 8.3 + 8 = 0, has equal roots. Find them. 9. (1.) Find the roots of the equation x'-6.3°+3.0--18=0 by

Cardan's method. (2.) Shew that this method is applicable only when two

roots of the proposed cubic are impossible. 10. Give Euler's solution of the biquadratie x*+px®+qx+r=0.

11. The roots of the equation x" +pra-'+qx9– + tx + u= 0 being a, b, c, &c. if

S, = a + b + c +
Se = a + 12 + C +

Son = a* +6++ . [(m) being any integer not <n;) prove that

Sn + pam-i+qSm-e+ +US=0. 12. The roots of the equation 2+pe+qx+r=0, are a, b, c,

transform it into one whose roots shall be a',v,c. 13. (1.) If (a) be an approximate root of the equation

.2 +p.r* +or=r,
so that a, +pa,' +9a;=r;;
prove that x = a,+

a,. (

very nearly;

rorr, + 2a," + pa, r or r, being used according as (a,) is > or < 1. (2.) Approximate by this formula to the value of (x) in the

equation 2 – 2x = 5.




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1. Trace the signs of the Sine, Cosine, Tangent, and Secant, through the circle.

2. Transform the formula, (cos. A)" + a. (cos. A)". (sin. B) + b.(cos. A). (sin. C)'+&c. where the radius 1, to an equivalent formula, where the radius =r; and prove the rule..

3. Given the sines and cosines of two arcs A and B; it is required to find sin. (A+B) and sin. (A – B). 4. Prove that, sin. (A+B). sin.(A-B) = (sin. A)-(sin. B)*

and cos. (A + B). cos. (A-B) = (cos. A)— (sin. B)* 1

I 5. If x + = 2 cos. A, prove that 2 + = 2 cos. 3A.

x3 6. Tan. (45° +A) = Tan. (45° - A) + 2 Tan. 2A. Prove this, and explain what is meant by a Formula of Verification. 7. Tan. A + cotan. A = 2 cosec. 2A.

" Tan. A cotan. A = 2 cotan. 2A. 8. In any triangle, the sum of any two sides : their difference :: the tangent of half the sum of the angles subtended by those sides : the tangent of half their difference.

9. Given the sine of 1', shew how the sines of all arcs from l' to 90°

may be found, rad. 1. 10. Given two sides of a triangle, and an angle opposite to one of them, solve the triangle; and shew the ambiguity in this case.

11. Given two sides and the included angle; solve the triangle.

12. Explain the method of finding the distance between two visible but inaccessible objects on an horizontal plane ; and shew how the requisite triangles are to be solved.

13. Two sides of a triangle and the angle included being given, find the area of the triangle.

14. The perimeter and the three angles of a triangle being given, find each of the sides.





sin. a

sin. c

1. Every plane section of a sphere is a circle.

2. The sum of the three angles of a spherical triangle is greater than two, and less than six, right angles.

3. The angles of a spherical triangle are A, B, C; the sides respectively opposite to them, (a), (b), (c); the rad. of the sphere =1. Prove the following theorems : I.

cos, a x cos. 6 Cos. C =

sin. a x sin. b
sin. A sin. B sin. C

sin. 6
III. Tang. } (A+1)=Cotan. C.

cos. (a-b)
cos. A (a+b)

sin. } (a-6)
Tang. } (A-B)=Cotan. } C.

sin. } (a+b) IV. If (a+b+c) = S

S sin.

) x . sin.' ACE


sin. a X sin. 6


sin. cos. ¿C=

sin. al X sin. b 4. What are the general theorems deduced by the application of the formulæ I. and Ill. to the polar triangle?

5. Prove Napier's rules for that case in wiich the complement of an angle is the middle part.

6. Given the obliquity of the ecliptic, the right-ascension and declination of a star; find the angle of position in terms of those quantities.

7. In a spherical triangle, two sides and the included angle are given :-Required the third side in a form suited to logarithmic computations?

8. Find the area of a spherical triangle.

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1. If all the sides of one spherical triangle be respectively equal to all the sides of another, then all the angles of the one are equal respectively to all the angles of the other.

2. Every section of a sphere is a circle. 3. Deduce an expression for the area of a spherical polygon of a sides.

4. Shew that, in the complete solution of all the cases of a rightangled spherical triangle, there must necessarily be six ambiguous equations : explain the modes by which apparent ambiguities are removed in all the other cases.

5. Shew how Napier's two rules may be applied to quadrantal triangles. Enumerate the circular parts, and write down the ten equations necessary to their solution.

6. In how many equations can we exhibit the solution of all the cases of oblique-angled spherical triangles ?

7. In any spherical triangle in which A, B, C, are the angles, and a, b, c, the opposite sides respectively: (1.) Given a b C, required c, in a form for logarithmic com

putation, without the intervention of A, B. (2.) Prove that

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x {20+log. sin. (S-6)+log. sin. (S-c)- log.sin. 6-log.sin.e}





1. Give a definition of the fluxion, or differential of a quantity, and from that definition shew how the differential of at may be found.

2. Differentiate the nth hyp. log. X.

3. Exemplify the infinitesimal analysis, by drawing a tangent to a circle at a given point.

4. Trace the curve whose equation is zo — y = ax®, and construct its asymptote.

5. If any number of lines, all terminating in a given point, and situated in the same plane, be given in magnitude and position: it is required to find the position of another line passing through the same point, such that the sum of the projections of all the former this last shall be a maximum.

0 6. Draw an asymptote to the spiral whose equation is cos.

lines upon


and find the angle contained between its apse and asymptote.

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7. Find the value of a


2* when a = 0, and also of

Na - /2ax-x? et esin. :

when x = 0.
X – sin.
8. Integrate the following differentials :

dxw x+11+ 2o and nx




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dx 9. Find the point of suspension when the time of an oscillation of a given straight line is the least possible.

10. Determine the centre of gyration of a rectangle revolving in its own plane round an axis given in position.

11. Find the area of the catenary.

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