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Full text of "Algebra; an elementary text-book for the higher classes of secondary schools and for colleges"




/*; 



ALGEBRA 



AN 



ELEMENTARY TEXT -BOOK 



Uniform with Part I 



PART II 



Completing the Work and containing an Index 

to both Parts. 

640 pp., post 8vo. 



BY THE SAME AUTHOR 

AN INTRODUCTION TO ALGEBRA 

For the Use of Secondary Schools and Technical 
Colleges. 

Third Edition. Crown 8vo. 

Or may he had in two separate Parts. 

I have kept the fundamental principles of the 
subject well to the front from the very beginning. 
At the same time I have not forgotten, what 
every mathematical (and other) teacher should 
have perpetually in mind, that a general proposi- 
tion is a property of no value to one that has not 
mastered the particulars. The utmost rigour of 
accurate logical deduction has therefore been less 
my aim than a gradual development of algebraic 
ideas. In arranging the exercises I have acted 
on a similar principle of keeping out as far as 
possible questions that have no theoretical or 
practical interest. — Preface. 



AGENTS 

America . . The Macmillan Company 

60 Fifth Avenue, New York 

Australasia . Oxford University Press 

205 Flinders Lane, Melbourne 

Canada . . The Macmillan Co. of Canada, Ltd. 

St. Martin's House, 70 Bond Street, Toronto, 2 

India . . . Macmillan Sc Company, Ltd. 
276 Hornby Road, Bombay 
294 Bow Bazar Street, Calcutta 
North Beach Road, Madras 




ALGEBEA 

AN ELEMENTAEY TEXT-BOOK 

FOR THE 

HIGHER CLASSES OF SECONDARY SCHOOLS 
AND FOR COLLEGES 



BY 



G. CHRYSTAL, M.A., LL.D. 

HONORARY FELLOW OF CORPUS CHRISTI COLLEGE, CAMBRIDGE ; 
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF EDINBURGH 



PART I. 

FIFTH EDITION 




A. & C. BLACK, Ltd. 

4, 5 & 6 SOHO SQUARE, LONDON, W.l. 
1926 



Printed in Great Britain. 



'■ I should rejoice to see . . . morphology introduced into tha 
elements of Algebra. '— Sylvester 



fa. i 






9j 



PtihlUhed July 1S86 

Reprinted, with corrections and additions, 1889 

New impressions 1893 and 1898 

Reprinted, with corrections and additions, 1904 and 1910 

Reprinted in 1920 and 1926 



PKEFACE TO THE FIFTH EDITION. 

In this Edition considerable alterations have been made 
in chapter xii. In particular, the proof of the theorem 
that every integral equation has a root has been amplified, 
and also illustrated by graphical considerations. 

An Appendix has been added dealing with the general 
algebraic solution of Cubic and Biquadratic Equations ; 
with the reducibility of equations generally ; and with the 
possibility of solution by means of square roots. As the 
theorems established have interesting applications in Ele- 
mentary Geometry, it is believed that they may find an 
appropriate place in an Elementary work on Algebra. 

G. CHEYSTAL. 

29th June 1904. 



PEEFACE TO THE SECOND EDITION. 

The comparatively rapid sale of an edition of over two 
thousand copies of this volume has shown that it has, to 
some extent at least, filled a vacant place in our educational 
system. The letters which I have received from many 
parts of the United Kingdom, and from America, containing 
words of encouragement and of useful criticism, have also 
strengthened me in the hope that my labour has not been 



VI PREFACE 

in vain. It would be impossible to name here all the 
friends who have thus favoured me ; and I take this oppor- 
tunity of offering them collectively my warmest thanks. 

The present edition has been thoroughly revised and 
corrected. The first chapter has been somewhat simplified ; 
and, partly owing to experience with my own pupils, partly 
in consequence of some acute criticism sent to me by Mr. 
Levett of Manchester, the chapters on Indices have been 
recast, and, I think, greatly improved. In the verification 
and correction of the results of the exercises I have been 
indebted in a special degree to the Rev. John Wilson, 
Mathematical Tutor in Edinburgh. 

The only addition of any consequence is a sketch of 
Horner's Method, inserted in chapter xv. I had originally 
intended to place this in Part II. ; but, acting on a sugges- 
tion of Mr. Hayward's, I have now added it to Part I. 

To help beginners, I have given, after the table of 
contents, an index of the principal technical terms used in 
the volume. This index will enable the student to turn up 
a passage where the " hard word " is either defined or other- 
wise made plain. 

G. CHKYSTAL. 

Edinburgh, 11th October 1889. 



PREFACE TO THE FIRST EDITION. 

The work on Algebra of which this volume forms the first 
part, is so far elementary that it begins at the beginning of 
the subject. It is not, however, intended for the use of 
absolute beginners. 

The teaching of Algebra in the earlier stages ought to 
consist in a gradual generalisation of Arithmetic ; in other 
words, Algebra ought, in the first instance, to be taught as 
Aritlimetica Universalis in the strictest sense. I suppose 
that the student has gone in this way the length of, say, the 
solution of problems by means of simple or perhaps even 
cpiadratic equations, and that he is more or less familiar 
with the construction of literal formulae, such, for example, 
as that for the amount of a sum of money during a given 
term at simple interest. 

Then it becomes necessary, if Algebra is to be any- 
thing more than a mere bundle of unconnected rules, to 
lay down generally the three fundamental laws of the 
subject, and to proceed deductively — in short, to introduce 
the idea of Algebraic Form, which is the foundation of all 
the modern developments of Algebra and the secret of analy- 
tical geometry, the most beautiful of all its applications. 
Such is the course followed from the beginning in this 
work. 



Vlll PREFACE 

As mathematical education stands at present in this 
country, the first part might be used in the higher classes 
of our secondary schools and in the lower courses of our 
colleges and universities. It will he seen on looking through 
the pages that the only knowledge required outside of 
Algebra proper is familiarity with the definition of the 
trigonometrical functions and a knowledge of their funda- 
mental addition-theorem. 

The first object I have set before me is to develop 
Algebra as a science, and thereby to "increase its usefulness 
as an educational discipline. I have also endeavoured so 
to lay the foundations that nothing shall have to be un- 
learned and as little as possible added when the student 
comes to the higher parts of the subject. The neglect of 
this consideration I have found to be one of the most 
important of the many defects of the English text -books 
hitherto in vogue. Where immediate practical application 
comes in question, I have striven to adapt the matter to 
that end as far as the main general educational purpose 
would allow. I have also endeavoured, so far as possible, 
to give complete information on every subject taken up, or, 
in default of that, to indicate the proper sources ; so that 
the book should serve the student both as a manual and 
as a book of reference. The introduction here and there of 
historical notes is intended partly to serve the purpose just 
mentioned, and partly to familiarise the student with the 
great names of the science, and to open for him a vista 
beyond the boards of an elementary text-book. 

As examples of the special features of this book, I may 
ask the attention of teachers to chapters iv. and v. With 
respect to the opening chapter, which the beginner will 



PREFACE IX 

doubtless find the hardest in the hook, I should mention 
that it was written as a suggestion to the teacher how 
to connect the general laws of Algebra with the former 
experience of the pupil. In writing this chapter I hud to 
remember that I was engaged in writing, not a book on the 
philosophical nature of the first principles of Algebra, but 
the first chapter of a book on their consequences. Another 
peculiarity of the work is the large amount of illustrative 
matter, which I thought necessary to prevent the vagueness 
which dims the learner's vision of pure theory ; this has 
swollen the book to dimensions and corresponding price 
that require some apology. The chapters on the theory of 
the complex variable and on the equivalence of systems of 
equations, the free use of graphical illustrations, and the 
elementary discussion of problems on maxima and minima, 
although new features in an English text- book, stand so 
little in need of apology with the scientific public that I 
offer none. 

The order of the matter, the character of the illustra- 
tions, and the method of exposition generally, are the result 
of some ten years' experience as a university teacher. I 
have adopted now this, now that deviation from accepted 
English usages solely at the dictation of experience. It 
was only after my own ideas had been to a considerable 
extent thus fixed that I did what possibly I ought to have 
done sooner, viz., consulted foreign elementary treatises. 
I then found that wherever there had been free considera- 
tion of the subject the results had been much the same. 
I thus derived moral support, and obtained numberless hints 
on matters of detail, the exact sources of which it would be 
difficult to indicate. I may mention, however, as specimens 



PREFACE 



of the class of treatises referred to, the elementary text- 
books of Baltzer in German and Collin in French. Anion" 
the treatises to which I am indebted in the matter of theory 
and logic, I should mention the works of De Morgan, Pea- 
cock, Lipschitz, and Serret. Many of the exercises have 
been either taken from my own class examination papers 
or constructed expressly to illustrate some theoretical point 
discussed in the text. For the rest I am heavily indebted 
to the examination papers of the various colleges in Cam- 
bridge. I had originally intended to indicate in all cases 
the sources, but soon I found recurrences which rendered 
this difficult, if not impossible. 

The order in which the matter is arranged will doubt- 
less seem strange to many teachers, but a little reflection 
will, I think, convince them that it could easily be justified. 
There is, however, no necessity that, at a first reading, the 
order of the chapters should be exactly adhered to. I think 
that, in a final reading, the order I have given should be 
followed, as it seems to me to be the natural order into 
which the subjects fall after they have been fully com- 
prehended in their relation to the fundamental laws of 
Algebra. 

With respect to the very large number of Exercises, 
I should mention that they have been given for the con- 
venience of the teacher, in order that he might have, year 
by year, in using the book, a sufficient variety to prevent 
mere rote-work on the part of his pupils. I should much 
deprecate the idea that any one pupil is to work all the 
exercises at the first or at any reading. We do too much 
of that kind of work in this country. 

I have to acknowledge personal obligations to Professor 



TREFACE XI 

Tait, to Dr. Thomas Muir, and to my assistant, Mr. R E. 
Allardice, for criticism and suggestions regarding the 
theoretical part of the work ; to these gentlemen and to 
Messrs. Mackay and A. Y. Fraser for proof reading, and 
for much assistance in the tedious work of verifying the 
answers to exercises. In this latter part of the work I 
am also indebted to my pupil, Mr. J. Mackenzie, and to 
my old friend and former tutor, Dr. David Rennet of 
Aberdeen. 

Notwithstanding the kind assistance of my friends and 
the care I have taken myself, there must remain many 
errors both in the text and in the answers to the exercises, 
notification of which either to my publishers or to myself 

will be gratefully received. 

G. CHRYSTAL. 



Edinburgh, 2&h June 1880. 



CONTENTS. 



CHAPTER I. 



FUNDAMENTAL LAWS AND PROCESSES OF ALGEBRA. 

PAGE 

Laws of Association and Commutation for Addition and Subtraction 2-7 

Essentially Negative Quantity in formal Algebra ... 8 

Properties of . . . . . . . . 11 

Laws of Commutation and Association for Multiplication . . 12 
Law of Distribution . . . . . . .13 

Laws of Association, Commutation, and Distribution for Division . 14-19 

Properties of 1 . . . . . . . . 17 

Synoptic Table of the Laws of Algebra .... 20 

Exercises I. ........ 22 

Historical Note ....... 24 



CHAPTER II. 

MONOMIALS — LAWS OF INDICES — DEGREE. 



Laws of Indices 

Theory of Degree, Constants and Variables 

Exercises II. . 



25-29 
30 
31 



CHAPTER III. 

THEORY OF QUOTIENTS — FIRST PRINCIPLES OF THEORY OF NUMBERS. 



Fundamental Properties of Fractions and Fundamental Operations 
therewith ...... 

Exercises III. ...... 

Prime and Composite Integers .... 

Arithmetical G.C.M. . 

Theorems on the Divisibility of Integers 

Remainder and Residue, Periodicity for Given Modulus 

Arithmetical Fractionality .... 

The Resolution of a Composite Number into Prime Factors is unique 
General Theorem regarding G.C.M., and Corollaries . 
The Number of Primes is infinite 
Exercises IV. . 



33-36 
36 
38 
39 
41 
42 
43 
44 
44 
47 
48 



XIV 



CONTENTS 



CHAPTER IV. 



DISTRIBUTION OF PRODUCTS ELEMENTS OF THE THEORY OF RATIONAL 



INTEGRAL FUNCTIONS. 

Generalised Law of Distribution ..... 

Expansion by enumeration of Products ; Classification of the Products 

of a given set of letters into Types ; S and II Notations 
Principle of Substitution 
Theorem regarding Sum of Coefficients 
Exercises V. 

General Theorems regarding the Multiplication of Integral Functions 
Integral Functions of One Variable 

Product of Binomials 

Binomial Theorem .... 

Detached Coefficients 

Addition rule for calculating Binomial Coefficients, with a General 
isation of the same 

x n±y7i as a Product .... 
Exercises VI. . 

Exercises \Il. .... 
Homogeneity .... 

General forms of Homogeneous Integral Functions 

Fundamental Property of a Homogeneous Function 

Law of Homogeneity 

Most general form for an Integral Function 
Symmetry ..... 

Properties of Symmetric and Asymmetric Functions 

Rule of Symmetiy .... 

Most general forms of Symmetric Functions 
Principle of Indeterminate Coefficients 
Table of Identities .... 
Exercises VIII. 



PAGE 

49 

51-54 
54 
55 
56 
57 

59-69 
60 
61 
64 

66 
68 
69 
70 

71-75 
72 
73 
74 
75 

75-79 
76 
77 
78 
79 
81 
83 



CHAPTER V. 

TRANSFORMATION OF THE QUOTIENT OF TWO INTEGRAL FUNCTIONS. 



Algebraic Integrity and Fractionality .... 

Fundamental Theorem regarding Divisibility . 

Ordinary Division-Transformation, Integral Quotient, Remainder 

Binomial Divisor, Quotient, and Remainder . 

Remainder Theorem ...... 

Factorisation by means of Remainder Theorem 
Maximum number of Linear Factors of an Integral Function of a; 
New basis for the Principle of Indeterminate Coefficients 
Continued Division ...... 



85 
86 
86-93 
93 
96 
97 
98 
99 
102 



CONTENTS 



XV 



One Integral Function expressed in powers of another . 

Expansion in the form Ao+Ai(aj — ai) + A%(x — «x)(a; — a2) + A$(x-ai) 

(x-(t2){x-a 3 )+ . . . 
Exercises IX. . 

CHAPTER VI. 

GREATEST COMMON MEASURE AND LEAST COMMON MULTIPLE. 

G.C.M. by Inspection .... 
Ordinary process for Two Functions . 
Alternate destruction of highest and lowest terms 
G.C.M. of any number of Integral Functions . 
General Proposition regarding the Algebraical G.C.M., 
regarding Algebraic Primeness 

L.C.M. 

Exercises X. . 

CHAPTER VII. 



PAOE 

105 

107 
108 



with Corollaries 



112 
113 

117 
119 

119 
122 
124 



FACTORISATION OF INTEGRAL FUNCTIONS. 

Tentative Methods . . . . . . .126 

General Solution for a Quadratic Function of a; . . 128-137 

Introduction of Surd and Imaginary Quantity . . 130-133 

Progression of Real Algebraic Quantity .... 130 

Square Root, Rational and Irrational Quantity . . . 132 

Imaginary Unit ....... 132 

Progression of Purely Imaginary Quantity .... 133 

Complex Quantity ....... 133 

Discrimination of the different cases in the Factorisation of 

ax' 2 + bx + c ....... 134 

Homogeneous Functions of Two Variables . . . .136 

Use of the Principle of Substitution . . . . .136 

Use of Remainder Theorem . . . . . .138 

Factorisation in general impossible . . . . .139 

Exceptional case of ax 2 + 2hxy + by 2 + 2gx + 2fy + c . . .140 

Exercises XI. . . . . . . . .142 

CHAPTER VIII. 

RATIONAL FRACTIONS. 

General Propositions regarding Proper and Improper Fractions 144-147 

Examples of Direct Operations with Rational Fractions . 147-150 

Inverse Method of Partial Fractions .... 151-159 

General Theorem regarding decomposition into Partial Fractions . 151 
Classification of the various species of Partial Fractions, with 

Methods for determining Coefficients . . . .153 

Integral Function expansible in the form 2{a u + ai(x- a)+ . . . + 

a r -. 1 (x-a) 1 — 1 }(x-pY(x-y) t . . . ..... 155 

Exercises XII. ....... 159 



XVI 



CONTENTS 



CHAPTER IX. 



FURTHER APPLICATIONS TO THE THEORY OF NUMBERS. 

TAOE 

Expression of an Integer as an Integral Factorial Series . . 163 

Expression of a Fraction as a Fractional Factorial Series . . 165 

Scales of Arithmetical Notation .... 167-175 

Expression of an Integer in a Scale of given Radix . . . 167 

Arithmetical Calculation in various Scales . . . .169 

Expression of any Fraction as a Radix Fraction . . .170 

Divisibility of a Number and of the Sum of its Digits by r - 1 ; the 

"Nine Test" ....... 174 

Lambert's Theorem . . . . . .176 

Exercises XIII. ....... 177 



CHAPTER X. 

IRRATIONAL FUNCTIONS. 

Interpretation of scW" ....... 180 

Consistency of the Interpretation with the Laws of Indices Examined 182 
Interpretation of a; . . . . . . .185 

Interpretation of x~ m ....... 186 

Examples of Operation with Irrational Forms . . 187-189 

Rationalising Factors ...... 189-198 

Every Integral Function of \Jp, \Jq, \Jr, &c, can be expressed in 
the linear form A + B\Jp + C\/q + 'D\Jr + . . . + ~E\/pq + . . . 
+Y"sfpqr+. . . . . . . . .193 

Rationalisation of any Integral Function of \Jp, \Jq, \Jr, &c. . 195 
Every Rational Function of \/p, \Jq, \/r, &c, can be expressed in 

linear form . . . . . . .196 

General Theory of Rationalisation .... 197-198 

Exercises XIV. ....... 199 

Historical Note' ....... 201 



CHAPTER XL 

ARITHMETICAL THEORY OF SURDS. 

Algebraical and Arithmetical Irrationality 

Classification of Surds ..... 

Independence of Surd Numbers 

Expression of \/{a + \/b) in linear form 

Rational Approximations to the Value of a Surd Number 

Extraction of the Square Root 
Square Root of an Integral Function of x 
Extraction of Roots by means of Indeterminate Coefficients 
Exercises XV. ...... 



203 
204 

205-207 
207 

210-215 
211 
215 
217 
218 



CONTENTS 



XV11 



CHAPTER XII. 

COMPLEX NUMBERS. 

PACK 

Independence of Heal and Imaginary Quantity . . . 221 

Two-folduess of a Complex Number, Argand's Diagram . . 222 

If x + yi = x' + y'i, then x = x', y = y' ..... 224 

Every Rational Function of Complex Numbers is a Complex Number 224-227 
If <(>(x + yi) = X + Yi, then $(x-yi)=X-Yi ; if <p(x + yi) — 0, then 

<f>(x-yi) = ....... 226 

Conjugate Complex Numbers ...... 228 

Moduli. ........ 229 

If x + yi = 0, then | x + yi | = ; and conversely . . . 229 

| <p(x + yi) | = \/{<P(x + yi)<P(x-yi)}; Particular Cases . . . 2l;0 
The Product of Two Integers each the Sum of Two Squares is the 

Sum of Two Squares ...... 230 

Discussion by means of Argand's Diagram . . . 232-236 

Every Complex Number expressible in the form r(cos 6 + i sin d) ; 

Definition of Amplitude ...... 232 

Addition of Complex Numbers, Addition of Vectors . . 233 
\z, + z 2 + . . . + z n \^\z 1 \ + \z 2 \ + . . . +\z H \ . . .234 
The Amplitude of a Product is the Sum of the Amplitudes of the 

Factors ; Demoivre's Theorem ..... 235 

Root Extraction leads to nothing more general than Complex 

Quantity 236-244 

Expression of ij{x + yi) as a Complex Number . . . 237 

Expression of Mx + yi) as a Complex Number . . . 238 

Every Complex Number has n nth roots and no more . . 240 

Properties of the nth roots of ±1 ..... 240 

Resolution of x"± A into Factors ..... 243 
Every Integral Equation has at least one root ; Every Integral 
Equation of the ?ith degree has n roots and no more ; Every 
Integral Function of the nth degree can be uniquely resolved 
into n Linear Factors ..... 244-250 

Upper and Lower Limits for the Roots of an Equation . . 247 

Continuity of an Integral Function of z . . . 248 

Equimodular and Gradient Curves of/(r) .... 248 

Argand's Progression towards a Root ..... 249 

Exercises XVI. 251 

Historical Note ....... 253 



CHAPTER XIII. 

RATIO AND PROPORTION. 

Definition of Ratio and Proportion in the abstract 
Propositions regarding Proportion 
Examples ..... 

Exercises XVII. .... 

VOL. I 



255 
257-264 
264-266 

267 

b 



XV111 



CONTENTS 



Ratio and Proportion of Concrete Quantities 

Definition of Concrete Ratio 

Difficulty in the case of Incommensurables 

Euclidian Theory of Proportion 
Variation .... 

Independent and Dependent Variables 

Simplest Cases of Functional Dependence 

Other Simple Cases . 

Propositions regarding "Variation" 
Exercises XVIII. 



CHAPTER XIV. 



PAOE 

268-27.3 
269 
270 
272 

273-279 
273 
274 
275 
276 
279 



ON CONDITIONAL EQUATIONS IN GENERAL. 

General Notion of an Analytic Function 
Conditional Equation contrasted with an Identical Equation 
Known and Unknown, Constant and Variable Quantities 
Algebraical and Transcendental Equations ; Classification of Integral 

Equations ..... 
Meaning of a Solution of a System of Equations 
Propositions regarding Determinateness of Solution 
Multiplicity of Determinate Solutions 
Definition of Equivalent Systems ; Reversible and Irreversible 

Derivations ..... 

Transformation by Addition and Transposition of Terms 
Multiplication by a Factor .... 
Division by a Factor not a Legitimate Derivation 
Every Rational Equation can be Integralised . 
Derivation by raising both sides to the same Power . 
Every Algebraical Equation can be Integralised ; Equivalence of the 

Systems, Pj = 0, P 2 = 0, . . . P„ = 0, and L, P, + L 2 P 2 + . . . + L n P„ = 0, 

P 2 = 0, . . . P„ = .... 

Examination of the Systems P = Q, R=S; PR=QS, R = S 
On Elimination .... 

Examples of Integralisation and Rationalisation 

Examples of Transformation 

Examples of Elimination 

Exercises XIX. .... 

Exercises XX. ..... 

Exercises XXI. .... 

CHAPTER XV. 

VARIATION OP A FUNCTION. 

Graph of a Function of one Variable . 
Solution of an Equation by means of a Graph 
Discontinuity in a Function and in its Graph 



281 
282- 

283 

283 

284 

286-288 

289 

289 
291 
292 
293 
294 
295 



296 
297 
298 
299-301 
302 
304 
305 
306 
308 



310 
313 
315 



CONTENTS 



XIX 



326 



PACK 

Limiting Cases of Algebraic Operation . 318-322 

Definition of the Increment of a Function .... 322 
Continuity of the Sum and of the Product of Continuous Functions . 323 
Continuity of any Integral Function ..... 324 
Continuity of the Quotient of Two Continuous Functions; Exception 324 
General Proposition regarding Continuous Functions . . 325 

Number of Roots of an Equation between given limits . . 326 

An Integral Function can change sign only by passing through the 

value ; Corresponding Theorem for any Rational Function 
Sign of the Value of an Integral Function for very small and for very 

large values of its Variable ; Conclusions regarding the Number 

of Roots ....... 

