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12. If ab(x? yo)+ xy(a? 62) 1

(a? +62)(2x2 + y2) Then (a? +62)(x2 + y2)>2ab(x2 - y2)+2.ry(a'-62),

and (a? - 2ab +62)x2 + (a? + 2ab +62)y2> 2xy(a? 62);

.: (ab)2x2 – 2(a - b)(a + b)xy + (a + b)y->0. .. (a - b)x - (a +b)y>0; (a - b)x> (a + b)y; and

a+b

-6 °

2

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If m

a+b _ (a")"+(6x)"_**+y* 5.

which is of the forms Section IV., Art. 13, p. 22. ?

x+y a" to (a)+(6") amn+bmn_(am)"+(b*)m_2"+y+

not of the required forms, except when m=r, a+b (a")+(6") x+y an+"

r; then

a” –b"_"Ły. ap+bA P a artu

1 6. a"-3"=a-(8m)*=an – 2cm. am-b=am - (31*)=a"-3.

Here a"- 200 and am - 2m have a common divisor when m and n are odd or even numbers. 7. Let am 3"=(am) – (0*)=x-Y ; 1

1 then an – bm=(am) m* — (64) mn=2"*- ymm, and the given expressions are reduced to the

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ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY.

In Twelve Sections, demy 8vo.

CONTENTS AND PRICES.

PRICE

3d.

SECTION I. Of Numbers, pp. 28
SECTION II. Of Money, pp. 52

6d.
SECTION III. Of Weights and Measures, pp. 28 3d.
SECTION IV. Of Time, pp. 24.....

3d. SECTION V. Of Logarithms, pp. 16

2d.
SECTION VI. Integers, Abstract, pp. 40

5d.
SECTION VII. Integers, Concrete, pp. 36
SECTION VIII. Measures and Multiples, pp. 16 .. 2d.
SECTION IX. Fractions, pp. 44

5d.
SECTION X.
Decimals, pp. 32

4d.
SECTION XI.
Proportion, pp. 32...

4d.
SECTION XII. Logarithms, pp. 32

6d.

...... 5d.

LONDON: SOLD AT THE NATIONAL SOCIETY'S DEPOSITORY,

WESTMINSTER.

NOTICE.

EACH section of the Arithmetic may be purchased separately; also the twelve sections together, done up in boards, with cloth

covers, at 48. 6d.

WORKS BY PERCIVAL FROST, M.A., FORMERLY FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE;

MATHEMATICAL LECTURER OF KING'S COLLEGE.

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NOTES AND ILLUSTRATIONS, AND A LARGE COLLECTION OF PROBLEMS, PRINCIPALLY INTENDED AS

EXAMPLES OF NEWTON'S METHODS;

ALSO

HINTS FOR THE SOLUTION OF THE PROBLEMS.

THIRD EDITION.

BY

PERCIVAL FROST, M.A.,
FORMERLY FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE;

MATHEMATICAL LECTURER OF KING'S COLLEGE,

The portion of Newton's Principia which is here presented to the Student contains the solution of the principal problem in celestial Mechanics, and must be interesting to all, even those who do not intend to follow out the more complicated problems of the Lunar and Planetary Theories.

The study of the geometrical methods employed by Newton cannot be too strongly recommended to a student who intends to pursue Mathematics, whether Pure or Applied, to the higher branches; for he will, under this training, be less likely to work in the dark when he uses more intricate machinery.

I have endeavoured in this work to explain how several of the results obtained in the Differential aud Integral Calculus can be represented in a geometrical form; and I have shown how, in a large class of problems, the geometrical methods are at least as good an 'open sesame' as the Differential Calculus.

In this, the third edition, I have given Solutions or Hints for the solutions of all the problems, in order that a student may, unaided by a tutor, have all the adrantages which the book supplies.

MACMILLAN AND CO. London and Cambridge

ELEMENTARY ALGEBRA,

WITH BRIEF NOTICES OF ITS HISTORY.

SECTION VII.

SIMPLE EQUATIONS.

BY ROBERT POTTS, M.A.,

TRINITY COLLEGE, CAMBRIDGE,
HON. LL.D. WILLIAM AND MARY COLLEGE, VA., U.S.

LONDON:
LONGMANS AND CO.,

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