al to - C (at – 6)? i 2 } =l+ 26 altch 2c Hence b altci cal+ch and ... =1. с al-ci bai-ch 5. In the verification, when w?=ab is found, (a +x)(6+x)_2ab + (a+b)ałbł. 2a!!! +a+b (ad +61)? (ał + )2 bi 6. Express the products of the factors in the numerators of the two fractions, as the sums of two squares, and 1+ (a - b)2 (c-d)? lab - 1) (od + 1)? 1 + ax + by 7. Suppose =+1, {(1+a? +6)(1 + x2 + y)} ? ?" then(1+ ax+by):=(1+a? +62)(1+x2 + y2), whence (a -- ~)* +(6-y)? +(ay – b«e)?=0. 8. The quotient is ✓(1-x2)/(1 - y?) – xy. (y+z)+(1-yz)? ; z+y (= + y)2 (1+z)? (1+ya) (1 )(?) 1+x? 1+z? 10. Since x2 + y2 + 2 + 2xyz=1, 2xyz +22=1 -- ? - y’, and adding x?y? to these equals, x?y2 + 2xyz +22=1 – x2 - y2 + x?y?, or (1 —- x2)(1-y?)=(2+xy)2. Similarly (1 - y)(1 - z)=(x + y) and (1 - x)(1 - z)=(y + 2). 1+y=, {{1+ 211+y") }'=x{2+). Similarly v{"+3241 +)}=y(+2), and = { (1+7°/17 + y) (1, '"}!==(x+y). 9 XLI. 1. See Art. 15, p. 16. 2. See Art. 16, p. 17. 5. The former is the greater or the less, according as x2 is greater or less than ab. 6. Here ał+alb is greater or less than abd +61, according as at(a+b) is greater or less than bl(a+b). 7. Since as1 – 62)}+(1 — a?)}<1, then a (1 – 62)* <1 – 6(1 - a2), and a? - a?l? <1 – 28(1 - a)!+62 - a2b, or 26(1-a2)! <(62 -a)+1, also 462 – 4a27% <(62 – a?)? + 2(62 — a?) +1, whence (62 +a%)2 --2(62+a?)+1> 0; .:.02 + a2 - 1>0 and a2 +62> 1. .: a+b> 2(ab)', similarly b+c> 2(60)}, a+c> 2(ac)}; For a +b> 2(ab)), c+d> 2(cd)!; .. xy=(a + b)(c+d)=(atct + bidt)? +(aldı - blck)? ; consequently xy> (atct + bld!)?, and atyt>abc + bidt. 10. If the sum of first and third expressions be taken to be greater than the second, the sum of the squares of the first and third will also be greater than the square of the second. From this inequality may be deduced that 2xʻy? + 2x2.2 + 2y2z2 – 2* - yt – 24 is greater than zero; and as this expression is homogeneous, like expressions may be deduced by assuming the sum of the first and second greater than the third, and the sum of the second and third greater than the first. See Section IV., XIII. Ex. 10, p. 32. 12. If Olfr? – ??) + ry(q? – 62) i (02 + b)(2+ y) <?' Then (a? +62)(x? + y2)> 2ab(t? – y?) + 2ry(a? – 62), and (a? - 2ab +62)ą? +(a? + 2ab +62)y?> 2cyla? - b?); .. (4-3)^x} - 2(a - b)(a+b)n +(a+b)? v?>0. a+ 1 a+b _(a")"=(6")*_2"+y", which is of the forms Section IV., Art. 13, p. 22. 5. ? x+y a"+ * (a")+(8") amit mn (ar)" +(6x) 14 dety not of the required forms, except when m=1, a” (a") + (6) x+y If =r; then a" #$"_?"+1". р 9 art 2+y m 1 1 6. a" – "=a" -(67)"=a* - 2*. an-b=a*- (5*)"=an —**. Here a* - 2* and an - 2* have a common divisor when m and n are odd or even numbers. 7. Let an - bu=(ao) - (*)=x-Y; 1 then a* - bm=(a*)*** - (*)***=2" -Y pen, and the given expressions are reduced to the 1 1 1 1 1 1 = When x=2z, then 22=2-, 1 10. Since x= a + b and 2 = a-US Then a? - 72 q? -7 2(a2 +62) 2(a2 +b) 2(3-m) 22 +22 1+ 2(a2 +62) a-b b (a2 (a - b) a+b) _a+b Sa+bon (a-6) 2-0 ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY. In Twelve Sections, demy 8vo. CONTENTS AND PRICES. . PRICE SECTION I. Of Numbers, pp. 28 3d. SECTION II. Of Money, pp. 52 6d. TION III. Of Weights and Measures, pp. 283d. 3d. 2d. 5d. 5d. 4d. SECTION XI. Proportion, pp. 32... 4d. SECTION XII. Logarithms, pp. 32 6d. 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. Demy 8vo.—Price 128. NEWTON'S PRINCIPIA, ' FIRST BOOK, SECTIONS I., II., III. WITH 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., 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 |