Propositions regarding Maxima and Minima . 

Continuity and Graphical Representation of f{x, y) ; Graphic Surface 

Contour Lines ...... 

f(x, y) = Q represents a Plane Curve .... 

Graphical Representation of a Function of a Single Complex Variabl 
Horner's Method for approximating to the Real Roots of ar 

Equation .... 
Multiplication of Roots by a Constant 
Increase of Roots by a Constant 
Approximate Value of Small Root 
Horner's Process 
Example 

Extraction of square, cube, fourth, . . 
Exercises XXII. 



roots by Horner's Method 



328 
330 

331 
334 

335 

338-346- 
338 
339 
340 
341 

342-345 
346 
347 



CHAPTER XVI. 

EQUATIONS AND FUNCTIONS OF FIRST DEGREE. 

Linear Equations in One Variable .... 

Exercises XXIII. ....... 

Linear Equations in Two Variables — Single Equation, One-fold Infin 
ity of Solutions ; System of Two, Various Methods of Solution 
System of Three, Condition of Consistency 

Exercises XXIV. ....... 

Linear Equations in Three Variables — Single Equation, Two-fold In- 
finity of Solutions ; System of Two, One-fold Infinity of Solu 
tions, Homogeneous System ; System of Three, in general Deter- 
minate, Homogeneous System, Various Methods of Solution 
S3'stems of more than Three .... 

General Theory of a Linear System . 

General Solution by means of Determinants . 

Exercises XXV. ....... 

Examples of Equations solved by means of Linear Equations . 

Exercises XXVI. ....... 



349, 


350 




351 


352- 


-364 




364 



365-372 
373 

374-376 
376 

379-383 
383 



XX 



CONTENTS 



PAGE 

Graph of ax + b ....... 385 

Graphical Discussion of the cases 6 = ; a = ; a = 0, b = . ». 388 

Contour Lines of ax + by + c ...... 389 

Illustration of the Solution of a System of Two Linear Equations . 390 
Cases where the Solution is Infinite or Indeterminate discussed 

graphically ....... 391 

Exceptional Systems of Three Equations in Two Variables . . 393 

Exercises XXVII. ...... 394 



CHAPTER XVII. 

EQUATIONS OF THE SECOND DEGREE 
ax 2 + bx + c = has in general just two roots 
Particular Cases .... 

General Case, various Methods of Solution 
Discrimination of the Roots 
Exercises XXVIII. .... 
Equations reducible to Quadratics, by Factorisation, by Integralisa 

tion, by Rationalisation 
Exercises XXIX., XXX. 
Exercises XXXI. 

Reduction by change of Variable ; Reciprocal Equations 
Rationalisation by introducing Auxiliary Variables 
Exercises XXXII. .... 

Systems with more than One Variable which can be solved by means 
of Quadratics .... 

General System of Order 1x2 

General System of Order 2x2; Exceptional Cases 

Homogeneous Systems 

Symmetrical Systems 

Miscellaneous Examples 
Exercises XXXIII. 
Exercises XXXIV. 
Exercises XXXV. 



396 
397 
398 
400 

401 

402-406 
406 
407 

408-413 
413 
413 



414-427 
415 
416 
418 
420 
425 
427 
429 
430 



CHAPTER XVIII. 

GENERAL THEORY OP INTEGRAL FUNCTIONS. 

Relations between Coefficients and Roots .... 431 
Symmetric Functions of the Roots of a Quadratic . . .432 
Newton's Theorem regarding Sums of Powers of the Roots of any 

Equation ........ 436 

Symmetric Functions of the Roots of any Equation . . 438 
Any Symmetric Function expressible in terms of certain elementary 

Symmetric Functions ...... 440 



CONTENTS 



XXI 



Exercises XXXYI. 

Special Properties of Quadratic Functions 

Discrimination of Roots, Table of Results 

Generalisation of some of the Results 

Condition that Two Quadratics have Two Roots in common 

Lagrange's Interpolation Formula . 

Condition that Two Quadratics have One Root in common 

Exercises XXXVII. . 

Variation of a Quadratic Function for real values of its Variable ; 

Analytical and Graphical Discussion of Three Fundamental Cases, 

Maxima and Minima 



PAGE 

415 

447-453 

447 
449 
450 
451 
452 
453 



458 
461 
462 
463 



Examples of Maxima and Minima Problems . 

General Method of finding Turning Values by means of Equal Roots . 

Example, y = x 3 - 9x ,2 + 24a; + 3 . ..... 

Example, y= (x 2 - Sx + 15)/* ...... 

General Discussion of y = (ax 2 + bx + c)j(a'x 2 + b'x + c'), with Graphs of 

certain Particular Cases ..... 464-467 

Finding of Turning Values by Examination of the Increment . 468 

Exercises XXXVIII .469 



CHAPTER XIX. 

SOLUTION OF PROBLEMS BY MEANS OP EQUATIONS. 

Choice of Variables ; Interpretation of the Solution . 
Examples ....••• 

Exercises XXXIX. ....•• 

CHAPTER XX. 

ARITHMETIC, GEOMETRIC, AND ALLIED SERIES. 

Definition of a Series ; Meaning of Summation ; General Term 
Integral Series . . . . ■ 

Arithmetic Progression 

Sums of the Powers of the Natural Numbers 

Sum of any Integral Series . 

Arithmetic-Geometric Series, including the Simple Geometric Series 
as a Particular Case 

Convergency and Divergency of Geometric Series 
Properties of Quantities in A. P., in G.P., or in H.P. 

Expression of Arithmetic Series by Two Variables 

Insertion of Arithmetic Means 

Arithmetic Mean of n given quantities 

Expression of Geometric Series by Two Variables 

Insertion of Geometric Means 

Geometric Mean of n given quantities 



471 

472-476 

476 



480 
482-488 

482 
484-487 

487 

489-494 
495 

496-502 
496 
497 
497 
499 
499 
500 



XX11 



CONTENTS 



Definition of Harmonic Series . 

Expression in terms of Two Variables 

Insertion of Harmonic Means 

Harmonic mean of n given quantities 

Propositions regarding A.M., G.M., and H. M. 
Exercises XL. ..... 

Exercises XLI. ..... 

Exercises XLII. .... 

CHAPTEE XXL 

LOGARITHMS. 

Discussion of a x as a Continuous Function of as 
Definition of Logarithmic Function 
Fundamental Properties of Logarithms 

Computation and Tabulation of Logarithms 
Mantissa and Characteristic . 
Advantages of Base 10 
Direct Solution of an Exponential Equation 
Calculation of Logarithms by inserting Geometric Means 
Alteration of Base .... 

Use of Logarithms in Arithmetical Calculation 

Interpolation by First Differences 

Exercises XLIII. .... 

Historical Note .... 

CHAPTER XXII. 



PAGE 
500 

501 
501 
501 
501 

502 
505 

507 



509 
511 
512 

513-519 
514 
515 

516 
517 
519 

519-523 



524 
527 
529 



THEORY OP INTEREST AND ANNUITIES. 

Simple Interest, Amount, Present Value, Discount . . 531-532 

Compound Interest, Conversion -Period, Amount, Present Value, 

Discount, Nominal and Effective Rate . . . 533-535 

Annuities Certain, Accumulation of Forborne Annuity, Purchase Price 
of Annuity, Terminable or Perpetual, Deferred or Undeferred, 
Number of Years' Purchase .... 536-540 

Exercises XLIV. ....... 540 



APPENDIX. 

Commensurable Roots, Reducibility of Equations 

Equations Soluble by Square Roots 

Cubic ...... 

Biquadratic, Resolvents of Lagrange and Descartes 
Possibility of Elementary Geometric Construction 
Exercises XLV. .... 



RESULTS OF EXERCISES. 



543-546 
546-548 
549 
550 
551 
553 

555 



INDEX OF PRINCIPAL TECHNICAL TEEMS 
USED IN PAET I. 



Addition Eule for binomial coefficients, 
66 

Afline of a complex number, 223 

Algebraic sum, 10 

Algebraical function, ordinary, 281 

Alternating function, 77 

Amount, 532 

Amplitude of a complex number, 236 

Annuity, certain, contingent, termin- 
able, perpetual, immediate, deterred, 
forborne, number of years' purchase, 
536 et seq. 

Antecedent of a ratio, 255 

Antilogaritlim, 518 

Argand-diagram, 222 

Argand's progression, 249 

Argument, 524 

Arithmetic means and arithmetic mean, 
497 

Arithmetic progression, 482 

Arithmetico-geometric series, 491 

Association, 3, 12 

Auxiliary variables, 380 

Base of an exponential or logarithm, 511 
Binomial theorem, 62 

Characteristic, 514 

Coefficient, 30 

Commensurable, 203 

Common Measure and Greatest Common 

Measure (arithmetical sense), 38, 39 ; 

algebraical sense, 111 
Commutation, 4, 12 
Complex number or quantity, 133, 221 
Conjugate complex numbers, 228 
Consequent of a ratio, 2f>5 
Consistent system of equations, 288 
Constant, 30 
Continued division, 102 
Continued proportion, 256 
Continuity of a function, 317, 323, 324, 

336 
Contour lines of a function, 333 
Couvergency of a series, 493 
Conversion-period (for interest), 533 

Degree, 30, 58 
Degree of an equation, 284 
Derivation of equations, 290 
Detached coefficients. 63 91 



Determiuateness of a system of equa- 
tions, 286 
Differences (first), 521 
Discontinuity of a function, 317 
Discount, 532 
Discriminant, 134, 141 
Distribution, 13, 49 
Divergency of a series, 493 
Divisibility (algebraical sense), 85 
Divisibility (arithmetical sense), 38 

Efjminant (or resultant), 415. 430 

Elimination, 298 

Equation, conditional, 282 

Equation and equality (identical), 22 

Equimodular curves, 248 

Equiradical surds, 204 

Equivalence of systems of equations, 

289 
Exponent, 25 
Exponential function, 509 
Exponential notation (exp a ), 511 
Extraneous solutions, 294 
Extremes and means of a proportion, 

256 

Factor (arithmetical sense), 38 
Fractional (algebraical sense), 30, 85 
Fractional (arithmetical sense), 43 
Freehold, value of, 539 
Function, analytical, 281 
Function, rational, integral, algebraical, 
58 

Geometric means and geometric mean, 

499 
Geometric series, 489 
Gradient curves, 248 
Graph of a function, 312, 333 
Graphical solution of equations, 313 
Greatest Common Measure (algebraical 

sense), 111 

Harmonic means and harmonic mean, 

501 
Harmonic series, 500 
Homogeneity, 71 
Homogeneous system of equations, 418 

Identity, identical, 22 
Imaginary unit and imaginary quantity, 
132 



xxm 



XXIV 



INDEX 



Increment of a function, 322 
Indeterminate coefficients, 79, 100 
Indeterminate forms, 318, 319, 320 
Indeterminateness of a system of equa- 
tions, 286 
Index, 25 

Infinite value of a function, 315 
Infinitely great, 318 
Infinitely small, 318 
Integral (algebraical sense), 25, 85 
Integral (arithmetical sense), 37 
Integral function, 58 
Integral quotient (algebraical sense), 86 
Integral series, 484 
Integralisation of equations, 296 
Integro-geometric series, 492 
Interest, simple and compound, 531, 533 
Interpolation, 524 
Inverse, 5, 14 

Irrationality (algebraic), 203, 240 
Irrationality (arithmetical), 203 
Irreducible case of cubic, 549 
Irreducible equation, 545 
Irreversible derivation, 290 

Laws of Algebra, fundamental, 20 
Least Common Multiple (algebraical 

sense), 122 
Limiting cases, 318 
Linear (algebraic sense), 138 
Linear irrational form, 193 
Logarithmic function, 511 

Manifoldness, 496 
Mantissa, 514 

Maxima values of a function, 330 
Mean proportionals, 256 
Minima values of a function, 330 
Modulus (arithmetical sense), 43 
Modulus of a complex number, 229 
Modulus of system of logarithms, 519 
Monomial function, 30 

Negative quantity, 9 
Nine-test, 175 

Operand and operator, 4 

Order of a symmetric function, 439 

Partial fractions, 151 

Pascal's triangle, 67 

Periodicity of integers, 43 

II-notation, 53 

l'rime (arithmetical sense), 38 

Primeness (algebraical), 120 

Principal, 532 

Principal value of a root, 182 



Proper fraction (algebraical sense), 86, 

144 
Proportion, 256, 269 
Proportional parts, 526 

Quantity, ordinary algebraic, 130 

Rate of interest, nominal and effective, 

535 
Ratio, 255, 268 
Rational (algebraic sense), 144 
Rational fraction, 144 
Rationalisation of equations, 296 
Rationalising factor, 190, 197 
Reciprocal equations, 410 
Reducibility of an equation, 545 
Remainder (algebraical sense), 86 
Remainder (arithmetical sense), 42 
Remainder theorem, 93 
Residue (arithmetical sense), 42 
Resolvent of a biquadratic, 550 
Resultant equation, 415 
Reversible derivation, 290 
Root of an equation, 284 
Roots of a function, 313 

Scales of notation, 168 

Series, 480 

Similar surds, 204 

S-notation, 53 

Solution of an equation, formal and 

approximate numerical, 284 
Substitution, principle of, 18 
Sum (finite) of a series, 481 
Sum (to infinity) of a series, 493 
Surd number, 132 
Surd number, monomial, binomial, &c, 

203, 204 
Symmetrical system of equations, 420 
Symmetry, 75 

Term, 30, 58 

Transcendental function, 282 
Turning values of a function, 330 
Type (of a product), 52 

Unity (algebraical sense of), 17 

Variable, 30 

Variable, independent and dependent, 

273 
Variation of a function, 311 
" Variation " (old sense of), 273, 275 

Weight of a symmetric function, 434 
Zero (algebraical sense of), 11, 14 



CHAPTEK I. 

The Fundamental Laws and Processes of Algebra 
as exhibited in ordinary Arithmetic. 

§1.1 The student is already familiar with the distinction 
between abstract and concrete arithmetic. The former is con- 
cerned with those laws of, and operations with, numbers that are 
independent of the things numbered ; the latter is taken up 
with applications of the former to the numeration of various 
classes of things. 

Confining ourselves for the present to abstract arithmetic, 
let us consider the following series of equalities : — 
2623 1023 2 623x3 + 1023x61 

ITT* - 3~~~~ 61x3 

70272 00i 

= = 384. 

183 

The first step is merely the assertion of the equivalence of 
two different sets of operations with the same numbers. The 
second and third steps, though doubtless based on certain simple 
laws from which also the first is a consequence, nevertheless 
require for their direct execution the application of certain rules, 
of a kind to which the name arithmetical is appropriated. 

We have thus shadowed forth two great branches of the higher 
mathematics: — one, algebra, strictly so called, that is, the theory of 
operation with numbers, or, more generally speaking, with quanti- 
ties ; the other, the higher arithmetic, or theory of numbers. These 
two science? are identical as to their fundamental laws, but differ 
widely in their derived processes. As is usual in elementary 
text-books, the elements of both will be treated in this work. 
VOL. I B 



2 REPRESENTATIVE GROUPS chap. 

§ 2.] Ordinary algebra is simply the general theory of those 
operations with quantity of which the operations of ordinary 
abstract arithmetic are a particular case. 

The fundamental laws of this algebra are therefore to be 
sought for in ordinary arithmetic. 

However various and complex the operations of arithmetic 
may seem, it appears on consideration that they are merely the 
result of the application of a very small number of fundamental 
principles. To make this plain we return for a little to the very 
elements of arithmetic. 

ADDITION, 

AND THE GENERAL LAWS CONNECTED THEREWITH. 

§ 3.] When a group of things, no matter how unlike, is con- 
sidered merely with reference to the number of individuals it 
contains, it may be represented by another group, the individuals 
of which are all alike, provided only there be as many individuals 
in the representative as in the original group. The members of 
our representative group may be merely marks (l's say) on a 
piece of paper. The process of counting a group may therefore 
be conceived as the successive placing of l's in our representa- 
tive group, until we have as many l's as there are individuals 
in the group to be numbered. This process of adding a 1 is 
represented by writing + 1. We may thus have 

+ 1, +1 + 1, +1 + 1 + 1, +1 + 1 + 1 + 1, &c, 

as representative groups or "numbers." As the student is of 
course aware, these symbols in ordinary arithmetic are abbreviated 

int0 1, 2, 3, 4,&c. 

Hence using the symbol " = " to stand for " the same as," or 
"replaceable by," or "equal to," we have, as definitions of 1, 2, 
3, 4, &c, 

1= +1, 

2= +1 + 1, 

3- +1 + 1 + 1, 

4= + 1 + 1 + 1 + 1, &c. 



I ASSOCIATION IN ADDITION 3 

And there is a further arrangement for abridging the repre- 
sentation of large numbers, which the student is familiar with as 
the decimal notation. With numerical notation we are not 
further concerned at present, but there is a view of the above 
equalities which is important. After the group +1 + 1 + 1 has 
been finished it may be viewed as representing a single idea to 
the mind, viz. the number " three." In other words, we may- 
look at +1 + 1 + 1 as a series of successive additions, or we 
may think of it as a whole. When it is necessary for any 
purpose to emphasise the latter view, we enclose +1 + 1 + 1 in 
a bracket, thus ( + 1 + 1 + 1) ; and it will be observed that pre- 
cisely the same result is attained by writing the symbol 3 in 
place of + 1 + 1 + 1, for in the symbol 3 all trace of the for- 
mation of the number by successive addition is lost. We might 
therefore understand the equality or equation 

3= +1 + 1 + 1 
to mean ( + 1 + 1 + 1)= +1 + 1 + 1, 

and then the equation is a case of the algebraical Law of 
Association. 

The full meaning of this law will be best understood by con- 
sidering the case of two groups of individuals, say one of three 
and another of four. If we wish to find the number of a group 
made up by combining the two, we may adopt the child's process 
of counting through them in succession, thus, 

+ 1 + 1 + 1 | +1 + 1 + 1 + 1 = 7. 
But by the law of association we may write for +1 + 1 + 1 

(+1 + 1 + 1), 
and for +1 + 1 + 1 + 1 

( + 1 + 1 + 1 + 1), 

and we have + (+ 1 + 1 + 1) + (+ 1 + 1 + 1 +1) = 7, 

or +3 + 4 = 7. 

It will be observed that we have added a + in each case be- 
fore the bracket, and it may be asked how this is justified. The 
answer is simply that setting down a representative group of 
three individuals is an operation of exactly the same nature as 



4 COMMUTATION IN ADDITION chap. 

setting down a group of one. The law of association for addition 
worded in this way for the simple case before us would be this : 
To set down a representative group of three individuals is the same 
as to set down in succession three representative individuals. 

The principle of association may be carried further. The 
representative group +3 + 4 may itself enter either as a whole 
or by its parts into some further enumeration : thus, 

+ 6 + ( + 3 + 4)= +6 + 3 + 4 
is an example of the law of association which the student will 
have no difficulty in interpreting in the manner already indi- 
cated. The ultimate proof of the equality may be regarded as 
resting on a decomposition of all the symbols into a succession 
of units. There is, of course, no limit to the complication of 
associations. Thus we have 

+ [( + 9 + 8) + { + 6 + ( + 5 + 3)}] + { + 6 + ( + 3 + 5)} 
" = + ( + 9 + 8) + { + 6 + ( + 5 + 3)} + 6 + ( + 3 + 5), 
= +9 + 8 + 6 + ( + 5 + 3) + 6 + 3 + 5, 
= +9 + 8 + 6 + 5 + 3 + 6 + 3 + 5, 

each single removal of a bracket being an assertion of the law 
of association. The student will remark the use of brackets of 
different forms to indicate clearly the different associations. 

§ 4.] It follows from the definitions 

3= + 1 + 1 + 1, 2 = +1 + 1, 
that +3 + 2= +2 + 3; 

and "by a similar proof we might show that 

+ 3 + 4 + 6= +3 + 6 + 4= +4 + 3 + 6, &c. ; 
in other words, the order in which a series of additions is arranged 
is indifferent. 

This is the algebraical Law of Commutation, and it will 
be observed that its application is unrestricted in arithmetical 
operations where additions alone are concerned. The statement 
of this law at once suggests a principle of great importance in 
algebra, namely, the attachment of the " symbol of operation " or 
" operator " to the number, or, more generally speaking, " subject " 
or " operand," on which it acts. Tims in the above equations 



I SUBTEACTION DEFINED 

the + before the 3 is supposed to accompany the 3 when it 
is transferred from one part of the chain of additions to another. 
The operands in +3, +4, and + 6 are already complex ; and 
it may be shown by a further application of the reasoning used 
in the beginning of this article that the operand may be complex 
to any degree without interfering with the validity of the com- 
mutative law ; for example, 

+ {+3 + ( + 2 + 3)} + ( + 6 + 8) 
= + ( + G + 8) + { + 3 + ( + 2 + 3)} , 

of which a proof might also be given by first dissociating, then 
commutating the individual terms +6, +8, '+ 3, &c, and then 
reassociating. 

The Laiv of Commutation, thus suggested by arithmetical considera- 
tions, is noiv laid down as a general law of algebra ; and forms a 
part of the definition of the algebraic symbol + .* 

SUBTRACTION. 

§ 5.] For algebraical purposes the most convenient course is 
to define subtraction as the inverse of addition ; or, as is more 
convenient for elementary exposition, we lay down that addition 
and subtraction are inverse to each other, t By this we mean 
that, whatever the interpretation of the operation + b may be, 
the operation - b annuls the effect of + b ; and vice versa. 

Thus, - is defined relatively to + by the equation 

+ a- b + b= + a (1), 

or + a + b-b- +a (2). 

These might also be written f f 



* See the general remarks in § 27. 

t Here we virtually assume that if x + a = y + a, then x = y. See Hankel, 
Vorlcsungen il. d. Complcxen Zahlen (Leipzig, 1867), p. 19. 

tt It may conduce to clearness in following some of the above discussions to 
remember that the primary view of a chain of operations written in any order 
is that the operations are to be carried out successively from left to right ; 
for example, if we think merely of the last addition, +2 + 3 + 5 + 6 in more fully 
expressive symbols means + (+2 + 3 + 5) + 6, that is, + 10 + 6; +a+b+c means 
+ ( + a + b) + c; +a~b + c means +(+a-b) + c; and so on. We may here re- 
mind the reader that, in ordinary practice, when + occurs before the first member 
of a chain of additions and subtractions, it is usually omitted for brevity. 



6 LAWS OF COMMUTATION AND ASSOCIATION chap. 

+ ( + a-b) + b= + a (T), 

+ ( + a + b)-b = + a (2'). 

From a quantitative point of view we might put the matter 
thus : the question, What is the result of subtracting b from a 1 
is regarded as the same as the question, What must be added to 
+ b to produce + a 1 and the quantity which is the answer to 
this question is symbolised by + a-b. Starting with the defini- 
tion involved in (1) and (2), and putting no restriction upon the 
operands a and b, or, what is the same thing from a quantitative 
point of view, assuming that the quantity + a-b always exists, we 
may show that the laws of commutation and association hold for 
chains of operations whose successive links are additions and sub- 
tractions. We, of course, assume the commutative law for addi- 
tion, having already laid it down as one of our fundamental laws. 
§ 6.] Since + a- c + c= + a by the definition of the mutual 
relation between addition and subtraction, we have 

a+b-c=a-c+c+b-c; 
= a - c + b + c - c, 

by law of commutation for addition ; 
= a-c + b (1), 

by definition of subtraction. 
Also a-b-c = a-c + c-b-c, 

by definition ; 
= a-c-b + c-c, 

by case (1) ; 
= a~c-b (2). 

by definition 

Equations (1) and (2) may be regarded as extending the law 
of commutation to the sign - .* We can now state this law 

fully as follows : — 

±a±b= ±b±a; 



* It might be objected here that it has not been shown that - c may come 
into the first place in the chain of operations. The answer to this would be 
that +a-c-b may either be a complete chain in itself or merely the latter 
part of a longer chain, say p + a - c - b. In the second case our proof would 
show that p+a-c-1>=p-c + a- b J and the nature of algebraic generality 



i FOR ADDITION AND SUBTRACTION 7 

or, in words, In any chain of additions and subtractions the different 
members may be written in any order, each with its proper sign 
attached. 

Here the full significance of the attachment of the operator 
to the operand appears. Thus in the following instance the 
quantities change places, carrying their signs of operation with 
them in accordance with the commutative law : — 

+ 3-2 + 1-1= +3 + 1-1-2, 
= +3-1 + 1-2, 
= -2-1 + 1+3. 

§ 7.] By the definition of the mutual relation between addi- 
tion and subtraction, we have 

a + ( + b-c)= + a + ( + b-c) + c-c, 

=a+b-c (1). 

Again, by the definition, 

p + b-c + c-b =p + b-b, 
=p. 
Hence a-( + b-c) = a-( + b-c) + b-c + c-b; 

= a-( + b-c) + ( + b-c) + c-b, 

by case (1) ; 
= a + c-b, 

by the definition ; 
= a-b+c (2), 

by the law of commutation already 
established. 
§ 8.] The results in last paragraph, taken along with those of 
§ 3 above, may be looked upon as establishing the law of associa- 
tion for addition and subtraction. This law may be symbolised 
as follows : — 

±(±a±b±c± &c.) = ± ( ± a) ± ( ± b) ± ( ± c) ± &c, 
with the following law of signs, 

+ ( + «)= + a, -( + «)= - a, 
+ ( - a) = - a, -(-«)=+ a. 

requires that +a-c-b should not have any property in composition which it 
has not per se. As to all questions of this kind see § 27. 



8 NON-ARITHMETICAL CASES chap. 

The same may be stated in words as follows : — If any number of 
quantities affected with the signs + or - occur in a bracket, the bracket 
may be removed, all the signs remaining the same if + precede the 
hracket, each + being changed into - and each - into + if -precede 
the bracket. 

In the above symbolical statement double signs ( ± ) have 
been used for compactness. The student will observe that with 
three letters 2x2x2x2, that is, 16, cases are included. Thus 
the law gives 

+ ( + a + b + c)= + a + b + c, 
+ (-a + b + c)= - a + b + c, 
-(-a + b + c)= + a-b - c, &c. 

§ 9.] It will not have escaped the student that, in the as- 
sumption that + a - b is a quantity that always exists, we have 
already transcended the limits of ordinary arithmetic. He will 
therefore be the less surprised to find that many of the cases 
included under the laws of commutation and association exhibit 
operations that are not intelligible in the ordinary arithmetical 
sense. 

If a = 3 and b = 2, 

then by the law of association and by the definition of sub- 
traction 

+ 3-2= +1 + 2-2, 
= +1, 
in accordance with ordinary arithmetical notions. 
On the other hand, if 

a = 2 and b = 3, 

then by the laws of commutation and association and by the 
definition of subtraction 

+ 2-3= +2-( + 2+l), 
= +2-2-1, 
= - 1 + 2 - 2, 
= - 1. 

Here we have a question asked to which there is no ordinary 
arithmetical answer, and an answer arrived at which has no 
meaning in ordinary arithmetic. 



I ESSENTIALLY NEGATIVE QUANTITIES 9 

Such an operation as + 2 - 3, or its algebraical equivalent, 
- 1, is to be expected as soon as we begin to reason about 
operations according to general laws without regard to the appli- 
cation or interpretation of the results to be arrived at. It must 
be remembered that the result of a series of operations may be 
looked on either as an end in itself, say the number of in- 
dividuals in a group, or it may be looked upon merely as an 
operand destined to take place in further operations. In the 
latter case, if additions and subtractions be in question, it must 
have either the + or the - sign, and either is as likely to occur 
and is as reasonably to be expected as the other. Thus, as the 
results of any partial operation, + 1 and - 1 mean respectively 
1 to be added and 1 to be subtracted. 

The fact that the operations may end in results that have no 
direct interpretation as ordinary arithmetical quantities need not 
disturb the student. He must remember that algebra is the 
general theory of those operations with quantity of which ordinary 
arithmetical operations are particular cases. He may be assured 
from the way in which the general laws of algebra are established 
that, when algebraical results admit of arithmetical meanings, 
these results will be arithmetically right, even when some of the 
steps by Avhich they have been arrived at may not be arithmetic- 
ally interpretable. On the other hand, when the end results 
are not arithmetically intelligible, it is merely in the first instance 
a question of the consistency of algebra with itself. As to what 
the application of such purely algebraical results may be, that 
is simply a question of the various uses of algebra • some of these 
will be indicated in the course of this treatise, and others will 
be met with in abundance by the student in the course of his 
mathematical studies. It will be sufficient at this stage to give 
one example of the advantage that the introduction of algebraic 
generality gives in arithmetical operations. + a - b asks the ques- 
tion what must be added to + b to give + a. If a = 3 and b — 2, 
the answer is 1 ; if a = 2 and b = 3, then, arithmetically speaking, 
there is no answer, because 3 is already greater than 2. But if 
we regard + a - b as asking what must be added to or subtracted 



10 REDUCTION OF AN ALGEBRAICAL SUM OHAP. 

from + b to get + a, then the evaluation of + a-b in any case 
by the laws of algebra will give a result whose sign will indicate 
whether addition or subtraction must be resorted to, and to what 
extent ; for example, if a = 3 and b = 2, the result is + 1, which 
means that 1 must be added ; if a = 2 and 5 = 3, the result is 
- 1, which means that 1 must be subtracted. 

§ 10.] The application of the commutative and associative 
laws for addition and subtraction leads us to a useful practical 
rule for reducing to its simplest value an expression consisting 
of a chain of additions and subtractions. 
We have, for example, 

+a-b+c+d-e-f+g 

= + a + c + d + g-b- e -/, 

= + (a + c + d + g) - (b + e + /), 

= + { + ( a + c + d + g)-(b + e+f)} (1), 

= - { + (b + e+f)-(a + c + d + g)} (2). 

If a+c+d+g be numerically greater than b + e+f, (1) is 
the most convenient form ; if a + c + d + g be numerically less 
than b + e + f, (2) is the most convenient. The two taken to- 
gether lead to the following rule for evaluating a chain of 
additions and subtractions : — * 

Add all the quantities affected ivith the sign + , also all those 
affected with the sign - ,• take the difference of the two sums and affix 
t/ie sign of the greater. 

Numerical example : — 

+3-5+6+8-9-10+2 

= + (3 + 6 + 8 + 2)-(5 + 9 + 10), 

= +19-24, 

= -(24-19)= -5. 

§ 11.] The special case +a-a deserves close attention. A 
special symbol, namely 0, is used to denote it. The operational 
definition of is therefore given by the equations 

+ a - a= - a + a = 0. 

In accordance with this Ave have, of course, the results, 

* Such a chain is usually spoken of as an "algebraical sum." 



I PROPERTIES OF H 

b + = b = b - 0, 
and + = - 0, 

as the student may prove by applying the laws of commutation 
and association along with the definition of 0. 

§ 12.] It will be observed that 0, as operationally defined, is 
to this extent indefinite that the a used in the above definition 
may have any value whatever. 

It remains to justify the use of the of the ordinary 
numerical notation in the new meaning. This is at once done 
when we notice that in a purely quantitative sense stands for 
the limit of the difference of two quantities that have been made 
to differ by as little as we please. 

Thus, if we consider a + x and a, 

+ (a + x) — a = + a-a + x = x. 
If we now cause the x to become smaller than any assignable 
quantity, the above equation becomes an assertion of the identity 
of the two meanings of 0. 



MULTIPLICATION. 

§ 13.] The primary definition of multiplication is as an ab- 
breviation of addition. Thus + a + a, + a + a + a, + a + a + a + a, 
&c, are abbreviated into + a x 2, + a x 3, + a x 4, &c. ; and, in 
accordance with this notation, + a is also represented by + a x 1. 
a x 2 is called the product of a by 2, or of a into 2 ; a is also 
called the multiplicand and 2 the multiplier. Instead of the 
sign x , a dot, or mere apposition, is often used where no am- 
biguity can arise. Thus a x 2, a. 2, and a2 all denote the same 
thing. 

.§ 14.] So long as a and b represent integral numbers, as is 

supposed in the primary definition of multiplication, it is easy to 

prove that 

a x b = b x a ; 

or, adopting the principle of attachment of operator and operand, 

with full symbolism (see above, § 4), 

x a x b = x b x a. 



12 COMMUTATION AND ASSOCIATION chap. 

The same may be established for any number of integers, for 

example, 

x a x b x c — x a x c x b = x b x c x a, &c. 

In other words, The order of operations in a chain of multiplication 
is indifferent. 

This is the Commutative Law for multiplication. 

§ 15.] We may introduce the use of brackets and the idea of 
association in exactly the same way as we followed in the case 
of addition. Thus in x a x ( x b x c) we are directed to multiply 
a by the product of b by c. The Law of Association asserts 
that this is the same as multiplying a by b, and then multiplying 
this product by c. Thus 

x a x ( x b x c) = x a x b x c. 

The like holds for a bracket containing any number of factors. 
In the case where a, b, c, &c, are integers, a proof of the truth 
of this law might be given resting on the definition of multi- 
plication and on the laws of commutation and association for 
addition. 

§ 16.] Even in arithmetic the operation of multiplication is 
extended to cases which cannot by any stretch of language be 
brought under the original definition, and it becomes important 
to inquire what is common to the different operations thus com- 
prehended under one symbol. The answer to this question, 
which has at different times greatly perplexed inquirers into the 
first principles of algebra, is simply that what is common is the 
formal laws of operation which we are now establishing, namely, 
the commutative and associative laws, and another presently to be 
mentioned. These alone define the fundamental operations of 
addition, multiplication, and division, and anything further that 
appears in any particular case (for example, the statement that 
| x | is \ of §) is merely a matter of some interpretation, 
arithmetical or other, that is given to a symbolical result demon- 
strably in accordance with the laws of symbolical operation. 

Acting on this principle we now lay down the laws of com- 
mutation and association as holding for the operation of multi- 
plication, and, indeed, as in part defining it. 



i FOR MULTIPLICATION LAW OF DISTRIBUTION 13 

§ 1 7.] The consideration of composite multipliers or com- 
posite multiplicands introduces the last of the three great laws 
of algebra. 

It is easy enough, if we confine ourselves to the primary 
definition of multiplication, to prove that 

+ ax( + b + c) = + a x b + a x c, 
+ a*( + b-c)= + a x b - a x c, 
( + a-b)x( + c-d)=+axc-axd-bxc + bxd. 

These suggest the following, which is called the Distributive 
Law : — 

The product of two expressions, each of which consists of a chain 
of additions and subtractions, is equal to the chain of additions and 
subtractions obtained by multiplying each constituent of the first expres- 
sion by each constituent of the second, setting down all the partial 
products thus obtained, and prefixing the + sign if the two constituents 
previously had like signs, the - sign if the constituents previously had 
unlike signs. 

Symbolically, thus : — 

(±a±b) x(±c±d) 

= (± a) x (± c) + (± a) x (± d) + (± b) x (± c) 

+ ( ± b) x ( ± d), 

with the following law of signs : — 

( + a) x ( + c) = + ac, ( + a) x ( - c) = - ac, 
( - a) x ( + c) = - ac, ( - a) x ( - c) - + ac. 

There are sixteen different cases included in the above equation, 
as will be seen by taking every combination of one or other of 
the double signs before each letter. 

Thus ( + a-b)( + c + d) 

= + ac + ad -be- bd ; 

(-a-b)(-c + d) 

= + ac- ad + bc-bd ; 
and so on. 

There may, of course, be as many constituents in each 

bracket as we please. If, for example, there be m in one 



14 PROPERTY OF 



CHAP. 



bracket and n in the other, there will be mn partial products 
and 2 m+n different arrangements of the signs. 

Thus ( + a-b + c)(-d + e) 

- - ad + bd - cd + ae - be + ce ; 
and so on. 

The distributive law, suggested, as we have seen, by the 
primary definition of multiplication, is now laid down as a law 
of algebra. It forms the connecting link between addition and 
multiplication, and, along with the commutative and associative 
laws, completes the definition of both these operations. 

§ 18.] By means of the distributive law we can prove another 
property of 0. For, if b be any definite quantity, subject without 
restriction to the laws of algebra, we have 

+ ba-ba= + bx( + a-a) = ( + a-a)x( + b), 

- -bx( + a-a) = ( + a-a)x(-b), 

whence = ( + i)x0 = 0x( + J) = (-J) x = 0x(-J) ; 
or briefly b x = x b = 0. 

DIVISION. 

§ 19.] Division for the purposes of algebra is best defined as 
the inverse operation to multiplication : that is to say, the 
mutual relation of the symbols x and -f- is defined by 

x a-^-b x b- x a (1), 



or 



* 



x a x b -r- b = x a (2). 



From a quantitative point of view, this amounts to defining 
the quotient of a by b, that is, a -i- b, as that quantity which, 
when multiplied by b, gives a. 

In a -f- b, a is called the dividend and b the divisor. Some- 
times a is called the antecedent and b the consequent of the 
quotient. 

Another notation for a quotient is very often used, namely, T or 

' b 

a/b. As this is the notation of fractions, and therefore has 
a meaning already attached to it in the case where a and b 
are integers, it is incumbent upon us to justify its use in another 

* See second footnote, p. 5. 



I QUOTIENT AND FRACTION 15 

meaning. To do this we have simply to remark that b times =-, 

that is, b times a of the bih parts of unity, is evidently a times 
unity, that is, a ; also, by the definition of a^-b, b times a-r-b is a. 

Hence we conclude that 7 is operationally equivalent to a-^-b in 

the case where a and b are integers. No further justification is 

necessary, for when either a or b, or both, are not integers, 7 

loses its meaning as primarily defined, and there is no obstacle 
to resrardinc; it as an alternative notation for a-^-b. 

In the above definition we have not written the signs + or - 
before a and b, but they were omitted simply for brevity, and one 
or other must be understood before each letter. We shall continue 
to omit them until the question as to their manipulation arises. 

§ 20.] Since division is fully defined as the inverse of multi- 
plication, we ought to be able to deduce all its laws from the 
definition and the laws of multiplication. 

We have* 

X a X b-i-C = xdvCXCxt-rC, 

by definition ; 

= x a-^-c x b x c-f-r, 

by law of commutation for 
multiplication ; 
— x a-^-c x b (1), 

by definition. 
Again, x a-^-b-^c - xdvCxo-f J-i-c, 

by definition ; 

= xa-^rC~bxc-^-c, 
by case ( 1 ) ; 

= XO-rC-fi (2), 

by definition. 
In this way we establish the law of commutation for division. 

* Here again the remark made in the third note at the foot of p. 5 applies, 
namely, axb-i-c primarily means, if we think only of the last operation, the 
same as (a x b) -4- c ; a^-bxc the same as (a-T-b)x c ; and so on. 

As in the case of + a, when x a comes first in a chain of operations, x i 
in practice usually omitted for brevity. 



is 



16 COMMUTATION AND ASSOCIATION IN MULTIPLICATION chap. 

Taking multiplication and division together and attaching 
the symbol of operation to the operand, we may now give the 
full statement of this law as follows : — 

In any chain of multiplications and divisions the order of the 
constituents is indifferent, provided the proper sign be attached to each 
constituent and move with it. 

Or, in symbols, for two constituents, 

*a*b = * b*a, 
there being 4 cases here included, for example, 

H-« xi= x b -~a, 

-i-a-i-b = -T-b-~a > and so on. 

§ 21.] By the definition of the mutual relation between 
multiplication and division, we have 

xax(x&-j-c)= x a x (^ x b -7- c) x c -7- c, 

= x a x b-r-c (1). 

Again, since 

x p x b-r-c x c— b = x p xb-h-b, 

= xp, 

therefore xa-i-(xb-7-c)= x a -f- ( x b -f- c) x b 4- c x c — b ; 

= xa-7-(xb-7-c)x(xb-~c)xc-7-b, 

by case (1) ; 
= x a x c-i-b, 

by definition ; 
= x a~b x c (2), 

by the law of commutation 
already established. 

These are instances of the law of association for division and 
multiplication combined, which we may now state as follows : — 

When a bracket contains a chain of multiplications and divisions, 
the bracket may be removed, every sign being unchanged if x precede 
the bracket, and every sign being reversed if ■— precede the bracket. 

Or, in symbols, for two constituents, 

l(lalb) = l(la)l{lb), 



I AND DIVISION PROPERTIES OF 1 17 

with the following law of signs : — 

x ( x a) - x a, x(Tfl)=Td, 
4- ( x rt) = 4- a, 4- ( 4- a) = • ". 

In the above equation eight cases are included, for example, 

x(-i-ox})= -4- a x 6, 

4- ( -r a x 6) = xfl-rJ, 

■4- ( 4- a 4- 6) = x a x b, 

and so on. 

§ 22.] Just as in subtraction we denote the special case 

+ a - a by a separate symbol 0, so in division we denote x a 4- a 

by a separate symbol 1 . From this point of view, 1 has a purely 

operational meaning, and we can prove for it the following laws 

analogous to those established for in § 11 ; — 

x a-r-a= -r- a x a = 1, 
6x 1=6 = 64-1, 

x 1 = 4- 1. 

Like 0, 1 has both a quantitative and a purely operational 
meaning. Quantitatively we may look on it as the limit of the 
quotient of two quantities that differ from each other by a 
quantity which is as small a fraction as we please of either. For 
example, consider a + x and a, then the equation 
(a + x) 4- a = a 4- a +z-r-a 

= 1 + X 4- H 

becomes, when x is made as small a fraction of a as we please, 
an assertion of the compatibility of the two meanings of 1. 

It should be noted that, owing to the one-sidedness of the 
law of distribution (that is, owing to the fact that in ordinary 
algebra 6 + ( x a -f-c) = x (b + a) 4- (6 + c) is not a legitimate trans- 
formation), there is no analogue for 1 to the equation 

b x = 0, 
which is true in the case of 0. 

§ 23.] If the student will now compare the laws of commuta- 
tion and association for addition and subtraction on the one hand 
and for multiplication and division on the other, he will find them 
to he formally identical. It follows, therefore, that so far as these 
VOL. I C 



18 PRINCIPLE OF SUBSTITUTION chap. 

laws are concerned there is virtually no distinction between addi- 
tion and subtraction on the one hand and multiplication and 
division on the other, except the accident that we use the signs 
+ and — in the one case and x and -r- in the other, — a conclusion 
at first sight a little startling. This duality ceases wherever the 
law of distribution is concerned. 

§ 24.] We have already been led to consider such expressions 
as + ( + 2) and + ( - 2), and to see that + a may, according to 
the value given to a, be made to stand for + ( + 2), that is, + 2, or 
+ ( - 2), that is, - 2. The mere fact that a particular sign, say 
+ , stands before a certain letter, indicates nothing as to its 
reduced or ultimate value ; the sign + merely indicates what 
has to be done with the letter when it enters into operation. 

In what precedes as to division, and in fact in all our general 
formulae, we may therefore suppose the letters involved to stand 
for positive or negative quantities at pleasure, without affecting 
the truth of our statement in the least. 

For example, by the law of distribution, 

(a - b) (c + d) = ac + ad -be - bd ; 

here we may, if we like, suppose d to stand for — d'. 
We thus have 

(a - 6) {c + ( - d')} = ac + a(- d') -bc-b{- d'), 

which gives, when we reduce by means of the law of signs 
proper to the case, 

(a - b) (c - d') = ac - ad' - be + bd', 

which is true, being in fact merely another case of the law of 
distribution, which we have reproduced by a substitution from 
the former case. This principle of substitution is one of the most 
important elements in the science ; it is this that gives to 
algebraic calculation its immense power and almost endless 
capability of development, 

§ 25.] We have now to consider the effect of explicit signs 
attached to the constituents of a quotient. As this is closely 
bound up with the operation of the distributive law for division, 
it will be best to take the two together. 



I DISTRIBUTION OF A QUOTIENT 19 

The full symbolical statement of this law for a dividend 
having two constituents is as follows : — 

(± a ±b) + (±c) = (±a) + (±c) + (±b) + (±c), 
with the following law of signs, 

( + a) -T- ( + c) - + a -f- c, ( + a) -f- ( - c) = - o-f- c, 

( - a) -r- ( + c) = -a-r-e, ( - a) -5- ( - c) = + a -f- c. 

Or briefly in words — 

In division the dividend may be distributed, the signs of the partial 
quotients following the same law as in multiplication. 

The above equation includes of course eight cases. It will 
be sufficient to give the formal proof of the correctness of the 
law for one of them, say 

( + a — 6)-t-( — c)= - a -f- e + 6 -f e. 

By the law of distribution for multiplication, we have 

( — a — c + b -T- c) x ( - c) = + (a -f- c) x t' - (b -i- c) x c ; 

- + a-b, 

by the definition. 

Hence ( + a— i)-r-( — c) = ( — a-r-c + J-r-c) * ( - c) -f- ( - c) ; 

= — a-f-e + 6-T-c, 

by the definition. 

§ 26.] The law of distribution has only a limited application 
to division, for although, as just proved, the dividend may be 
distributed, the same is not true of the divisor. Thus it is not 
true in general that 

a ~ (b + c) = a -v- b + a ~ c, 
or that a -f- (b — c) = a ■— b — a -f- c, 

as the student may readily satisfy himself in a variety of ways. 

§ 27.] As we have now completed our discussion of the 
fundamental laws of ordinary algebra, it may be well to insist 
once more upon the exact position which the}^ hold in the 
science. To speak, as is sometimes done, of the proof of these 
laws in all their generality is an abuse of terms. They are 
simply laid down as the canons of the science. The best evi- 
dence that this is their real position is the fact that algebras ai'e 



20 POSITION OF THE FUNDAMENTAL LAWS chap. 

in use whose fundamental laws differ from those of ordinary 
algebra. In the algebra of quaternions, for example, the law 
of commutation for multiplication and division does not hold 
generally. 

What we have been mainly concerned with in the present 
chapter is, 1st, to see that the laws of ordinary algebra shall be 
self-consistent, and, 2nd, to take care that the operations they lead 
to shall contain those of ordinary arithmetic as particular cases. 

In so far as the abstract science of ordinary algebra is con- 
cerned, the definitions of the letters and symbols used are simply 
the general laws laid down for their use. When we come to the 
application of the formulae of ordinary algebra to any particular 
purpose, such as the calculation of areas, for example, Ave have 
in the first instance to see that the meanings we attach to the 
symbols are in accordance with the fundamental laws above 
stated. When this is established, the formulae of algebra become 
mere machines for the saving of mental labour. 

§ 28.] We now collect, for the reader's convenience, the 
general laws of ordinary algebra. 

Definitions connecting the Direct and Inverse 

Operations. 



Addition and subtraction- 

+ a — b + b= + a, 
+ a + b — b= + a. 



Multiplication and division- 

x a -f- b x b = x a, 

x. a * b-i-b - x a. 



For addition and subtrac 
tion — 



Law of Association. 

For multiplication and divi- 



sion — 



±(±a±b)= ± ( ± a) ± ( ± b), 

with the following law of signs :— 

The concurrence of like signs gives the direct sign ; 
The concurrence of unlike signs the inverse sign. 



SYNOPTIC TABLE OF LAWS 



21 



Thus— 
+ ( + a) = + a, + ( - a) = - a, \ x ( x a) = x a, x ( -h a) = -s- a, 

- ( - a) = + a, - ( + a) = - a. j -=- ( -=- a) = x a, -f- ( x a) = -4- a, 

Law of Commutation. 



For addition and subtrac- 



tion- 



sion- 



For multiplication and divi- 
the operand always carrying its own sign of operation with it. 



±a±b= ±b±a, 



Properties of and 1. 





±5 + 

+ 



+ a — a, 
±b-0 = ±b, 

-0. 



1 = x a~-a, 
*bx 1 = *£>-=- 1 = *&, 

Xl= -T-l. 



Law of Distribution. 
For multiplication — 

(± a ± b) x (± c ± d) = + (± a) x (± C ) + (± a) x (± d) 
+ (±b)x(±c) + (±b)x(± d), 

with the following law of signs : — 

If a partial product has constituents with like signs, it must 
have the sign + ; 

If the constituents have unlike signs, it must have the 
sign - . 

Thus— 

+ ( + a) x ( + c ) = +ax c, + ( + a) x ( - c) = -axe, 
+ (-a)x(-c)= + a x c, +(-a)x( + c)= -axe. 

Property of 0. 

0x6 = 5x0-0. 
For division — 

( ± a ± b) -I- ( ± c) = + ( ± a ) -f ( ± e) + ( ± 6) -*- ( ± c), 
with the following law of signs : — 



22 



EXERCISES I 



CHAP. 



If the dividend and divisor of a partial quotient have like 
signs, the partial quotient must have the sign + ; 

If they have unlike signs, the partial quotient must have tho 



sign - . 



Thus— 

+ ( + a) — ( + c)= + a~-c, +( + a)-r-(-c) = 

+ .( - a) -7- ( - c) = + a + c, + ( - a) 4- ( + c) = 

N.B. — The divisor cannot be distributed. 



- a -f- c, 

— a-T-c 



Property of 0. 

-r- 6 = 0. 

N.B. — Nothing is said regarding &-r-0. This case will be 
discussed later on. 



The reader should here mark the exact signification of the 
sign = as hitherto used. It means "is transformable into by 
applying the laws of algebra, without any assumption regarding 
the operands involved." 

Any " equation " which is true in this sense is called an 
"Identical Equation," or an "Identity"; and must, in the first 
instance at least, be carefully distinguished from an equation the 
one side of which can be transformed into the other by means of 
the laws of algebra only when the operands involved have particular 
values or satisfy some particular condition. 

Some writers constantly use the sign = for the former kind 
of equation, and the sign = for the latter. There is much to 
be said for this practice, and teachers will find it useful with 
beginners. We have, however, for a variety of reasons, adhered, 
in general, to the old usage ; and have only introduced the 
sign = occasionally in order to emphasise the distinction in cases 
where confusion might be feared. 

Exercises I. 

[In working this set of examples the student is expected to avoid quoting 
derived formulae that he may happen to recollect, and to refer every step to 
the fundamental principles discussed in the above chapter.] 



i EXERCISES I 23 

(1.) Point out in what sense the usual arrangement of the multiplication 
of 365 by 492 is an instance of the law of distribution. 

(2.) I have a multiplying machine, but the most it can do at one time is 
to multiply a number of 10 digits by another number of 10 digits. Explain 
how I can use my machine to multiply 13693456783231 by 46381239245932. 

(3.) To divide 5004 by 12 is the same as to divide 5004 by 3, and then 
divide the quotient thus obtained by 4. Of what law of algebra is this an 
instance ? 

(4.) If the remainder on dividing X by a be R, and the quotient P, and 
if we divide P by 6 and find a remainder S, show that the remainder on 
dividing N by ab will be aS + R. 

Illustrate with 5015+-12. 

(5.) Show how to multiply two numbers of 10 digits each so as to obtain 
merely the number of digits in the product, and the first three digits on 
the left of the product. 

Illustrate by finding the number of digits, and the first three left-hand 
digits in the following : — 

1st. 3659893456789325678 x 342973489379265 ; 
2nd. 2 64 . 

(6.) Express in the simplest form — 

-(-(-(-( • • • (-D • • • )))), 

1st. Where there are 2« brackets ; 

2nd. "Where there are 2?i + 1 brackets ; n being any whole number whatever. 
(7.) Simplify and condense as much as possible — 

2a- {3a-[a-(b-a)]}. 



(8.) Simplify— 






1st. 


3 [4 -5[6- 7(8 -9.10-11)]}, 




2nd. 
(9.) Simplify— 


*tt-t»-M-*-A-A)]}- 




l-(2-(3- 


(4- . . . (9-(10-ll)) . . . ))). 




(10.) Distribute the 


following products : — 1st. {a + b) x (a + b) ; 


2nd 



(r*-6)x(a + 6) ; 3rd. (3a- 66) x (3a + 66) ; 4th. (Ja- £6) x (Ja + £6). 
(11.) Simplify, by expanding and condensing as much as possible — 

{(m + l)a+(n + l)b}{{m-l)a + (n-l)b} 
+ {{m + l)a-(n + l)b} {{m-l)a-(n-l)b}. 
(12.) Simplify- 

K)K)+K)K> 

(13.) Simplify— 

H)H)H)-H)B)H> 

(14.) Expand and condense as much as possible — 







24 HISTORICAL NOTE chap, i 

Historical Xote. — The separation and classification of the fundamental laws 
of algebra has been a slow process, extending over more than 2000 years. It is 
most likely that the first ideas of algebraic identity were of geometrical origin. 
In the second book of Euclid's Elements (about 300 B.C.), for example, we have a 
series of propositions which may be read as algebraical identities, the operands 
being lines and rectangles. In the extant works of the great Greek algebraist 
Diophantos (350 ?) we find what has been called a syncopated algebra. He uses 
contractions for the names of the powers of the variables ; has a symbol f- to 
denote subtraction ; and even enunciates the abstract law for the multiplication 
of positive and negative numbers ; but has no idea of independent negative quan- 
tity. The Arabian mathematicians, as regards symbolism, stand on much the 
same platform ; and the same is true of the great Italian mathematicians Ferro, 
Tartaglia, Cardano, Ferrari, whose time falls in the first half of the sixteenth 
century. In point of method the Indian mathematicians Aryabhatta (476), 
Brahmagupta (598), Bhaskara (1114), stand somewhat higher, but their works 
had no direct influence on Western science. 

Algebra in the modern sense begins to take shape in the works of Regiomon- 
tanus (1436-1476), Rudolff (about 1520), Stifel (1487-1567), and more particularly 
Viete (1540-1603) and Harriot (1560-1621). The introduction of the various 
signs of operation now in use may be dated, with more or less certainty, as 

follows : - and apposition to indicate multiplication, as old as the use of the 

Arabic numerals in Europe; + and -, Rudolff 1525, and Stifel 1544; =, 
Recorde 1557 ; vinculum, Viete 1591 ; brackets, first by Girard 1629, but not 
in familiar use till the eighteenth century ; < > , Harriot's Praxis, published 
1631 ; x , Oughtred, and ^-, Pell, about 1631. 

It was not until the Geometry of Descartes appeared (in 1637) that the im- 
portant idea of using a single letter to denote a quantity which might be either 
positive or negative became familiar to mathematicians. 

The establishment of the three great laws of operation was left for the present 
century. The chief contributors thereto were Peacock, De Morgan, D. F. Gregory, 
Hankel, and others, working professedly at the philosophy of the first principles ; 
and Hamilton, Grassmann, Peirce, and their followers, who threw a flood of light 
on the subject by conceiving algebras whose laws differ from those of ordinary 
algebra. To these should be added Argand, Cauchy, Gauss, and others, who 
developed the theory of imaginaries in ordinary algebra. 



CHAPTEE II. 
Monomials — Laws of Indices—Degree. 

THEORY OF INDICES. 

§ 1.] The product of a number of letters, or it may be num- 
bers, each being supposed simple, so that multiplication merely 
and neither addition nor subtraction nor division occurs, is called 
an integral term, or more fully a rational integral monomial (that 
is, one-termed) algebraical function, for example, ax3x6nxa 
xxxxxyxbxb. 

By the law of commutation we may arrange the constituents 
of this monomial in any order we please. It is usual and con- 
venient to arrange and associate together all the factors that are 
mere numbers and all the factors that consist of the same letter ; 
thus the above monomial would be written 

(3 x 6) x (a x a) x (b x b) x (x x x x x) x y. 

3x6 can of course be replaced by 18, and a further contrac- 
tion is rendered possible by the introduction of indices or ex- 
ponents. Thus a x a is written a 2 , and is read " a square," or 
"a to the second power." Similarly b x b is replaced by b*, and 
x x x x xhy x 3 , which is read " x cube," or " x to the third power." 
We are thus led to introduce the abbreviation x 11 for x x x x x x . . . 
where there are n factors, n being called the index or exponent* 
while x n is called the nth power of x, or x to the nth. power. 

§ 2.] It will be observed that, in order that the above defini- 
tion may have any meaning, the exponent n must be a positive 

* In accordance with this definition x 1 of course means simply x, and is 
usually so written. 



26 RATIONAL INTEGRAL MONOMIAL chap. 

integral number. Confining ourselves for the present to this 
case, we can deduce the following "laws of indices." 

I. (a) a m x a n = a m+n , 

and generally a™ x a n x aP x . . . = a m+n+p+ • • • 

(J3) — = a m ~ n if m>n, 

1 



a n 



a n-m 



if m < n. 



II. (a m ) n = a mn = (ft' 1 )' 

III. (a) \ab) m = a m b m , 

and generally (ahc . . . ) m = a m b m c m . . 



W 



¥ 1 ' 



To prove I. (a), we have, by the definition of an index, 
a m x a n = (a x a x a . . . m factors) x (a x a x a . . . n factors), 
= a x a x a . . . m + n factors, by the law of association, 
= a ,n+n , by the definition of an index. 

Having proved the law for two factors, we can easily extend 
it to the case of three or more, 

for a m x a n x a p = (a m x a 11 ) x a p , by law of association, 

_ a m+n x a p^ \)y case already proved, 
_ a (m+n)+p^ by case already proved, 
_ a m+n+p . 

and so on for any number of factors. 

In words this law runs thus : The product of any number of 
powers of one and the same letter is equal to a power of that 
letter whose exponent is the sum of the exponents of these 
powers. 

To prove I. (fi), 

a m 

— = (a x a x . . ,m factors) -7- (ft x ft x . . . n factors), 

by definition of an index, 



II 



LAWS OF INDICES 27 



I 

= a x a x a . . . m factors -~ a -f- a -j- . . . w divisions, 

by law of association. 

Now if m > ?i we may arrange these as follows : — 

a m 

— = (a x a x . . . m — n factors) x (a 4- a) x (a -J- a) ... w factors, 

l>y laws of commutation and association, 
= a x a x . . . m - « factors, by the properties of division, 
= a m ~ n . 

If wi < n, the rearrangement of the factors may be effected 
thus : — 



— = -7- (a y. a x ... n- m factors) x (a -f- a) x (a -r a) . . . wi factors 

= -ha"-™, 
1 



,71-Ml 



It is important to notice 'that I. (ft) can be deduced from 
I. (a) without any further direct appeal to the definition of an 
index. Thus, if m > n, so that m - n is positive, 

gm-n x a n _ a m-n)+n ^ J. ( a ) } 



Hence 

Therefore, by the definition of x and -J- : 



a m " n x a n -4- a n = a'" -f- a". 



Again, if m < n, so that n - m is positive, 

a m x ft n-m _ (jm+to-^ ty J ^ 

= a n , by the laws of + and - 

Hence 

a m x a n - m _j_ fl n - m _ a « _l_ (f n - j» 

Therefore, by the definition of x and ~ , 

a m = a n ~a n - m . 
Hence, by the laws of x and -f- , 

a m -f- a n = a n -f- a n ' m -~ a n , 
= (a n -7-a n )-r-a n - m t 
= 1 -r-a n - m . 



28 LAWS OF INDICES 



CHAP. 



To prove II., 
(a m ) n = a m x a m x . . . n factors, by definition, 

= (a x a x . . .7)i factors) x (a x a x . . . m factors) 
x . . ., n sets, by definition, 

ran factors, by law of association. 



= a x a x 



= a mn , by definition. 

To prove III. (a), 
(ab) m = (ab) x (ab) x . . . m factors, by definition, 

= (a x a x , . . m factors) x (6 x b x . . . m factors), 

by laws of commutation and association, 
= a m b m , by definition. 

Again, (abc) m = {(ab)c} m , 

= (ab) m c m , by last case, 

= (a™b m )c">, by last case, 

= a m b m c m , and so on. 
Hence the with power of the product of any number of letters 
is equal to the product of the wth powers of these letters. 

To prove III. (ft), 

(a\ m 
-A = (a ~ b) x(a~b)x. . . m factors, by definition, 

= (a x a x . . . m factors) -H (b x b x . . . m factors), 

by commutation and association, 
= a m -^ b m , 

_ a m 

In words : The mth power of the quotient of two letters is 
the quotient of the mt\\ powers of these letters. 

The second branch of III. may be derived from the first 
without further use of the definition of an index. Thus 

(d\ " l /(l \ " l 

v x bm = [b x h ) ' by IIL (a) > 

= a''\ by definition of x and ~- . 
Hence /a\ m 

t 1 x b 1 " + b m - a 11 ' -J- b"> 



that is, 



ii LAWS OF INDICES 29 

§ 3.] In so far as positive integral indices are concerned, the 
above laws are a deduction from the definition and from the 
laws of algebra. The use of indices is not confined to this case, 
however, and the above are laid down as the laws of indices 
generally. The laws of indices regarded in this way become in 
reality part of the general laws of algebra, and might have 
been enumerated in the Synoptic Table already given. In this 
respect, they are subject to the remarks in chap, i., § 27. The 
question of the meaning of fractional and negative indices is 
deferred till a later chapter, but the student will have no diffi- 
culty in working the exercises given below. All he has to do is 
to use the above laws whenever it is necessary, without regard 
to any restriction on the value of the indices. 

§ 4.] The following examples are worked to familiarise the 
student with the meaning and use of the laws of indices. At 
first he should be careful to refer each step to the proper law, 
and to see that he takes no step which is not sanctioned by 
some one of the laws of indices, or by one of the fundamental 
laws of algebra. 

Example 1. 

(a 3 b 2 c 5 ) x (a 5 b s c u )+(a 4 b 3 c 15 ) 

= a s a 5 b-b*c 5 c u 4- « 4 4- b 3 4- c ls , by commutation and association, 

= « ! ^'^" 4- « 4 4- b 3 -f- c 15 , by law of indices, I. (a), 

= (a 3 + 5 4- a*) x (b-+ 6 -f- b 3 ) x (c 5 +" -=- c 15 ), by commutation and 

association. 
= «W-<x6W-3 xc s+n-i5 j by law of indices, I. (/3), 
= a*bh. 

Example 2. 

(15-cVV) 2 '< (t^ttkY 

= 15V)V)V) a x nf£p> b )' laws of indices, III. (a) and III. (£), 

~(3x4) 2 (^vr y (a) ' 

32 x 523,3,-6-10 

= ~3 2 x 4--yy T ' by l - (a) and n -> 

= 3- 4- 3 2 x i>- 4- 4- x x s x if 4 1/ x z ]0 4- z™, 
= 5-~i 2 xx g 4-y 2 , 






30 ALGEBRAICAL INTEGRALITY AND FRACTIONALITY chap. 

THEORY OF DEGREE. 

§ 5.] The result of multiplying or dividing any number of 
letters or numbers one by another, addition and subtraction 
being excluded, for example, 3 x a x x x b -~ c — y x d, is called a 
(rational) monomial algebraical function of the numbers and letters 
involved, or simply a term. If the monomial either does not 
contain or can be so reduced as not to contain the operation of 
division, it is said to be integral; if it cannot be reduced so as 
to become entirely free of division, it is said to be fractional. In 
drawing this distinction, division by mere numbers is usually 
disregarded, and even division by certain specified letters may be 
disregarded, as will be explained presently. 

§ 6.] The number of times that any particular letter occurs 
by way of multiplication in an integral monomial is called the 
degree (or dimension) of the monomial in that particular letter ; 
and the degree of the monomial in any specified letters is the 
siim of its degrees in each of these letters. For example, the 
degree of 6 x a x a x x x x x x x y x y, that is, of 6a 2 x 3 y 2 , in a is 2, 
in x 3, in y 2, and the degree in x and y is 5, and in a, x, and y 7. 

In other words, the degree is the sum of the indices of the 
named letters. The choice of the letters which are to be taken 
into account in reckoning the degree is quite arbitrary ; one 
choice being made for one purpose, another for another. When 
certain letters have been selected, however, for this purpose, it 
is usual to call them the variables, and to call the other letters, 
including mere numbers, constants. The monomial is usually 
arranged so that all the constants come first and the variables 
last; thus, x and y being the variables, we write 32a 2 bcx 3 y' ; and 
the part 32a 2 bc is called the coefficient. 

In considering whether a monomial is integral or not, division 
by constants is not taken into account. 

§ 7.] The notion of degree is an exceedingly important one, 
and the student must at once make himself perfectly familiar 
with it. lie will find as he goes on that it takes to a large 
extent in algebra the same place as numerical magnitude in 
arithmetic. 



IT 



NOTIONS AND LAWS OF DEGREE 31 



The following theorems are particular cases of more general 
ones to be proved by and by. 

The degree of the product of two or more monomials is the sum 
of their respective degrees. 

If the quotient of two monomials be integral, its degree is the excess 
of the degree of the dividend over that of the divisor. 

For let A = cx[i/ l ":"ni> . . . 

At tjl Ml' »' )>' 

= cx y z u' ... 

where c and c are the coefficients, x, ij, z, u . . .the variables, and 
/, m, v,p . . ., /', m', ri,p' ■ ■ ■ are of course positive integral numbers. 
Then the degree, d, of A is given by d = I + m + n +p + . . ., and 
tbe degree, d', of A' by d' = 1' + m' + n + p + . . . 

But A x A' = (cxhf n z n uP . . .) x (c'x y m z n u p . . .) 

/ a l+V m+»i' n+n! p+v' 

= (c x c)x y z ir . . . 
the degree of which is (I + V) + (m + m') + (w + n) + (p + p') . . ., 
that is, (l + m + n+p . . .) + (/' + m! + n' +_// + . . . ), that is, 
d + d', which proves the first proposition for two factors. The 
law of association enables us at once to extend it to any number 
of factors. 

Again, let Q = A -=- A', and let Q be integral and its degree 8. 
Now we have, by the definition of division, Q x A' = A. Hence, 
by last proposition, the degrees of A and A' being d and d', as 
before, we have d = 8 + d', and thence 8 = d - d'. 

As an example, let A = G///, A' = T.r 7 // 3 , then A x A' = 
42x*y, and A 4- A' = fc 2 /. The degree of A x A' is 24, that is, 
14 + 10 ; that of A-j- A' is 4, that is, 14 - 10. 

The student will probably convince himself most easily of 
the truth of the two propositions by considering particular cases 
such as these ; but he should study the general proof as an 
exercise in abstract reasoning for on such reasoning he will have 
to rely more and more as he goes on. 

Exercises II. 

[Wherever it is possible in working the following examples, the student 
should verify the laws of degree, §§ 5-7.] 

11.) Simplify— 5 7 x 12 4 x 32-' x (3 2 x 4 2 x 5) 2 

(3 x 15 x 2 3 ) 10 



32 



EXERCISES II 



CHAP II. 



(2. 
(3. 



(4. 

(5.: 

(6.: 
(7.: 
(8.; 

(9. 

(io.; 
(ii. 

(12. 
(13. 

(14. 
(15. 

(16.; 

(17. 



"Which is greater, (2 2 ) 2 or 2'- ? Find the difference between them, 
Simplify — 



2- 



Simplify- 



/ \a 4 b 3 x 3 y 3 J ' 



2(2 2 ) 2 
36W6W 3 
81a 4 b 3 <? * 
Express in its simplest form — 
c y \ 2 / a*Vh? 
<a 3 bWtf) X \aWc 3 x 3 y' 
Simplify— /45« 3 &V\ 2 /2i3a 4 b 4 c 4 x\ 2 



Simplify — 
Simplify — 

Simplify — 
Simplify — 
Simplify — 



27a 2 6 2 c ) *\ 180a 2 bc J 
,„ * x*y 9 /3xV\ 2 



{.rhfz-j 7 X (;//l~Y) 7 X (A 5 /) 



S) 7 * 



gas/ Va;?// 

X"'J 



aj^J x (y 



J («p-?)« x (a^- r y-p \ J 
I (a r +») r -« / 

(z«- 6 x ^- c ) a x C^) 



/r 



(a! a Xic ) o -T-(jc o + c ) c 

Simplify— /x^\p+*^ (xp^\v"-!<1 

Simplify — 

| f £*Y x /^' Y" I _i_ {(^ x ( X m)m} x f^m)* x (jgljw^ 



Prove that- 



{yz)^(zxYP(xy)P" 



_ {xyz)P+^-^ 



(yt~ l z r - 1 )P{z r -' i xP- 1 Y{xP- l yi- l Y xPifz r 
Distribute the product — 



i V i 

aP + r- a» + r- 



Distribute — 

If ?n. = ft*, n = aM, a z ={mfn x Y ; show that a:?/z = lc 



CHAPTEE III. 

Fundamental Formula relating to Quotients or 
Fractions, with Applications to Arithmetical 
Fractions and to the Theory of Numbers. 

OPERATIONS WITH FRACTIONS. 

§ 1.] Before proceeding to cases where the fundamental 
laws are masked by the complexity of the operations involved, 
we shall consider in the light of our newly-acquired principles 
a few cases with most of which the student is already partly 
familiar. He is not in this chapter to look so much for new 
results as to exercise his reasoning faculty in tracing the opera- 
tion of the fundamental laws of algebra. It will be well, how- 
ever, that he should bear in mind that the letters used in the 
following formula? may denote any operands subject to the laws 
of algebra ; for example, mere numbers integral or fractional, single 
letters, or any functions of such, however complex. 

§ 2.] Bearing in mind the equivalence of the notations -, 

ajb, and a —■ b, the laws of association and commutation for 
multiplication and division, and finally the definition of a 
quotient, we have 

y b = O) -*- (P b ) ^xa-rii-rJ, 

= a -7- b -r-p x p, 
— a-r-b; 

that is, ^-7 = 7- 

pb b 

VOL. I D 



34 ALGEBRAICAL ADDITION OF FRACTIONS chap. 

Kead forwards and backwards this equation gives us the 
important proposition that ice may divide or multiply the numerator 
and denominator of a fraction by the same quantity without altering 
its value. 

§ 3.] Using the principle just established, and the law of 
distribution for quotients, we have 



a 
±- 

6 


1 


± qa 
qb 


pb 

qb' 






± qa 


±pb 



qb 
that is, To add or subtract two fractions, transform each by multiplying 
numerator and denominator so that both shall have the same denomi- 
nator, add or subtract the numerators, and write underneath the 
common denominator. 

The rule obviously admits of extension to the addition in 
the algebraic sense (that is, either addition or subtraction) of any 
number of fractions whatever. 

Take, for example, the case of three : — 

b d f bdf bdf bdf Yb ' 

± adf ± cbf ± ebd , , , , . ^ . . 

= J ,,; , by law ol distribution. 

bdf 

The following case shows a modification of the process, which 
often leads to a simpler final result. Suppose b - Ic, q- lr; then, 
taking a particular case out of the four possible arrangements 
of sign, 

a p a p 
b q Ic lr' 
_ ar pc 
Icr Ire 
_ ar - pc 
Icr 

Here the common denominator Icr is simpler than bq, which is 
Ihr. 

The same result would of course be arrived at by following 



in MULTIPLICATION AND DIVISION OF FRACTIONS 35 

the process given above, and simplifying the resulting fraction 
at the end of the operation, thus : — 

a p air - pic . 

(ar - pc)l 
= far ' 

by using the law of distribution in the numerator, and the laws 
of association and commutation in the denominator ; 

ar -cp 

§ 4.] The following are merely particular cases of the laws 
of association and commutation for multiplication and division : — 



©«(a)->+«>"<H-A 



= a -J- b x c -f- d, 
= a x c-7-b —el, 
= (ac) -t- (bd), 

ac 
= bd' 

or, in words, To multiply two fractions, multiply their numerators 
together for the numerator, and the denominators together for the 
denominator of the product. 
Again, 

© + ©-<«■»> +<•+<>. 

= a x d -7- b -f- c, 
- (ad) -7- (be), 
ad 

= Tc' 



also 

\C 



■© 



by last case. In words : To divide one fraction by another, invert 
the latter and then midtiply. 

§ 5.] In last paragraph, and in § 2 above, we have for 



36 EXERCISES III CHAr. 

simplicity omitted all explicit reference to sign. In reality we 
have not thereby restricted the generality of our conclusions, for 
by the principle of substitution (which is merely another name 
for the generality of algebraic formulae) we may suppose the p, 
for example, of § 2 to stand for - w, say, and we then have 

( - o>)a a 
C^jb = b ' 

that is, taking account of the law of signs, 

— wa a 

~-ub = b' 
and so on. 



Exercises III. 
(1.) Express in its simplest form — 





K» V 2 

x — y y-x 


(2.) Express in its 


simplest form — 




a h 




a-b b-a 


(3.) Simplify- 


P + Q P-Q 




P-Q P + Q' 


where 


T> = x + y, Q=x~y, 


(4.) Simplify— 


1 afl-tf 




x + y 

1+ 

x + y 


(5.) Simplify— 


1 1 1 
ab ac be 




a?-(b-c? 



(6.) Simplify- 



( h " \ I h * \ 

\ a + bj \ a-b) 



(7.) Simplify— _1_ _J_ 1x 

, 8 . )Siml , % - fr^ + iy^ hl y 

(9.) Simplify— fa b\/f0 &\ 

\b a) • \b* a 2 )' 



Ill ARITHMETICAL INTEGRALITY AND DIVISIBILITY 37 

(10.) Simplify— a{a -b)- b{n + b) 



a+b a- b 

(11.) Simplify— 1-x 1+a: 

l+x + x 2 1-x + x"' 

(12.) Simplify- x 

1+a . {x + lf-x * 
1 x 2 + x+l ' 

1 + x 
(13.) Simplify— 2_/l 1\ 

a- + b 2 ab\a b) 

w+ (k ' 

(14.) Show that &_ (a 2 -a 2 ) 2 (a 2 -^) 2 

a 2 J a+ a 2 (a a -ft 2 ) W-ft 2 ") 

is independent of x. 

(15.) Simplify— «■ 



6--° 



' -J 



(16.) Simplify— 1 

1 



a -2b 



a -2b- 



a -2b 
(17.) Simplify- a + b 

1 

a + b + 



1 
a ~b + 



a + b 



APPLICATIONS TO THE THEORY OF NUMBERS. 

§ 6.] In the applications that follow, the student should look 
somewhat closely at the meanings of some of the terms employed. 
This is necessary because, unfortunately, some of these terms, 
such as integral, factor, divisible, &c, are used in algebra generally 
in a sense very different from that which they bear in ordinary 
arithmetic and in the theory of numbers. 

An integer, unless otherwise stated, means for the present a 
positive (or negative) integral number. The ordinary notion of 
greater and less in connection with such numbers, irrespective of 
their sign, is assumed as too simple to need definition.* When 

* This is a very different thing from the algebraical notion of greater 
and less. See chap. xiii. , § 1. It may not he superfluous to explain 



38 PRIME AND COMPOSITE INTEGERS chap. 

an integer a can be produced by multiplying together two others, 
b and c, b and c are called factors of a, and a is said to be exactly 
divisible by b and by c, and to be a multiple of b or of c. Since 
the product of two integers, neither of which is unity, is an 
integer greater than either of the two, it is clear that no integer 
is exactly divisible by another greater than itself. 

It is also obvious that every integer (other than unity) has 
at least two divisors, namely, unity and itself; if it has more, it 
is called a composite integer, if it has no more, a prime integer. 
For example, 1, 2, 3, 5, 7, 11, 13, . . . are all prime integers, 
whereas 4, 6, 8, 9, 10, 12, 14 are composite. 

If an integer divide each of two others it is said to be a 
common factor or common measure of the two. If two integers 
have no common measure except unity they are said to be prime 
to each other. It is of course obvious that two integers, such as 
6 and 35, which are prime to each other need not be themselves 
prime integers. We may also speak of a common measure of more 
than two integers, and of a group of more than two integers 
that are prime to each other, meaning, in the latter case, a set 
of integers no two of which have any common measure. 

§ 7.] If we consider any composite integer N, and take in 
order all the primes that are less than it, any one of these either 
will or will not divide N. Let the first that divides N be a, 
then N = «N], where N, is an integer ; if N, be also divisible by a 
we have Ni = «N 2 , and N = fl(</N 2 ) = a 2 N 2 ; and clearly, finally, 
say N = a*N a , where N a is either 1 or no longer divisible by a. 
N a (if not =1) is now either prime or is divisible by some 
prime >a and <N a , and, a fortiori, <N, say b ; we should on 
the last supposition have N« = Z^N^, where N /3 <N a , and so on. 
The process clearly must end with unity, so that we get 

N = aW 

where a, b, . . . are primes, and a, /3, . . . positive integers. It 

here the use of the inequality symbols 4=, >, <, >, <f ; they mean 
respectively "is not equal to," "is greater than," "is less than," "is not 
greater than," "is not less than." Iustead of > , <t we may use <,> which 
may b; read "is equal to or less than," " is equal to or greater than." 



in ARITHMETICAL G.C.M. 39 

is to be observed that a", b , . . . are powers of primes, and 
therefore, as we shall prove presently, prime to each other. It 
is therefore always possible to resolve every composite integer into factors 
that are powers of primes ; and we shall presently show that this 
resolution can be effected in one way only. 

§ 8.] If a be divisible by c, then any integral multiple of a, say ma, 
is divisible by c; and, if a and b be each divisible by c, then the algebraic 
sum of any integral multiples of a and b, say ma + nb, is divisible by c. 

For by hypothesis a - ac and b = fir, where a and ft are in- 
tegers, hence ma = viae - (ma)c, where ma is an integer, that is, ma 
is divisible by c. And ma + nb = mac + nfic = {ma + nfi)c, where 
ma + n(3 is an integer, that is, ma + nb is divisible by c. The 
student should observe that, by virtue of the extension of the 
notion of divisibility by the introduction of negative integers, 
any of the numbers in the above proposition may be negative. 

§ 9.] From the last article we can deduce a proposition which 
at once gives us the means of finding the greatest common measure 
of two integers, or of proving that they are prime to each other. 

If a =pb + c, where a, b, c, p are all integers, then the G.C.M. of 
a and b is the G.C.M. of b and c. 

To prove this it is necessary and it is sufficient to show — 
1st, that every divisor of b and c divides a and b, and, 2nd, that 
every divisor of a and b divides b and r. 

Since a = pb + c, it follows from § 8 that every divisor of b 
and c divides a, that is, every divisor of b and c divides a and b. 

Again, since a = pb + c, it follows that c — a —pb • hence, again 
by § 8, every divisor of a and b divides c, that is, every divisor 
of a and b divides b and c. Thus the two parts of the proof are 
furnished. 

Let now a and // be two numbers whose G.C.M. is required ; 
they will not be equal, for then the G.C.M. would be either of 
them. Let b denote the less, and divide a by b, the quotient 
beings and the remainder c, where of course c<b* Next divide 
b by c, the quotient being q, the remainder d ; then divide c by 
d, the quotient being r, the remainder e, and so on. 

* For a formal definition of the remainder see § 11. 



40 ARITHMETICAL G.C.M. 



CHAP. 



Since a > b, b>c, c> d, d>e, &c, it is clear that the re- 
mainders must diminish down to zero. We thus have the 
following series of equations : — 

a =pb + c 
b = qc + d 
c = rd + e 



I = vm + n 
m = ten. 
Hence the G.C.M. of a and b is the same as that of b and c, which 
is the same as that of c and d, that is, the same as that of d and e, 
and finally the same as that of m and n. But, since m = wn, the 
G.C.M. of m and n is n, for n is the greatest divisor of n itself. 
Hence the G.C.M. of a and b is the divisor corresponding to the 
remainder in the chain of divisions above indicated. 

If n he different from unity, then a and b have a G.C.M. in 
the ordinary sense. 

If n he equal to unity, then they have no common divisor 
except unity, that is, they are prime to each other. 

§ 10.] It should be noticed that the essence of the foregoing 
process for finding the G.C.M. of two integers is the substitution 
for the original pair, of successive pairs of continually decreasing 
integers, each pair having the same G.C.M. All that is necessary 
is that j), q, r, . . . be integers, and that a, b, c, d, e, ... be in 
decreasing order of magnitude. 

The process might therefore be varied in several ways. 
Taking advantage of the use of negative integers, we may some- 
times abbreviate it by taking a negative instead of a positive 
remainder, when the former happens to be numerically less than 
the latter. 

For example, take « = 4323, & = 1595, 
we might take 4323 = 2 x 1595 + 1133 

or 4323 = 3x1595-462; 

the latter is to be preferred, because 462 is less than 1133. In practice the 
negative sign of 462 may be neglected in the rest of the operation, which may 
be arranged as follows, for the sake of comparison with the ordinary process 
already familiar to the student : — 



Ill 



riUME DIVISORS 



41 



1595)4323(3 
4785 



462)1595(3 
1386 



209)462(2 
418 

44)209(5 
220 

11)44(4 
44 • 

G.C.M.=11. 

By means of the process for finding the (x.C.M. we may prove 
the following proposition, of whose truth the student is in all 
probability already convinced by experience : — 

If a and b be prime to each other, and h any integer, then any 
common factor of ah and b must divide h exactly. 

For, since a and b are prime, we have by § 9, 



a=pb + c 
b = qc + d 
c = rd + e 

I - vm + 1 



>(1). Hence < 



r ah- pbh + ch 
bh = qch + dh 
eh = rdh + eh 

Ih = vmh + h 



•(2). 



Now, since any common factor of ah and b is a common 
factor of ah and bh, it follows from the first of equations (2) that 
such a common factor divides ch exactly, and by the second that 
it also divides dh exactly, and so on ; and, finally, by the last of 
equations (2), that any common factor of ah and b divides h 
exactly. 

In particular, since b is a factor of itself, we have 

Cor 1. If b divide ah exactly and be prime to a, it must divide h 
exactly. 

Cor. 2. If a' be prime to a and to b and to c, &c, then it is 
prime to their product abc . . . 

For, if a' had any factor in common with abc . . ., that is, 
with a(bc . . .), then, since a' is prime to a, that factor, by the 
proposition above, must divide be . . . exactly ; hence, since a' 



42 



REMAINDER AND RESIDUE 



CHAP. 



is prime to b, the supposed factor must divide c . . . exactly, and 
so on. But in this way we exhaust all the factors of the pro 
duct, since all are prime to a'. Hence no such factor can exist, 
that is, a' is prime to abc . . . 

An easy extension of this is the following : — 
Cor. 3. If all the integers a', b', c', . . . be prime to all the integers 
a,b,c,. . ., then the product a'b'c' . . . is prime to the product abc . . . 
A particular case of which is 

Cor. 4. If a' be prime to a (and in particular if both be primes), 
then any integral power of a' is prime to any integral power of a. 

§ 11. J It is obvious that, if a and b be two integers, we can 
in an infinite number of ways put a into the form of qb + r, where 
q and r are integers, for, if we take q any integer whatever, and 
find r so that a - qb = r, then a = qb + r. 

There are two important special cases, those, namely, where 
we restrict r to be numerically less than b, and either (1) positive 
or (2) negative. In each of these cases the resolution of a is 
always possible in one way only. For, in case 1, if qb be the 
greatest multiple of b which does not exceed a, then a - qb — r, 
where r <b ; hence a = qb + r ; and in case 2, if qb be the least 
multiple of b which is not less than a, then a - q'b- - r', where 
r' < b. Also the resolution is unique ; for suppose, in case 1, that 
there were two resolutions, another being a = \b + p, say; then 
qb + r = xb + p, therefore r — p - (x - q)b ; hence r - p is divisible 
by b ; but, r and p being each positive, and each numerically <b, 
r - p is numerically less than b, and therefore cannot be divisible 
by b. Hence there cannot be more than one resolution of the 
form 1. Similar reasoning applies to case 2. 

r and r' are often spoken of as the least positive and negative 
remainders of a with respect to b. When the remainder is spoken 
of without qualification the least positive remainder is meant. If 
a more general term is required, corresponding to the removal 
of the restriction r numerically < b, the word residue is used. 

It is obvious, from the definitions laid down in § 6, that a is 
or is not exactly divisible by b according as the least remainder of a 
with respect to b does or does not vanish. 



ill ARITHMETICAL FRACTIONALITV 43 

The student will also prove without difficulty that if the re- 
mainders of a and of a' with respect to b be the same, then a - a' is 
divisible by b ; and conversely. 

Cor. If q be a fixed integer (sometimes spoken of as a modulus), 
then every other integer can be expressed in one or other of the forms 

bq, bq+ 1, bq+ 2, . . ., bq + (q- I), 

where b is an integer. 

For, as we have seen, we can put any given integer a into 
the form bq + r, where r~^>q, and here r must have one of the 
values 0, 1, 2, . . ., (q- \). 

Example. Take q = 5, then 

= 0.5, 1 = 0.5 + 1, 2 = 0.5 + 2, 3 = 0.5 + 3, 4 = 0.5+4; 

5 = 1.5, 6 = 1.5 + 1, 7 = 1.5 + 2, 8 = 1.5 + 3, 9 = 1.5 + 4; 

10 = 2.5, 11 = 2.5 + 1, 12 = 2.5 + 2, 13 = 2.5 + 3, 14 = 2.5 + 4; 
and so on. 

It should be noticed that, since bq + (q - 1) = (b + \)q- 1, 
bq + (q - 2) = (b + l)q - 2, &c, we might put every integer into 
one or other of the forms 

bq, bq± 1, bq± 2, . . . , &c. 
For example, 

8 = 2.5-2, 9 = 2.5-1, 10 = 2.5, 11 = 2.5 + 1, 12 = 2.5 + 2. 

The above principle, which may be called the periodicity of 
the integral numbers with respect to a given modulus, is of great 
importance in the theory of numbers. 

§ 12.] When the quotient a/b cannot be expressed as an 
integer, it is said to be fractional or essentially fractional ; if a> b, 
a/b is called in this case an improper fraction; if a<b, a proper 
fraction. 

Hence no true fraction, proper or improper, can be equal to an 
integer. 

Every improper fraction a/b can be expressed in the form q + rjb, 
where q is an integer and rjb a proper fraction. For, if r be the 
least positive remainder when a is divided by b, a = qb + r, and 
ajb = (qb + r)jb = q + rjb, where q and r are integers and r < b. 

If two improper fractions ajb and a'jb' be equal, their integral 
parts and their proper fractional parts must be equal separately. For, 



44 THEOREM REGARDING G.C.M. CHAP. 

if this were not so, we should have, say ajb = q + rjb, a'jb' 
= q 1 + r'/b', and q + rjb = q' + r'jb' ; whence q - q = r'/b' - r/b = 
(r'6 - rb')/bb'. Now r'b < b'b and rb' < bb', hence r'b - rb' is numeric- 
ally < 66'. In other words, the integer q - q' is equal to a proper 
fraction, which is impossible. 

§ 13.] We can now prove that an integer can be resolved info 
factors which are powers of primes in one way only. 

For, since the factors in question are powers of primes, the}'' 
are prime to each other. Let, if possible, there be two such 
resolutions, namely, a'b'c . . . and a"b"c" ... of the same integer N. 
Since a'b'c! . . . = a"h"c" . . . , therefore a'b'c' ... is exactly divisible 
by a". Now, since a" is a power of a prime, it will be prime to 
all the factors a', b', c, . . . save one, say a', which is a power of 
the same prime. Moreover, such a factor as a' (that is, a power 
of the prime of which a" is a power) must occur, for, if it did 
not, then all the factors of a'b'c' . . . would be prime to a", and 
a" could not be a factor of N. It follows, then, that a' must be 
divisible by a". 

Again, since a"b"c" . . .= a'b'c' . . ., therefore a"b"c" ... is divis- 
ible by a', and it follows as before that a" is divisible by a'. 

But, if two integers be such that each is divisible by the 
other, they must be equal (§ 6) ; hence a" = a'. 

Proceeding in this way we can show that each factor in the 
one resolution occurs in the other. 

§ 14.] Every remainder in the ordinary jwocess for finding the 

G.C.M. of two positive integers a and b can be expressed in the form 

± (Art - Bo), where A and B are positive integral numbers. The 

upper sign being used for the 1st, 3rd, 5th, &c, and the lower for the 

2nd, 4th, &c, remainders. 

For, by the equations in § 9, we have successively — 

(i); 
(2); 



(3); 



c = 


+ 


{a-pb} 






d = 


h- 


- qc = b - 


q(a - pb), 




- 


- 


[qa - (1 


+ pq)b] 




e = 


c - 


- rd, 






= 


{a 


- pb] + ? 


•{qa-(\ + 


pq)b], 


= 


+ 


{(l+gr) 


a - (j) + r 


+pqr)b] 



in THEOREMS REGARDING G.C.M. 45 

and so on. It is evident in fact that, if the theorem holds for 
any two successive remainders, it must hold for the next. Now 
equations (1), (2), and (3) prove it for the first three remainders; 
hence it holds for the fourth ; hence for the fifth ; and so on. 

In the chapter on Continued Fractions, a convenient process 
will be given for calculating the successive values of A and B 
for each remainder. In the meantime it is sufficient to have 
established the existence of these numbers, and to have seen a 
straightforward way of finding them. 

Cor. 1. Since g, the G.C.M. of a and b, is the last remainder, we 
can always express g in the farm — 

g = ± (Aa - Bb) (4), 

where A and B are positive integers. 

Cor. 2. If a be prime to b, g = 1 ; hence, If a and b be two 
integers prime to each otlier, two positive integers, A and B, can 
always be found such that — 

Aa - Bb = ±1 (5). 

X.B. — It is clear that A must be prime to B. For, since ajg 
and b/g are integers, I and m say, we have, from (4), 

1 = ±(Al- Bm) ; 
hence, if A and B had any common factor it would divide 1 (by 
§ 8 above). 

Cor. 3. From Cor. 1 and § 8 we see that every common factor 
of a and b must be a factor in their G.C.M. 

A result which may be proved otherwise, and will probably 
be considered obvious. 

Cor. 4. Hence, To find the G.C.M. of more tlian two integers a, b, 
c, d, . . ., we must first find g the G.C.M. of a and 6, then g the 
G.C.M. of g and c, then g" the G.C.M. of g and d, and so on, the last 
G.C.M. found being the G.C.M. of all the given integers. 

For every common factor of a, b, c must be a factor in a and 
b, that is, must be a factor in g ; hence, to find the greatest com- 
mon factor iii a, b, c, we must find the greatest common factor 
in g and c ; and so on. 

From Cor. 2 we can also obtain an elegant proof of the 
conclusions in the latter part of § 10. 



46 EXAMPLES chap. 

Example 1. To express the G.C.M. of 565 and 60 in the form A565 - B60. 
We have 565 = 9x60 + 25, 60 = 2x25 + 10, 25 = 2x10 + 5, 10 = 2x5. 
Hence the G.C.M. is 5, and we have successively 

25 = 565-9x60; 
10=60-2(565-9x60} 

= - {2x565-19x60} ; 
5 = 25-2x10 
= 565 - 9 x 60 + 2 { 2 x 565 - 19 x 60 } 
= 5x565-47x60. 

Example 2. Show that two integers A and B can be found so that 

5A-7B = 1. 

We have 7 = 1x5 + 2, 5 = 2x2 + 1; whence 2 = 7-5, 1 = 5-2(7-5) 
= 3x5-2x7. 

Hence A = 3, B = 2 are integers satisfying the requirements of the 
question. 

Example 3. If a, b, c, d, . . . be a series of integers whose G.C.M. is g, 
show that integers (positive or negative) A, B, C, D, . . . can be found 
such that 

g = Aa + Bb + Cc + T)d + . . . 

(Gauss's Disquisitioncs Arithmetics, Th. 40). 

Find A, B, C, D, when a=36, 6 = 24, c=18, ^=30. 

This result may be easily arrived at by repeated application of corollaries 
1 and 4 of this article. 

Example 4. The proper fraction p/ab, where a is prime to b, can be de- 
composed, and that in one way only, into the form 

a' V , 

- + -T-1; 
a b 

where a' and b' are both positive, a' <a, V <b, and k is the integral jiart of 
a'/a + b'/b ; that is to say, or 1, according to circumstances. 

Illustrate with 6/35. 

Since a is prime to b, by Cor. 2 above, 

Aa -B6=±l ; 
multiplying this equation by ^j/ab, we have 

±p h~ =JL h »)■ 

o ' a ab ' 

If the upper sign has to be taken, resolve pk and^B as follows (§ 11) : — 

pA = lb + b' (b' positive <b), 

pB—ma~a' (a' positive <a). 

Then (1) becomes 

V 7 «' b' ' 

£-=:l-m+-+ T (2. 

ab a b 

Now, since p/ab is a proper fraction, the integral part on the right-hand side 
of (2) must vanish ; hence, since the integral part of a'/a + b'/b cannot exceed 
1 , we must have I - m = 0, or I - m — - 1 . 



in NUMBER OF PRIMES INFINITE 47 

If the lower sign has to be taken in (1), we have merely to take the 
resolutions 

pA = lb-b' (b' positive <b), 
pB = 7)ia + a' (a' positive <a), 

and then proceed as before. We leave the proof that the resolution is unique 
to the ingenuity of the reader. 

Illustration. 35 = 5 x 7. 

Now 3x5-2x7 = 1 (see Example 2 above) ; 

whence — = —(3 x 5 - 2 x 7), 

.55 .35 

_18 12 

~7 5' 
2x7+4 3x5-3 

~ 7 5 * 

4 3 

-2 +r -8 + |, 

■W-j. 

5 7 

N.B. — If negative numerators are allowed, it is obvious that p/ab can 
always be decomposed (sometimes in more ways than one) into an algebraic 
sum of two fractions a'/a aud b'/b, where a' and V are numerically less than 
a and b respectively. For example, we have 6/35 = 3/5 - 3/7 = 4/7 - 2/5. 

Example 5. If the n integers a, b, c, d. . . . be prime to each other, the 
proper fraction pjabed . . . may be resolved in one way only into the form 

a 8 y 8 , 

abed 

where a, 8, y, 5, . . . are all positive, a<a, 8<b, y<c, $<d, . . and £ 
has, according to circumstances, one or other of the integral values 

0, 1, 2, . . „»-l. 

(Gauss's Disquisitiones Arithmetical, Th. 310). 

This may be established by means of Example 3. 

Example 6. Work out the resolution of Example 5 for the fraction 
10729/17017. 

§ 15.] We conclude this chapter with a proposition which is 
as old as Euclid (ix. 20),* namely — 

The number of prime integers is infinite. 

For let a, /3, y, . . ., k he any series of prime integers what- 
soever, then we can show that an infinity of primes can be 
derived from these. 

In fact the integer a/3y . . . k + 1 is obviously not exactly 

* Most of the foregoing propositions regarding integral numbers were 
known to the old Greek geometers. 



48 EXERCISES IV 



CHAP. Ill 



divisible by any one of the primes a, (3, y, . . ., k. It must 
therefore either be itself a prime different from any one of the 
series a, (3, y, . . ., k, or it must be a power of a prime or a 
composite integer divisible by some prime not occurring among 
a, (3, y, . . ., k. We thus derive from a, /?, y, , . ., k at least 
one more prime, say A. Then from a, (3, y, . . ., k, A we can in 
like manner derive at least one more prime, \x ; and so on ad 
infinitum.'" 

Exercises IV. 

(1.) If the two fractions A/B, a/b be equal, and the latter be at its lowest 
terms, prove that A = /xa, B = /Jib, where /x is an integer. 

(2.) Prove that the sum or difference of two odd numbers is always even ; 
the sum or difference of an odd and an even number always odd ; the product 
of any number of odd numbers always odd ; the quotient of one odd number 
by another always odd, if it be integral. 

(3. ) If a be prime to b, then — 

1st. (a + b) n and (a-b) m have at most the G.C.M. 2 m ; 
2nd. a m + b m and a m -b m have at most the G.C.M. 2 ; 
3rd. a + b and a 2 + b" - ab have at most the G. C. M. 3. 

(4.) The difference of the squares of any two odd numbers is exactly 
divisible by 8. 

(5.) The snm of the squares of three consecutive odd numbers increased 
by 1 is a multiple of 12. 

(6. ) If each of two fractions be at its lowest terms, neither their sum nor 
their difference can be an integer unless the denominators be equal. 

(7.) Resolve 45738 and 297675 into their prime factors. 

(8.) Find the G.C.M. of 54643 and 91319, using negative remainders 
whenever it is of advantage to do so. 

(9.) Trove that the L. CM. of two integers is the quotient of their product 
by the G.C.M. 

(10.) If pi, g-i, g 3 be the G.C.M. 's, h, h, h the L.C.M.'s, of b and c, c and 

a, a and b respectively, G the G.C.M., and L the L.C.M., of the three a, b, 

c, show that 

, , T abcG 
1st. L = ; 



2nd. I;= jm , 
I* V \9tfsgi 



a). 

(11.) When x is divided by y, the quotient is u and the remainder u ; 
show that, when x and uy are divided by v, the remainders are the same, and 
the quotients differ by unity. 

* On this subject see Sylvester, Nature, vol. xxxviii. (1888), p. 261. 



CHAPTER IV. 

Distribution of Products— Multiplication of Rational 
Integral Functions— Resulting General Principles. 

GENERALISED LAW OF DISTRIBUTION. 

§ 1.] We proceed now to develop some of the more important 
consequences of the law of distribution. This law has already been 
stated in the most general manner for the case of two factors, 
eacb of which is the sum of a series of terms : namely, we multiply 
every term of the one factor by every term of the other, and set 
down all the partial products thus obtained each with the sign 
before it which results from a certain law of signs. 

Let us now consider the case of three factors, say 
(a + b + c + . . .)( a ' + b' + c' + . . .)( a " + b" + c" + . . .). 
First of all, we may replace the first two factors by the process 
just described, namely, we may write 

(aa' + ah' + ac' + . . . + ba' + bb' + bc' + . . .) (a" + b" + c" + . . .). 
Then we may repeat the process, and write 

aa'a" + aa'b" + aa'c" + . . . 
+ ab'a" + ab'b" + ab'c" + . . . 
+ ac'a" + ac'b" + ac'c" + . . . 



+ baa" + ba'b" + ba'c" + . . . &c, 
where the original product is finally replaced by a sum of 
partial products, each of three letters. AYe have simplified the 
matter by writing + before every term in the original factors, but 
the proper application of the law of signs at each step will pre- 
sent no difficulty to the student. 

VOL. I E 



50 GENERALISED LAW OF DISTRIBUTION chap. 

The important thing to remark is that Ave might evidently 
have arrived at the final result by the following process, which 
is really an extension of the original rule for two factors : — 

Form all possible partial products by taking a term from each 
factor (never more than one from each) ; determine the sign by the law 
of signs (that is, if there be an odd number of negative terms in the 
partial product, take the sign - ; if an even number of such vr none, 
take the sign + ). Set down all the partial products thus obtained. 

Cor. The number of terms resulting from the distribution of a 
product of brackets which contain I, m, n, . . . terms respectively is 
I x m x n x . . . For, taking the first two brackets alone, since 
each term of the first goes with each term of the second, the 
whole number of terms arising from the distribution of these is 
I x m. Next, multiplying by the third bracket, each of the I x m 
terms already obtained must be taken with each of the n terms 
of the third. We thus get (I x m) x n, that is, I x m x n terms. 
By proceeding in this way Ave establish the general result. 

It should be noted, hoAvever, that all the terms are supposed 
to be unlike, and that no condensation or reduction, owing to like 
terms occurring more than once, or to terms destroying each 
other, is supposed to be made. Cases occur in § 2 below in 
which the number of terms is reduced in this Avay. 

If the student have the least difficulty in folloAving the aboA T e, 
he "will quickly get over it by working out for himself the results 
stated below, first by successive distribution, and then by apply- 
ing the law just given. 

(a + b)(c + d)(e+f) 

— ace + acf+ ade + adf+ bee + bcf+ bde + bdf 

(2x2x2 = 8 terms) , 
(a-b)(c-d)(e-f) 

— ace - acf- ade + adf- bee + br/+ bde - bdf ; 
(a-b)(c-d)(e+f+g) 

— ace + acf + acg - ade - adf - adg - bee - bef - beg + bde + bdf+ bdg 

(2x2x3 = 12 terms). 

§ 2.] It Avas proved above that in the most general case of 
distribution the number of resulting terms is the product of the 
numbers of terms in the different factors of the product. An 
examination of the particular cases where reductions may be 



iv ENUMERATION OF PRODUCTS 51 

afterwards effected will lead us to some important practical 
results, and will also bring to notice certain important principles. 

Consider the product (a + b) (a + b). By the general rule the 
distribution will give 2x2 = 4 terms. We observe, however, 
that only two letters, a and 6, occur in the product, and that 
only three really distinct products of two factors, namely, a x a, 
a x b, b x b, that is, a 3 , ab, b 3 , can be formed with these ; hence 
among the four terms one at least must occur, more than once. 
In fact, the term a x b (or b x a) occurs twice, and the result of 
the distribution is, after collection, 

(a + b)(a + b) = a 2 + 2ab + b\ 
This may of course be written 

(a + b) 2 = a 2 + 2ab + b 2 (1). 

Similarly (a - bf = a 2 - 2ab + b 2 (2). 

In the case (a + b) (a — b) = a - b 2 (3), 

the term ab occurs twice, once with the + and again with the - 
sign, so that these two terms destroy each other when the final 
result is reduced. 

Before proceeding to another example, let us write down all 
the possible products of three factors that can be made with two 
letters, a and b. These are a 3 , a 2 b, ab 2 , b 3 , four in all. 

Hence in the distribution of (a + b) 3 , that is, of (a + b) (a + b) 
(a + b), which by the general rule would give 2x2x2 = 8 terms, 
only four really distinct terms can occur. Let us see what terms 
recur, and how often they do so. a 3 and b 3 evidently occur each 
only once, because to get three a's, or three b's, one must be 
taken from each bracket, and this can be done in one way only. 
a"b may be got by taking b from the first bracket and a from 
each of the others, or by taking the b from the second, or from 
the third, in all three ways ; and the same holds for ab 3 . Thus 
the result is 

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (4). 

In a similar way the student may establish for himself that 
(a - b) 3 = a 3 - Za 2 b + 3ab 2 - b 3 (5), 

(a ± by = a* ± ia 3 b + 6a 3 b 2 ± lab 3 + b* (6), 



52 COUNTING OF RECURRENCES chap. 

and, remembering that the possible binary products of three 
letters, a, b, c, are a 2 , b 2 , c 2 , be, ca, ab, six in number, that — 

(a + b + c)*= a 2 + b 2 + c 2 + 2bc + 2ca + 2ab (7), 

(a + b- c) 2 = a 2 + b 2 + c 2 -2bc-2ca + 2ab (8), 

&c. 
The ternary products of three letters, a, b, c, are a 3 , a 2 b, a 2 c, 
ab 3 , ac 2 , abc, b 3 , b 2 c, be 2 , c 3 . The enumeration is made more certain 
and systematic by first taking those in which a occurs thrice, 
then those in which it occurs twice, then those in which it occurs 
once, and, lastly, those in which it does not occur at all.* 

Bearing this in mind, the student, by following the method 
we are illustrating, Avill easily show that 

(a + b + c) 3 = (a + b + c) (a + b + c) (a + b + e), 

= a 3 +b 3 + c 3 + 3b 2 c + Uc 2 + 3c 2 a + 3ca 2 

+ 3a% + 3ab 2 + 6abc (9), 

from which again he may derive, by substituting (see chap, i., 
§ 24) - c for c on both sides, the expansion of (a + b - c) 3 , and so 
on. He should not neglect to verify these results by successive 
distributions, thus : — 

(a + b + c) 3 = (a + b + cf(a + b + c) 

= (a 2 + b 2 + c 2 + 2bc + 2ca + 2ab) (a + b + c\ 
= a 3 + ab 2 + ac 2 + 2abc + 2ca 2 + 2a 2 b 
+ a% + b 3 + be 2 + 2b 2 c + 2abc + 2ab 2 
+ ca 2 + b 2 c + c 3 + 2bc 2 + 2c 2 a + 2abc 
= &c. 

It is by such means that he must convince himself of the 
coherency of algebraical processes, and gain for himself taste and 
skill in the choice of his methods. 

* There is another way of classifying the products of a given degree which 
is even more important and which the student should notice, namely, according 
to type. All the terms that can be derived from one another by interchanges 
among the variables are said to be of the same type. For example, consider 
the ternary products of a, b, c. From a 3 we derive, by interchange of b and a, b 3 ; 
from this again, by interchange of b and c, <? : no more can be got in this way, 
so that a 3 , b 3 , c 3 form one ternary type ; b"c, be", ca, car, a"b, ab", form another 
ternary type ; and abc a third. Thus the ternary products of three variables 
fall into three types. 



IV S AND li NOTATIONS 53 

Let us consider one more case, namely, (b + c) (c + a) (a + b). 
Here even all the ten permissible ternary products of a, b, c cannot 
occur, for a 3 , b a , c 3 are excluded by the nature of the case, since 
a occurs in only two of the brackets, and the same is true of b 
and c. In fact, by the process of enumeration and counting of 
recurrences, we get 

(b + c)(c + a) (a + b) = be + b 2 c + ca 2 + c 2 a + ab 2 + a 2 b + 2abc (10). 

In the product (b - c) (c - a) (a - b) the term abc occurs twice 
with opposite signs, and there is a further reduction, namely, 

(b - c) (c - a) (a - b) = be 2 - b 2 c + ca 2 - c 2 a + ab 2 - a'b (11). 

2 Notation. — Instead of writing out at length the sum of all the terms of 
the same type, say bc + ca + ab, the abbreviation Hbc is often used ; that is to 
say, we write only one of the terms in question, and prefix the Greek letter 
2, which stands for "sum," or, more fully, "sum of all terms of the same 
type as." The exact meaning of 2 depends on the number of variables that 
are in question. For example, if there be only two variables, a and b, then 
"Lab means simply ab ; if there be four variables, a, b, c, d, then Hab means 
ab + ac + ad + bc + bd + cd. Again, if there be two variables, a, b, 2« 2 6 means 
a?b + ab~; if there be three, a, b, e, 1a?b means a 2 b + ab 2 + a*c + ac 2 + b 2 c + bc*. 
Usually the context shows how many variables are understood ; but, if this 
is not so, it may be indicated either by writing the variables under the 2, 
thus 2a6, or otherwise. 

abed 

This notation is much used in the higher mathematics, and will be found 
very useful in saving labour even in elementary work. For example, the 
results (4), (9), and (10) above may be written— 

{a + b) 3 = Za s + 3?,a-b; 
(a + b + c) 3 = 2<t 3 + 32a 2 6 + 6abc ; 
(b + c) (c + a) (a + b) = ~2,a-b + 2abc. 
By means of the ideas explained in the present article the reader should 
find no difficulty in establishing the following, which are generalisations of 
(1) and (9) :— 

(a + b + c + d+ . . . ) 2 = 2a 2 + 22«& (12), 

(a + b + c + d+ . . . ) 3 = 2« 3 + 32« 2 & + 62aZ>c (13), 

the number of variables being any whatever. 

IT Notation. — There is another abbreviative notation, closely allied to the 
one we have just been explaining, which is sometimes useful, and which often 
appears in Continental works. If we have a product of terms or functions of 
a given set of variables, which are all different, but of the same type (that is, 
derivable from each other by interchanges, see p. 52), this is contracted by 
writing only one of the terms or functions, and prefixing the Greek letter II, 
which stands for "product of all of the same type as." Thus, in the case of 
three variables, a, b, c, 



54 



PRINCIPLE OF SUBSTITUTION 



CHAP. 



lla-b means a"b x air x arc x ac 2 x b'c x be" ; 
H(b + c) means (b + c) {c + a) (a + b) ; 
ri fb + c\ ( b + c \ ( c + a \ ( a + b 

u {W+r?) means [FT?) [* + *) {*+-& 
and so on. 

"We might, for example, write (10) above — 

nib + c) = Zb 2 e + 2abc. 
§ 3.] Hitherto we have considered merely factors made up of 
letters preceded by the signs + and - . The case where they are 
affected by numerical coefficients is of course at once provided 
for by the principle of association. Or, what comes to the same 
thing, cases in which numerical coefficients occur can be derived 
by substitution from such as we have already considered. For 
example — 

(3a + 2b) 3 = {(3a) + (2b)} 3 

= (3a) 3 + 3(3a) 2 (2b) + 3(3a)(2b) 2 + (2b) 3 , 
whence, by rules already established for monomials, 

= 27 a 3 + 54a 2 b + 36ab 2 + 8b 3 . 
(a-2b + 5c) 2 = {(a) + (-2b) + (5c)} 2 

= (af + ( - 2b) 2 + (5c) 2 + 2( - 2b) (5c) + 2(5c)(a) + 2(a) ( - 2b) 
= a 2 + ib 2 + 25c 2 - 205c + lOca - iab. 

The student will observe that in the final result the general 
form by means of which this result was obtained has been lost, 
so far at least as the numerical coefficients are concerned. 

§ 4.] It is very important to notice that the principle of 
substitution may also be used to deduce results for trinomials 
from results already obtained for binomials. Thus from (a + b) 3 = 
a 3 + 3a 2 b + 3ab 2 + b 3 , replacing b throughout by b + c, we have 
{a + (b + c)} 3 = a 3 + 3a\b + c) + 3a(b + cf + (b + c) 3 
= a 3 + 3a 2 b + 3a 2 c 

+ 3a(b 2 + 2bc + c 2 ) 
+ b 3 + 3b 2 c + 3bc 2 + c 3 ; 
whence (a + b + c) 3 = a 3 + b 3 + c 3 + 3b 2 c + 3bc 2 + 3c 2 a + 3ca 2 

+ 3a 2 b + 3ab 2 + Qabc. 
By association of parts of the factors, and by partial distri- 
bution in the earlier parts of a reduction, labour may often be 
saved and elegance attained. 



iv SUMS OF COEFFICIENTS DO 

For example — 

{a + b + c - d) (a - b + c + d) 

= {(a+e) + (6 - d)} {(a + c) - (b - d)) ; 
= (a + c) 2 -(b-dy 2 , 

by formula (3) above ; 
= (a 2 + 2ac + c 2 ) - (b n - - 2bd + cP) ; 
=a?-P+(*-cP+2ac+2bd. 
Again, 

(a + b + c) (b + c - a) (c + a - b) (a + b - c) 

= {(b + c) + a}{(b + c)-a}{a-{b-c)}{a + (b-c)}; 
= {(b+c)*-a?}{a?-(b-c)*\, 

by a double application of formula (3) ; 
= Ji 2 + 26c + c" - a-} {a- - b 2 + 2bc - c 2 } ; 
= {2bc + {b"- + c" - a-)} {2bc - (b 2 + c 2 - a")} ; 
= {2bcf-{b 2 + c--a 2 )\ 

by formula (3); 
= 46V - (6 4 + c ' + a 4 + 26 2 c 2 - 2c V - 2a 2 6 2 ) ; 
= 26 2 c 2 + 2c 2 a 2 + 2a 2 6 2 - a* - ¥ - c\ 
a result which the student will meet with again. 

§ 5.] There is an important general theorem which follows 
so readily from the results established in §§ 1 and 2 that we may 
give it here. If all the terms in all the factors of a product be 
simple letters unaccompanied by numerical coefficients and all affected 
with the positive sign, then the sum of the coefficients in the distributed 
value of the product will be I x m x n x . . ., where I, m, n, . . . are 
the numbers of the terms in the respective factors. 

This follows at once from the consideration that no terms 
can be lost since all are positive, and that the numerical co- 
efficient of any term in the distribution is simply the number of 
times that that term occurs. 

Thus in formula? (4), (G), and (10) in § 2 above we have 

1+3 + 3 + 1 =2x2x2, 

1+4 + 6 + 4 + 1 =2x2x2x2, 
1 + 1 + 1 + 1 + 1 + 1 + 2 = 2x2x2, 

&c. 
In formulae (8) and (11) of § 2, and in the formula? of § 3, 
the theorem does not hold on account of the appearance of 
negative signs and numerical coefficients. 

The following more general theorem, which includes the one 
just stated as a particular case, will, however, always apply : — 
The algebraic sum of the coefficients in the expansion of any 



56 



EXERCISES V 



CHAP. 



product may be obtained from the product itself by replacing each of 
the variables by 1 throughout all the factors. 

Thus, in the case of 

(a + b - c) 2 = a 2 + b 2 + c 2 - 2bc - 2ca + 2ab, 
we have (1 + 1 - l) 2 = 1 = 1 + 1 + 1 - 2 - 2 + 2. 

The general proof of the theorem consists merely in this — 
that any algebraical identity is established for all values of its 
variables : so that we may give each of the variables the value 1. 
When this is done, the expanded side reduces simply to the 
algebraic sum of its coefficients. 

Exercises V. 
(1.) How many terms are there in the distributed product (a^+a^) 
(h + b 2 + b 3 ) ( Cl + 0., + % + c 4 ) (ih + d. 2 + d 3 + d 4 + d 5 ) ? 

Distribute, condense, and arrange the following : — 
(2.) (x+y){x-y)( a ?-y*)(a?+y*)*. 
(3.) ( x ^ + ,f){x i -f-)(x i + y i ). 
(4.) { x + y)\x-yf. 
(5.) (x + 2y)\x-2y)\ 

(6.) (b + c){c + a){a + b)(b-e)(c-a)(a-b). 
(7.) (x- + x + l) 3 . 
(8.) (3a + 26-l) 3 . 

(9.) ( x *+x + l + - + K 
\ X x- 

(10.) (a + b + cy, and {a-b-c)\ 

(11.) Write down all the quaternary products of the three letters x, y, z ; 
point out how many diiferent types they fall into, and how many products 
there are of each type. 

(12.) Do the same thing for the ternary products of the four letters a, b, c, d. 

(13.) Find the sum of the coefficients in the expansion of (2a + 3b + Ac) 3 . 

Distribute and condense the following, arranging terms of the same type 
together : — 

x . y z \ f x y 



(H.) ~ + -^- 



o - c c- a a- b 



b+c c+a 



a + bj 



(15.) [x + y + z)--x{y + z-x)-y(z + x-y)-z(x + y-z). 

(16. ) {b-c) {b + c- a) + {c- a) (c + a -b) + {a- b)(a + b-c). 

(17.) (b + c)(y + z) + (c + a)(z + x) + (a + b)(x + y)-(a + b + c)(x + y + z). 

(18.) 2u(b + c-a)m(b+c-a).* 



* "Wherever in this set of exercises the abbreviative symbols S and IT are 
used, it is understood that three letters only are involved. The student who 
finds difficulty with the latter part of this set of exercises, should postpone 
them until he has read the rest of this chapter. 



IV 



EXERCISES V 57 



Show that 

(19.) (x + yy = 2(.r- + y") (X + y) 2 - (.'." - y- )-. 
(20.) 
a i( x _ l)i _ 4a 3 b(x - a)(x - b) 3 + 6a"b"{x - a)-(x - b)- - ial?{x - af{x- b) + b*{x-a)* 
= (a* - ia 3 b + 6a"b- - iab 3 + 6*)a^. 
(21.) (x- - ay") (.,■>"- - ay'-) = {xx'±ayy'f - a{xy , ±yxf ; 
(a- 2 - ay-f = (x 3 + Zaxtff - a[Zxhj + ay 3 )' ; 
(x~ - By 2 - C; 2 + BCu-) (x' 2 - By' 2 - Cz' 2 + BGu'-) 
= {xx' + Byy'±C(zz' + Bmi')\ 2 -B{a'y' + x'y±C(uz' + u'z)} 2 

-Q{xz' -Byu'±{zx' -Buy')\ 2 + BC{yz' -xu'±{%ix' -zy')) 2 . 

Lagranrjc. 

The theorems (21.) are of great importance in the theory of numbers ; they 
show that the products and powers of numbers having a certain form are 
numbers of the same form. They are generalisations of the formula; numbered 
V. in the table at the end of this chapter. 

Distribute, condense, and arrange — 

(22.) 2a26c-II(6 + c). 

(23.) 2a(2a 2 + 26c) + 2r/2a 2 - 2(6 + cf. 

(24. ) (6 - e) (6 + cf + (c - a) (c + a) 3 + {a-b) (a + bf. 

(25.) Distribute 

{{a + b)x*-abxy + {a-b)y 2 ){{a-b)x 2 + abxy + (a + b)y 2 }; 

and arrange the result in the form 

Ax* + Bx?y + Cx 2 y 2 + Dxy 3 + Ey*. 

Show that 

(26. ) { 3? -y 3 + Bxy(2x + y)} 3 + { y 3 -x 3 + 3xy(2y + x)} 3 

=27xy{x + y) (x 2 + xy + y 1 ) 3 . 

(27.) Z{2(x 2 + xy + y 2 )(x 2 + xz + z 2 )-(y 2 + yz + z 2 ) 2 }=Z{Zyz} 2 . 

(28.) i|2a 2 (6 + c) 2 + 2a6c2a} = {26c} 2 . 

(29.) 2(a-6)(a-c)={2a 2 -26c}. 

. (3«6c - 26 3 - a 2 d) 2 + 4(ac - b 2 ) 3 _ (Mtb - 2c 3 - d 2 a) 2 + i( db - c 2 ) 3 
(30.) ^ -_ d . 2 



general theory of integral functions. 

§ 6.] As we have now made a beginning of the investigation 
of the properties of rational integral algebraical functions, it will 
be well to define precisely what is meant by this term. 

We have already (chap, ii., § 5) defined a rational integral 
algebraical term as the product of a number of positive integral 
powers of various letters, x, y, z, . . ., called the variables, multi- 
plied by a coefficient, which may be a positive or negative number, 
or a mere letter or function of a letter or letters, but must not 
contain or depend upon the variables. 



58 GENERAL NOTIONS REGARDING chap. 

A rational integral algebraical function is the algebraical sum of 
a series of rational integral algebraical terms. Thus, if x, y, z, ... be 
the variables, /, m, n, . . ., /', m', n', . . ., I", m", n", . . . positive in- 
tegral numbers, and C, C, C", . . . coefficients as above defined, 
then the type of such a function as we have defined is 

Cx l y m z n • ■• + Casty*'* 1 ' . . . + C"x l "y m "z n " ...+ &c. 
For shortness, we shall, when no ambiguity is to be feared, speak 
of it merely as an " integral function." 

To fix the notion, we give a few special examples. Thus 
(a) dx 3 + dxy + 2y 2 is an integral function of x and y ; 
(/3) ax 2 + bxy + cy-, a, b, c being independent of x and y, is an integral 

function of x and y ; 
(7) 3X 3 - 2.t 2 + 3x + 1 is an integral function of x alone ; 

OS If - % 

{5) - + T + - -1 is an integral function, if x, y, z be regarded as the 

a b c ° 

variables ; but is not an integral function if the variables be 
taken to be x, y, z, a, b, c, or a, b, c alone. 

Each term has a "degree," according to the definition of 
chap, ii., § 6, which is in fact the sum of the indices of the vari- 
ables. The decrees of the various terms will not in general be 
alike ; but the degree of an integral function is defined to be the 
degree of the term of highest degree that occurs in it. 

For example, the degree of (a) above in x and y is the 3rd, of (/3) the 
2nd in x and y and the 1st in a, b, c, of (7) in x the 3rd, of (5) in x, y, z the 1st. 

§ 7.] From what has already been shown in this chapter it 
appears that, in the result of the distribution of a product of any 
number of integral functions, each term arises as the product of 
a number of integral terms, and is therefore itself integral. 
Moreover, by chap, ii., § 7, the degree of each such term is the 
sum of the degrees of the terms from which it arises. Hence 
the following general propositions : — 

The product of any number of integral functions is an integral 
function. 

The highest * term in the distributed product is the product of the 

* By " highest term " is meant term of highest degree, by " lowest term " 
term of lowest degree. If there be a term which does not contain the vari- 
ables at all, its degree is said to be zero, and it of course would be the lowest 
term in an integral function, for example, +1 in (7) above. 



IV 



INTEGRAL FUNCTIONS 59 



highest terms of the several factors, and the lowest term is the product 
of their lowest terms. 

The degree of the product of a number of integral functions is the 
sum of tlie degrees of the several factors. 

Every identity already given in this chapter, and all those 
that follow, will afford the student the means of verifying these 
propositions in particular cases. It is therefore needless to do, 
more than call his attention to their importance. They form, it 
may he said, the corner-stones of the theory of algebraic forms. 

INTEGRAL FUNCTIONS OF ONE VARIABLE. 

§ 8.] The simplest case of an integral function is that where 
there is only one variable x, As this case is of great importance, 
we shall consider it at some length. The general type is 

2W n +Pn-iX n ~ 1 + . . . +p v r.+p , 
where p , p lf . . ., p n are the various coefficients and n is a posi- 
tive integral number, which, being the index of the highest term, 
is the degree of the function. The function has in general n + 1 
terms, but of course some of these may be wanting, or, which 
amounts to the same thing, one or more of the letters p ,p u . . .,p n 
may have zero value. 

§ 9.] When products of integral functions of one variable 
have to be distributed, it is usually required at the same time to 
arrange the result according to powers of as, as in the tj-pical 
form above indicated. We proceed to give various instances of 
this process, using in the first place the method described in 
the earlier part of this chapter. The student should exercise 
himself by obtaining the same results by successive distribution 
or otherwise. 

In the case of two factors (x + a) (x + b), we see at once that 
the highest term is x 2 , and the lowest ab. A term in x will be 
obtained in two ways, namely, ax and bx ; hence 

(x + a) (x + b) = x 2 + (a + b)x + ab (1). 

This virtually includes all possible cases ; for example, putting - a for a 

we get 

(x + ( - a)) {x + b) =x- + (( - a) + b)x + ( - a)b, 

= x 2 + {-a + b)x - ab. 



60 DISTRIBUTION OF (x - ff x ) (x - C( 2 ) . . . (x - a n ) CHAP. 

Similarly (a: - a) (x - b) = or + ( - a - b)x + ab, 

= o? -(a + b)x + ab. 
(x - a) (x - a) = o;" + ( - a t a)x+ a 2 , 
= or - lax + a", &c. 
Cases in which numbers occur in place of a and b, or in which x is affected 
with coefficients in the two factors, may be deduced by specialisation or other 
modification of formula (1), for example, 

U'-2)(.r+3)=.r 2 + (-2 + 3)z + (-2)( + 3), 
= or + x-6. 



(px + q) (rx + s) =p ( x + 



«\-'*+i 



=prx~+pri - + -)x+pr — , 
=prx 2 + (rq +ps)x + qs, 

which might of course be obtained more quickly by directly distributing the 
product and collecting the pow y ers of x. 

In the case of three factors of the first degree, say (x + a x ) 
(x +- a 2 ) (x + a 3 ), the highest term is .r 3 ; terms in x 2 are obtained 
by taking for the partial products x from two of the three brackets 
only, then an a must be taken from the remaining bracket ; we 
thus get a x x 2 , a 2 x 2 , a^z 2 ; that is, {a x + a B + a 3 )x 2 is the term in x 2 . 
To get the term in x, x must be taken from one bracket, and a's 
from the two remaining in every possible way ; this gives 
(a^a 2 + a,{a z + a 2 a 3 )x. The last or absolute term is of course a^a^ 
Thus (x + rtj) (x + a.,) (x + a 3 ) 

= x 3 + (a, + a 2 + a 3 )x 2 + (rt,a 2 + a,a 3 + a 2 a 3 )x + a l a i a 3 (2). 

By substitution all other cases may be derived from (2), for 
example, 

(x - a,) (x - a 2 ) (x - a 3 ) 

= x 3 - (a, + a 2 + a 3 )x 2 + (a a a a + a,a 3 + a 2 a 3 )x - a x a 2 a 3 (3) ; 
(x + 1) (x + 2) (x - 3) = x - 7x - G, and so on. 

After what has been said it is easy to find the form of the dis- 
tribution of a product of n factors of the first degree. The result is 

(x + «,) (x + a 2 ) . . . (x + a n ) 

= x n + P.a:' 1 - 1 + P^'- 2 + . . . + P„_,a; + P n (4), 



IV 



BINOMIAL THEOREM 61 



where P x signifies the algebraic sum of all the a's, P 2 the alge- 
braic sura of all the products that can be formed by taking two 
of them at a time, P 3 the sum of all the products three at a time, 
and so on, P n being the product of them all. 

§ 10.] The formula (4) of § 9 of course includes (1) and (2) 
already given, and there is no difficulty in adapting it to special 
cases where negative signs, &c, occur. The following is par- 
ticularly important : — 

= X n - P,':"- 1 + P^- 2 -... + (- l)"- 1 ^-!* + ( - l)»P n (I)- 

Here T t P 2 , &c, have a slightly different meaning from that 
attached to them in § 9 (4) : P 3 , for example, is not the sum 
of all the products of -a 1} - a 2 , . . ., - a n , taken three at a 
time, but the sum of the products of + a 1} + a 2 , . . ., + a n , taken 
three at a time; and the coefficient of a; n ". 3 is therefore - P 3 , 
since the concurrence of three negative signs gives a negative 
sign. As a special case of (1) let us take 

(x - a) (x - 2a)(x - Sa)(x - 4a) = x* - P r x 3 + F 2 x 2 - P^ + P 4 . 

Here P, = a + 2a + 3a + 4a = 10a, 

P 2 = 1 x 2a 8 + 1 x 3a" + 1 x 4a 2 + 2 x 3« 2 + 2 x 4a 2 + 3 x 4a 2 

= 35 a 2 , 
P 3 - 2 x 3 x 4a 3 + 1 x 3 x 4a + 1 x 2 x 4a 3 + 1 x 2 x 3a 3 

- 50a 3 , 
P 4 = 1 x 2 x 3 x 4a' = 24a 4 . 

So that (x - a) (x - 2a) (x - 3a) (x - 4a) 

- x - Wax 3 + 35a 2 / - 50a 3 * + 24«\ 

§ 11.] Another important case of § 9 (4) is obtained by 
making a l = a 2 = a 3 = . . . = a n , each = a say. The left-hand side 
then becomes (x + a) n . Let us see what the values of P n P 2 , . . ., 
P n become. Pj obviously becomes na, and P n becomes a n . Con- 
sider any other, say P r ; the number of terms in it is the number 
of different sets of r things that Ave can choose out of n things. 
This number is, of course, independent of the nature of the 
things chosen ; and, although we have no means as yet of calcu- 
lating it, we may give it a name. The symbol generally in use 



62 BINOMIAL COEFFICIENTS chap. 

for it is n C r , the first suffix denoting the number of things chosen 
from, the second the number of things to be chosen. Again, 
each term of P r consists of the product of r letters, and, since in 
the present case each of these is a, each term will be a r . All 
the terms being equal, and there being n C r of them, we have in 
the present case P r = n C/t r . Hence 

(x + a) n = x n + naz n - 1 + n C 2 a 2 x n ~ 2 + n Q 3 a z x n ~ z + . . . + a n ; 

or, if we choose, since n C\ — n > rfin = 1, we may write 

(x + a) n = x n + nC^x 71 ' 1 + n C 2 a 2 x n ~ 2 + ... + n O n . l cfl- 1 x + n C n a n (1). 

This is the " binomial theorem " for positive integral exponents, and 
the numbers n C M n C 2 , n C 3 , . . . are called the binomial coefficients of 
the nth order. They play an important part in algebra j in fact, 
the student has already seen that, besides their function in the 
binomial expansion, they answer a series of questions in the 
theory of combinations. When we come to treat that subject 
more particularly we shall investigate a direct expression for n C r 
in terms of n and r. Later in this chapter we shall give a pro- 
cess for calculating the coefficients of the different orders by 
successive additions. 

By substituting successively -a, +1, and - 1 for a in (1) 
we get 

(x - a) n = x n - n G x ax n ~ 1 + n C 2 a 2 x n - 2 - n G 3 a 3 x n - 3 + . . . 

+ ( - 1)W W (2) ; 

(x+l) n = x n + n C 1 x n - l + n C.p: n - 2 + . . .+„C n (3); 

(x-l) n = x n - n C 1 x n - 1 + n CfP-*-. . . + (-l)\C n (4); 

and an infinity of other results can of course be obtained by 
substituting various values for x and a. 

§ 12.] In expanding and arranging products of two integral 
functions of one variable, the process which is sometimes called 
the long rule for multiplication is often convenient. It consists 
simply in taking one of the functions arranged according to 
descending powers of the variable and multiplying it successively 
by each of the terms of the other, beginning with the highest 
and proceeding to the lowest, arranging the like terms under 
one another. Thus we arrange the distribution of 



iv LONG MULTIPLICATION 63 

(x a +2x 2 +2x+l)(x 2 -x+l) 
as follows : — a? + 2x 2 + 2x + 1 

x 2 - x + 1 



x + 2x* + 2x a + of 
- x 4 - 2/ - 2x 2 - x 

+ z* + 2x 2 + 2x + 1 





x* + 


x* + x 3 + 


x 2 


+ 


x + 


1 




or again 


(px* 


+ qx + r)(rx 2 


+ 


qx+p) 






px 2 


+ qx 


+ r 












rx* 


+ qx 


+ p 












prx* 


+ qrx 3 
+ pqx 3 


2 2 

+ r x 

2 2 

+ qx 








+ qrx 








+ px 








+ pqx 


+ pr 



prx K + (pq + qr)x 3 + (jf + q 2 + r 2 )x 2 + (pq + qr)x + pr. 

The advantage of tins scheme consists merely in the fact that 
like powers of x are placed in the same vertical column, and that 
there is an orderly exhaustion of the partial products, so that 
none are likely to be missed. It possesses none of the funda- 
mental importance which might be suggested by its prominent 
position in English elementary text-books. 

§ 13.] Method of Detached Coefficients. — When all the powers 
are present a good deal of labour may be saved by merely 
writing the coefficients in the scheme of § 12, which are to be 
multiplied together in the ordinary way. The powers of x can 
be inserted at the end of the operation, for we know that the 
highest power in the product is the product of the highest powers 
in the two factors, and the rest follow in order. Thus we may 
arrange the two multiplications given above as follows : — 

1+2+2+1 
1-1 + 1 

1+2+2+1 
-1-2-2-1 

+1+2+2+1 

1+1+1+1+1+1; 



61 



DETACHED COEFFICIENTS 



CHA1". 



•whence 

(x 3 + 2.c 2 + 2x + l)(x 2 - x + 1) = X s + x* + x 3 + %• + x + 1. 
Again, 



p 


+ 2 


+ r 


r 


+ 1 


+ 1> 


pr 


+ qr 


+ r~ 




+ M 





+ qr 

+ pq + pr 



pr + (qr + pq) + (p 2 + q 2 + r 2 ) + (pq + qr) + pr ; 
whence 

(px 2 + qx + r)(rx 2 + qx +p) 

- prx* + (pq + qr)x 3 + (p 2 + q 2 + r 2 )x~ + (pq + qr)x + pr. 

The student should observe that the use of brackets in the 
last line of the scheme in the second example is necessary to 
preserve the identity of the several coefficients. 

It has been said that this method is applicable directly only 
when all the powers are present in both factors, but it can be 
made applicable to cases where any powers of x are wanting by 
introducing these powers multiplied by zero coefficients. For 
example — 

(x i -2x"+l)(x i + 2x 2 +l) 

= (%* + Ox 3 - 2x 2 + Ox + l)(x* + Ox 3 + 2x 2 + Ox + 1) ; 

1+0-2+0+1 
1+0+2+0+1 



whence 



1+0-2+0+1 
+0+0+0+0+0* 
+2+0-4+0+2 

+0+0+0+0+0* 
+1+0-2+0+1 

1 +0+0+0-2+0+0+0+1 

x* + O.c 7 + 0x° + 0/ - 2x* + 0x a + 0/ + Ox + 1 ; 

(x* - 2x 2 + 1)(^ 4 + 2x 2 + 1) = x - 2x* + 1. 



iv DETACHED COEFFICIENTS 65 

The process might, of course, be abbreviated by omitting the 
lines marked *, which contain only zeros, care being taken to 
place the commencement of the following lines in the proper 
columns ; and, in writing out the result, the terms with zero 
coefficients might be omitted at once. With all these simplifica- 
tions, the process in the present case is still inferior in brevity 
to the following, which depends on the use of the identities 
(A + B) (A - B) = A 2 - B 2 , and (A + B) 2 = A 2 + 2AB + B 2 . 

(x* - 2x* + 1) (x* + 2x* + 1) = {(x* + 1) - 2x*}{(x< + 1) + 2x 2 } 

= (x* + I) 2 - (2xJ 
= x + 2x* + 1 - 4a; 4 
= x*-2x*+l. 

The method of detached coefficients can be applied with ad- 
vantage in the case of integral functions of two letters which are 
homogeneous (see below, § 17), as will be seen by the following 
example : — 

(J - xy + f) (x 3 - 2x*y + 2xf - f), 
1-2+2-1 
1-1 + 1 



1-2+2-1 
-14-2-2 + 1 
+1-2+2-1 

1-3 + 5-5 + 3-1, 
= x" - Sx*y + Ssfy* - bx*y* + 2>x>f - y. 

If the student Avill work out the above distribution, arrange 
his work after the pattern of the long rule, and then compare, 
he will at once see that the above scheme represents all the 
essential detail required for calculating the coefficients. 

The reason of the applicability of the process is simply that 
the powers of x diminish by unity from left to right, and the 
powers of y in like manner from right to left. 

We shall give some further examples of the method of 
detached coefficients, by using it to establish several important 
results. 

VOL. ] F 



66 PASCAL'S ARITHMETICAL TRIANGLE chap. 

§ 14.] Addition Rule for calculating the Binomial Coefficients. 
We have to expand (x + l) 2 , (x + l) 3 , . . ., (x + l) n . Let us 
proceed by successive distribution, using detached coefficients. 

1 + 1 (The coefficients of x + 1 ), 

1 + 1 



1 + 1 

+ 1 + 1 

1 + 2 + 1 (The coefficients of (x + l) 2 ), 

1 + 1 



1 + 2 + 1 

+1+2+1 

1 + 3 + 3+1 (The coefficients of (x + l) 3 ). 

The rule which here becomes apparent is as follows : — 
To obtain the binomial coefficients of any order from those of the 
previous order — 1st, Write down the first coefficient of the previous order; 
2nd, Add the second of the previous order to the first of the same ; 
3rd, Add the third of the previous order to the second of the same ; 
and so on, taking zeros when the coefficients of the previous order run 
out. We thus get in succession the first, second, third, &c, coefficients 
of the new order. For example, those of the fourth order are 

1 + (1 + 3) + (3 + 3) + (3 + 1) + (1 + 0), 
that is, 1+4 +6 +4 +1, 

which agrees with the result obtained by a different method 
above, § 2 (6). 

We have only to show that this process is general. Suppose 
we had obtained the expansion of (x + l) n , namely, using the nota- 
tion of § 11, 

(., + l)» = x n + n C lX n-l + n C 2 .^- 2 + n C 3 .^- 3 + . . . + n C»_ r T + n C n . 
Hence 
(z+ l) n + 1 = (a;+ l)' l x(.c+ 1) 

= (^ + n C l .r' l - 1 + n C li ^- 2 + . • .+*C n _ 1 a> + n C n )(a;+l) 



IV PASCAL'S TRIANGLE GENERALISED 67 

using detached coefficients, we have the scheme 

1 + n *~>i + nv-'g + n^3 + • • • + tv^n-i "i~ iv^n 
1 + 1 

I + »^i + n\- -2 + 7V~ n 

1 + (1 + n C,) + ( U C, + M C 2 ) +....+ ( H C n _, + n C„) + ( n G n + 0). 
Hence (s+l)»+ 1 

= af+» + (1 + W C>" + (.0, + „cgaf»-» + („C, + »C,)*»- a + . . ., 
in which the coefficients are formed from the coefficients of the 
nth. order, precisely after the law stated above, namely, 

This law is therefore general, and enables us whenever we 
know the binomial coefficients of any rank to calculate those of 
the next, from these again those of the next, and so on. A 
table of these numbers (often called Pascal's Triangle) carried to 
a considerable extent is given at the end of this chapter, among 
the results and formulae collected for reference there. 

§ 15.] "We may calculate the powers of x 3 + x z + x + 1 by 
means of the following scheme, in which the lines of coefficients 
of the constantly-recurring multiplier, namely, 1 + 1 + 1 + 1, are 
for brevity omitted. 



Tower. 




























1st. 


1 


+ 1 


+ 


1 


+ 


1 




















+ 1 


+ 


1 


+ 


1 


+ 


1 


















+ 


1 


+ 


1 


+ 


1 


+ 


1 










1 


+ 2 






+ 


1 


+ 


1 


+ 


1 + 


1 
1 






2nd. 


+ 


3 


+ 


4 


+ 


3 


+ 


2 + 








+ 1 


+ 


2 


+ 


3 


+ 


4 


+ 


3 + 


2 + 


1 










+ 


1 


+ 


2 


+ 


3 


+ 


4 + 


3 + 


2+ 1 






1 








+ 


1 


+ 


2 


+ 


3 + 


4 + 


3+ 2+ 1 




3rd. 


+ 3 


+ 


6 


+ 


10 


+ 


12 


+ 


12 + 


10 + 


6+3+1 








+ 1 


+ 


3 


+ 





+ 


10 


+ 


12 + 


12 + 


10+ 6+ 3 + 


1 








+ 


1 


+ 


3 


+ 


6 


+ 


10 + 


12 + 


12 + 10+ 6 + 


3 + 1 












+ 


1 


+ 


o 


+ 


6 + 


10 + 


12 + 12 + 10 + 


6 + 3 + 1 



4th. 1 + 4 + 10 + 20 + 31 + 40 + 44 + 40 + 31 + 20+10 + 4+1 

and so on. 



68 X n ± if AS A PRODUCT chap. 

The rule clearly is— Jo get from the coefficients of any order 
the rth of the succeeding, add to the rth of that order the three preced- 
ing coefficients, taking zeros when the coefficients required by the ride do 

not exist. 

The rule for calculating the coefficients of the powers of 
x n + x 71 ' 1 + x 1l ~ 2 + . . .+x+l is obtained from the above by 
putting n in place of 3. 

These results may be regarded as a generalisation of the pro- 
cess of tabulating the binomial coefficients. They are useful in 
the Theory of Probability. 

§ 16.] As the student will easily verify, we have 

(x-y)(x 2 + xy + f) = x 3 -y 3 (1), 

(as + y) (x 2 - xy + f) = x 3 + y 3 (2). 

The following is a generalisation of the first of these : — 
If n be any integer, 

(a; - y) (x n - 1 + x n ~ % y + x n ~ 3 y 2 + . . . + xy n ~ 2 + y n ~ a ), 

1 + 1 + 1 +. . .+ 1 + 1 
1-1 

1 + 1 + 1 +. . . + 1 + 1 
-1-1-. . .-1-1-1 



1 + + + . . . + + 0-1 

- x n - y n (3). 

Again, n being an odd number, 

(x + y) (x n ~ 1 - x n ~hj + x n - 3 y 2 -- • • ~ Xl f " 2 + V n ' *)> 
( - sign going with odd powers of y) 
1 - 1 + 1 - . . . - 1 + 1 
1 + 1 



1-1+1-. . .-1+1 
+ 1-1+. . . + 1-1 + 1 

1 + + + . . . + + 0+1 

■ x tl + y n (4). 



iv EXERCISES VI 69 

And, similarly, n being an even number, 

(x + y) (x n ~ l - x n ~ 2 y + x n ~ hf - . . . 4 xf ' 2 - if ~ l ) 

= x n -if (5). 

The last two may be considered as generalisations of (2) and of 
(x + if) (x - y) = x 2 - if respectively. 

Exercises VI. 

(1. ) The variables being x, y, z, point out the integral functions among the 
following, and state their degree : — 

(a) Zx"- + 2xy+Sy~ ; 
, m 3 2 3 
x- xy y- 
(7) a?yz + y-zx + z"xy + x 3 + y 3 + z 3 ; 

.„ x-y-z* a?}/ 2 * oW 
X'yz xyz xyz- 

Distribute the following, and arrange according to powers of x : — 

(2.) 6{aj-J(a ! -l)}{a!-S(a!-l)}+20{a!-|(a!-l)}{ a ;-|(a;-l)}. 

x(x + l)(x + 3) a;(a;+l)(2a:+l) 
(3.) g . 

(4.) {(aj-2)(a:-3)+<a;-3)(a;-l) + (a:-l)(a!-2)} 

x {(a;+2)(a;+3) + (a;+3J(a;+l) + (a:+l)(ar+2)}. 

(5.) {x + a} {x- + (b + c)x + bc} {x 3 - (a + b + c)x" + (be + ca + ab)x - abc] . 
(6.) {(x+p)(x-q)(x + l)}{(x-p)(x + q)(x-l)}. 
(7. ) (z 2 - y") (x* - 2f) (a? - 3/) (a? - 4tf) (a? - 5y- ). 

(8.) {az + (6-c)y} {te + (c-a)?/} {<.-.>■ + («-%} ;* and show that the 
sum of the coefficients of x-y and y 3 is zero. 

(9.) Show that 

(x + 4«) 4 - 1 Oa(x + £«) 3 + 35a 2 (.r + £a) 2 - 50a 3 (a; + |a) + 24a 4 
= (x 2 -i« 2 )(ar-|a 2 ). 
(10.) Show that 

(- + ^)(-+^)(- + fl/)-(- + ^)(- + ?2/)(- + ^ 
_ a;y(a;-2/)(g-r)(r-jp)(p- g) 

Distribute and arrange according to powers of x, the following : — 

(11.) {(i+c)a! s +(c+a)aj+(a+J)}-{(&-c)ar J +(c-a)a;+(ffl-&)}. 
(12.) (x n '-x + l){x 2 + x + l)(x' 1 -2x + l){x i + 2x + l). 



* In working some of these exercises the student will find it convenient to 
refer to the table of identities given at the end of this chapter. 



70 EXERCISES VI, VII chap. 

(13.)-! 5x 2 - ix(x -y) + {x -y) 2 } (2s + By). 
(14. ) (2x 2 - Zxy + 2y 2 ) (2x 2 + Bxy + 2if). 
(15.) {(x-+x+l)(x--x + l)(x--l)}\ 
(16.) (x 3 -x~ + x-l)-{x* + x~ + x+l) 2 . 

(17.) {lx*-fr? + lx+%)(ix*+fr?-hx+i)- 

(18.) (x* - ax*y + abx-y" + bxy 3 + y*) (ax 2 - abxy + by"). 

(19.) (x 2 + ax + b 2 ) 3 + (x 2 + ax- b 2 ) 3 + (x 2 - ax + b 2 ) 3 + (x 2 -ax- b 2 ) 3 . 

(20.) (x i -2a 2 x 2 + a*)\ 

(21.) (.r 4 -a 5 ) 3 . 

(22.) (3*+|)'. 

(23.) (a + bx 2 )*. 

(24.) {(^ + 2/ 3 )(^-2/ 3 )} 9 . 

(25.) (l+z + a: 2 + a; 3 + ai 4 ) 3 . 

(26.) Calculate the coefficient of a; 4 in the expansion of (1+x + x 2 ) 8 . 
(27. ) Calculate the coefficient of X s in (1 - 2x + 3x 2 + 4x?- x A )\ 
(28.) Show that 

{a + b) 3 (a 5 + b 5 ) + 5ab(a + b) 2 (a 4 + 6 4 ) + I5a 2 b 2 (a + b) {a 3 + b 3 ) 
+ B5a 3 b 3 (a 2 + b 2 ) + 70a i b i =(a + b)\ 
(29.) Show that 

«Ci + n C 2 + „C 3 + . . . +„C„ = 2"-1; 
1 + «Co + n Ci + . . . = »Cj + n C-3 + n Cs + . . . ; 

(30. ) There are five boxes each containing five counters marked with the 
numbers 0, 1, 2, 3, 4 ; a counter is drawn from each of the boxes and the 
numbers drawn are added together. In how many different ways can the 
drawing be made so that the sum of the numbers shall be 8 ? 

(31.) Show that 
(x-y) 2 (x n - 2 + x n - 3 y+ . . . +xy n - 3 + y n - 2 ) = x n -x n - l y-xy n - 1 + ij n . 



Exercises VII. 

Distribute the following, and arrange according to descending powers 
of x: — 

(1.) (3.i' + 4)(4;c + 5)(5;e + 6)(6a; + 7). 

(2. ) (px + q-r) (qx + r-p) (rx +p - q). 

(3. ) [x - a) (x - 2a) [x - 3a) {x - 4a) (x + a) (x + 2a) (x + 3a) (x + 4a). 

(4. ) (x 3 + 2,x 2 + 2x + 1) (or 3 - 3.x- 2 + Bx - 1 ). 

(5. ) (ix 3 + \x 2 + \x + f ) (Ix 3 + \x 2 + kx + 1 ). 

(6.) (x-#{x*-ix+l)(x+lHx*+fr+i). 

7. l-x 2 + -x+ 7 ) (-x 2 + 1 x + - -JX- + -X+-). 

(8.) (2aj-3)» 

(9.) {{x+y)(x*-xy+y*)}*. 

(10.) (a; 2 -l) 4 (u; + l) 10 . 



iv HOMOGENEOUS FUNCTIONS 71 

(11.) In the product (x + a)(x+b)(z+e), -r- disappears, and in the product 
(x- a)(x+b)(x + c), x disappears; also the coefficient of X in the former is 
equal to the coefficient of a; 2 in the latter. Show that a is either or 1. 

Prove the following identities : — 

(12.) (b-c)(x-a) 2 +(c-a){x-b) 2 +(a-b){x-c) 2 + (b-c)(c-a){a-b) = 0. 

(13. ) 2(2a - b - c) (h - 6) (h -c) = 2(6 - c)-{h - a). 

(14.) (s-af+(s-b) 3 + (s-c)* + 3abc = s 3 , 
where 2s = a + b + c. 

(15.) (s-a)*+(8-b) 4 +(8-c)* 

= 2(s - b)%s - c) 2 + 2(s - c) 2 (s - a) 2 + 2(s - a) 2 (s - b) 2 , 
where 3s = a + b + c. 

(16.) (as+bc)(bs+ ca) (cs + ab) = (b + c) 2 (c + a) 2 (a + 6) 2 , where s - a + b + c. 

(17 '. ) s(s - a - d)(s - d-b)(s - c - d) = (s - a)(s -b)(s - c) {s - d) - abed, 
where 2s = a + b + c + d. 

(18. ) 16(8 - a) (s -b)(s- c) (s - d) = 4(6c + ad) 2 - (6 2 + c 2 - a 2 - d 2 ) 2 , 
where 2s=a+b + c+d. 

(19. ) 2(6 - c) 6 =3n(i - cf + 2(2« 2 - 26c) 3 . 

(20.) If U„ = (6- c) n + (c -«)" + (« -6)», then 

U n+3 - (a 2 + b 2 + c 2 -bc-ca- ab)V n+1 - (b - e) (e - a) (a - 5)U„= 0. 
(21.) If pi = a + b + c, p 2 = bc + ca + ab, p 3 = abc, s n = a n + b n + c n , show that 
Si =lh , s 2 -piSi - 2p- 2 , s s =piS- 2 - paSi + 3p 3 , 

S» =PlS„-i -p-fin-2 +PsS n -3- 

(22. ) If j)-2= (b -c)(e-a) + (e- a) (a - 6) + (a - 6) (6 - c), 
2i 3 =(b-c){c-a){a-b), 
s H = (b- c) n + (c- a) n + {a- 6)» 
show that 

*2 = - S^pa , «3 = 3ps , Si = 2ps?, s s = - 5p»ps, 
s fi = - 2pa 8 f 8pg 8 , s 7 = 7p- 2 ps 5 20S7S3 = 21s 5 2 . 



Homogeneity. 

§ 17.] An integral function of any number of variables is said 
to be " Homogeneous " when Vie degree of every term in it is the same. 
In such a function the degree of the function (§ G) is of course 
the same as the degree of every terra, and the number of terras 
which (in the most general case) it can have is the number of 
different products of the given degree that can he formed with 
the given number of variables. If there be only two variables, 
and the degree be n, we have seen that the number of possible 
terms is n + 1. 



72 HOMOGENEOUS FUNCTIONS chap. 

For example, the most general homogeneous integral functions of x and y 
of the 1st, 2nd, and 3rd degrees are * 

Az+By (1), 

Ax 2 + Bxy + Cy 2 (2), 

Ax 3 + Bx 2 y + Cxy 2 + Vy 3 ( 3), 

A, B, C, &c. , representing the coefficients as usual. 
For three variables the corresponding functions are 

Ax + By + Cz (4), 

A* 2 + By 1 + Cz 2 + Vyz + Esse +¥xy (5), 

Ax 3 + Bif + Cz 3 + Vyz 2 + P'tfz + Qzx 2 + Q,'z 2 x + Bxy 2 + Wx 2 y + Sxyz ( 6 ), 

&c, 

As the case of three variables is of considerable importance, we shall in- 
vestigate an expression for the number of terms when the degree is n, 

We may classify them into — 1st, those that do not contain x ; 2nd, those 
that contain x ; 3rd, those that contain x 2 ; . . . ; n + lth, those that contain x n . 

The first set will simply be the terms of the ?ith degree made up with 
y and z, n + 1 in number ; the second set will be the terms of the (?i — l)th 
degree made up with y and z, n in number, each with x thrown in ; the third 
set the terms in y and 2 of («-2)th degree, n-l in number, each with x 2 
thrown in ; and so on. Hence, if N denote the whole number of terms, 

N = (7!+l)+?l + (?l-l)+ . . .+2 +1. 

Reversing the right-hand side, we may write 

N= 1 +2+ 3 + . . . +n + (n + l). 

Now, adding the two left-hand and the two right-hand sides of these equali- 
ties, we get 

2N=(»+2) + (»+2) + (» + 2)+. . . +(h + 2) + (« + 2); 

= (n + l){n + 2), 

since there are n + 1 terms each =n + 2. 
Whence N=4(»+l)(»+2). 

For example, let n = 3 ; N=£(3+ 1) (3 + 2)= 10, which is in fact the number 
of terms in (6), above. 

In the above investigation we have been led incidentally to sum an 
arithmetical series (see chap, xx.) ; if we attempted the same problem for 4, 
5, . . ., 7n variables, we should have to deal with more and more complicated 
series. A complete solution for a function of the ?i.th degree in m variables 
will be given in the second part of this work. 

* Homogeneous integral functions are called binary, ternary, &c, accord- 
ing as the number of variables is 2, 3, &c. ; and quadric, cubic, &c, according 
as the degree is 2, 3, &c. Thus (3) would be called a binary cubic ; (5) a 
ternary quadric ; and so on. 



IV HOMOGENEOUS FUNCTIONS 73 

The following is a fundamental property of homogeneous 
functions : — If each of the variables in a homogeneous function of the 
nth degree be multiplied by the same quantity p, the result is the same 
as if the function itself were multiplied by p n . 

Let us consider, for simplicity, the case of three variables ; 
and let 

F = kxPtftf + A'«py tf' + . . ., 

where p + q + r =p' + q + r' = &c, each = n. 

If we multiply x, y, z each by p, we have 

F = A(px)*(py)i(pzy + A'(pr)P'(pyy( P :y + . . . } 
= ApP+<i+ r xPy ( }z r + A'pP'+<i'+ r 'xP'f'z r '+ . . ., 

by the laws of indices. Hence, since p + q + r = p + q + r' = &c. 
= n, we have 

F' = p n { AxPy?z r + A'xP'y<i'z r ' +...}, 

= p n F, 

which establishes the proposition in the present case. The 
reasoning is clearly general.* 



* This property might be made the definition of a homogeneous function. 
Thus we might define a homogeneous function to be such that, when each 
of its variables is multiplied by p, its value is multiplied by p n ; and define n 
to be its degree. If we proceed thus, we naturally arrive at the idea of homo- 
geneous functions which are not integral or even rational ; and we extend the 
notion of degree in a corresponding way. For example, (.c 3 - y 3 )/(x + y) is 
a homogeneous function of the 2nd degree, for ( (px) 3 - {py) 3 )j{ (px) + (py) ) 
= p i (x 3 - y 3 )/(x + y). Similarly \Z(- t ' 3 + V 3 )' l/(* 2 + V") are homogeneous functions, 
whose degrees are f and -2 respectively (see chap, x.) Although these ex- 
tensions of the notions of homogeneity and degree have not the importance of 
the simpler cases discussed in the text, they are occasionally useful. The 
distinction of homogeneous functions as a separate class is made by Euler in 
his Introductio in Analysin fnfinitorum (1748), (t. i. chap, v.), in the course 
of an elementary classification of the various kinds of analytical functions. 
He there speaks, not only of homogeneous integral functions, but also of 
homogeneous fractional functions, and of homogeneous functions of fractional 
or negative degrees. 



74 LAW OF HOMOGENEITY chap. 

Example. 

Consider the homogeneous integral function 3.>j 2 - 2xy + y 2 , of the 2nd 
degree. We have 

3(/w) 2 - 2( P x) ( P y) + (p2/) 2 = 3pV - 2p"xy + p*y*, 
= P *(dx i -2vy+y 2 )> 
in accordance with the theorem above stated. 

The following property is characteristic of homogeneous integral 
functions of the first degree. 

If for the variables x, y, z, . . . toe substitute Xx, + /j..r 2 , ky x 
+ ay at Xz x + fxz 2 , . . . respectively, the result is the same as that 
obtained by adding the results of substituting x„-y Xi z lt . . . and x 2 , 
y,, z 2 , . . . respectively for x, y, z, . . . in the function, after multi- 
plying these results by X and fi respectively. 

Example. 

Consider the function Ax + By + Cz. 

We have 

A(X.ri + fix.,) + B(\2/! + ^ 2 ) + C(Xzi + ftna) 

= AXa'x + BXyi + C\Sj + Afix? + Bp.y- 2 + Cfiz* 
= \( Aa?i + Byx + Czi) + fi{Ax. 2 + By- 2 + Cz»). 

This property is of great importance in Analytical Geometry. 

§ 18.] Law of Homogeneity. — Since every term in the product 
of two homogeneous functions of the mth and ?ith degrees re- 
spectively is the product of a term (of the mth degree) taken 
from one function and a term (of the «th degree) taken from 
the other, we have the following important law : — 

The product of two homogeneous integral functions, of the mth and 
nth degrees respectively, is a homogeneous integral function of the 
(m + n)th degree, 

The student should never fail to use this rule to test the 
distribution of a product of homogeneous functions. If he finds 
any term in his result of a higher or lower degree than that 
indicated by the rule, he has certainly made some mistake. He 
should also see whether all possible terms of the right degree are 
present, and satisfy himself that, if any are wanting, it is owing 
to some peculiarity in the particular case in hand that this is so, 
and not to an accidental omission. 

The rule has many other uses, some of which will be illus- 
trated immediately. 



IV 



SYMMETRICAL FUNCTIONS 75 



§ 19.] If the student has fully grasped the idea of a homo- 
geneous integral function, the most general of its kind, he will 
have no difficulty in rising to a somewhat wider generality, 
namely, the most general integral function of the nth. degree in 
in variables, unrestricted by the condition of homogeneity or 
otherwise. 

Since any integral term whose degree does not exceed the 
nth may occur in such a function, if we group the terms into such 
as are of the Oth, 1st, 2nd, 3rd, . . . , nth degrees respectively, 
we see at once that we obtain the most general type of such a 
function by simply writing down the sum of all the homogeneous 
integral functions of the m variables of the Oth, 1st, 2nd, 3rd, . . ., 
nth degrees, each the most general of its kind. 

For example, the most general integral function of x and y of the third 
degree is 

A + Bx + Cy + Dx- + Exy + Fy 2 + Gx 3 + Hx-y + Ixrf + hf. 

The student will have no difficulty, after what has been done 
in § 17 above, in seeing that the number of terms in the general 
integral function of the ?tth degree in two variables is 

J(n+l)(n + 2). 

Symmetry. 

§ 20.] There is a peculiarity in certain of the functions we 
have been dealing with in this chapter that calls for special notice 
here. This peculiarity is denoted by the word " Symmetry "; and 
doubtless it has already caught the student's eye. What we 
have to do here is to show how a mathematically accurate 
definition of symmetry may be given, and how it may be used 
in algebraical investigations. 

1st Definition. — An integral function* is said to be symmetrical 
with respect to any two of its variables when the interchange of these 
two throughout the function leaves its value unaltered. 

* As a matter of fact these definitions and much of what follows are 
applicable to functions of any kind, as the student will afterwards learn. 
According to Baltzer, Lacroix (1797) was the first to use the term Symmetric 
Function, the older name having been Invariable Function. 



76 VARIOUS KINDS OF SYMMETRY chap. 

For example, 2a + Bb + 3c 

becomes, by the interchange of b and c, 

2a + 3c +36, 

which is equal to 2a + Sb + Be by the commutative law. Hence 2a + 3b + 3c is 
symmetrical with respect to b and c. The same is not true with respect to 
a and b, or a and c ; for the interchange of a and b, for example, would 
produce 2b + Sa + 3c, that is, 3a + 26 + 3c, which is not in general equal to* 
2a + 36 + 3c. 

2nd Definition. — An integral function is said to be symmetrical 
(that is, symmetrical with respect to all its variables) when the interchange 
of any pair whatever of its variables would leave its value unaltered. 

For example, Sx + Sij + Bz is a symmetrical function of x, y, z. So are 
yz + zx + xy and 2(x 2 + y- + z 2 ) + Zxyz. Taking the last, for instance, if we 
interchange y and z, we get 

2(x° + z- + y-) + 3xzy, 
that is, 2(x 2 + y 2 + z 2 ) + Sxyz, 

and so for any other of the three possible interchanges. 

On the other hand, x' 2 y + y-z + z-x is not a symmetrical function of x, y, z, 
for the three interchanges x with y, x with z, y with z give respectively 

y-x + x-z + z~y, 

o o o 

z-y + y-x + x-z, 

x-z + z-y + ifx, 

and, although these are all equal to each other, no one of them is equal to the 
original function. It will be observed from this instance that asymmetrical 
functions have a property — which symmetrical functions have not — of assuming 
different values when the variables are interchanged: thus x 2 y + y-z + z~x is 
susceptible of two different values under this treatment, and is therefore a 
two-valued function. The study of functions from this point of view has 
developed into a great branch of modern algebra, called the theory of substitu- 
tions, which is intimately related with many other branches of mathematics, 
and, in particular, forms the basis of the theory of the algebraical solution of 
equations. (See Jordan, Traiti des Substihitions, and Serret, Cours d'Alg&bre 
Superieure. ) 

All that we require here is the definition and its most elementary con- 
sequences. 

3rd Definition. — A function is said to be collaterally symmetrical 

\ X X X ) 

iii ttco sets of variables - ' " 2 ' ' r, each of the same number, 

I a,,a. 2 ,. . ., a n ) 

* It may not be amiss to remind the student that for the present "equal 
to" means "transformable by the fundamental laws of algebra into." 



IV 



RULE OF SYMMETRY 77 



ivhen the simultaneous interchanges of two of the first set and of the 
corresponding two of the second set leave its value unaltered. 

For example, a?x + bhj + <?z 

and (b + c)x + (c + a)y + (a + b)z 

are evidently symmetrical in this sense. 

Other varieties of symmetry might be defined, but it is 
needless to perplex the student with further definitions. If he 
fully master the 1st and 2nd, he will have no difficulty with the 
3rd or any other case. At first he should adhere somewhat 
strictly to the formal use of, say, the 2nd definition ; but, after 
a very little practice, he will find that in most cases his eye will 
enable him to judge without conscious effort as to the symmetry 
or asymmetry of any function.* 

§ 21.] From the above definitions, and from the meaning of 
the word " ecpial " in the calculation of algebraical identities, we 
have at once the following 

Rule of Symmetry. — The algebraic sum, product, or quotient of 
two symmetrical functions is a symmetrical function. 

Observe, however, that the product, for example, of two 

asymmetrical functions is not necessarily asymmetrical. 

Thus, a + b + c and bc + ca + ab being both symmetrical, their product, 
(a + b + c) ( be + ca + ab) = b"c + be 2 + c"a + ca- + orb + ab 2 + 3abc, 
is symmetrical. 

Again, a-bc and ab-c 2 are both asymmetrical functions of a, b, c, yet their 
product, 

(<rbc) x (ab' 2 c") — a 3 b 3 c 3 , 
is a symmetrical function. 

§ 22.] It will be interesting to see what alterations the 
restriction of symmetry will make on some of the general forms 
of integral functions written above. 

Since the question of symmetry has nothing to do with 
degree, it can only affect the coefficients. Looking then at the 

* There is a class of functions of great importance closely allied to sym- 
metrical functions, which the student should note at this stage, namely, those 
that change their sign merely when any pair of the variables are interchanged. 
Such functions are called "alternating." An example is [y — z)(z-x) (x — y). 
Obviously the product or quotient of two alternating functions of the same 
set of variables is a symmetric function. The term Alternating Function is 
due to Cauchy (1812). 



78 APPLICATION OF THE RULE chap. 

homogeneous integral functions of two variables on page 72, we 

see that, in order that the interchange of x and y may produce 

no change of value, we must have A = Bin§17(l); A = C in 

(2) ; A = D and B = C in (3). 

Hence the symmetrical homogeneous integral functions of x and y of 1st, 
2nd, 3rd, &c, degrees are 

Ax + Ay (1), 

Ax 2 + Bxy + Ay 2 (2), 

Ax* + Bx 2 y + Bxy 2 + Ay 3 (3), 

&c 
The corresponding functions of x, y, z are 

Ax + Ay + Az (4), 

Ax 2 + Ay 2 + Az 2 + Byz + Bzx + Bxy (5), 

Aa? + Ay 3 + Az* + Pyz 2 + ?y 2 z + Bzx" + Bz 2 x + Fxy 2 + T?x 2 y + Sxyz (6), 

&c, 
The most general symmetrical integral function of x, y of the 3rd degree 
will be the algebraic sum of three functions, such as (1), (2), and (3), together 
with a constant term, namely, 

F + Ax + Ay + Bx 2 + Cxy + By- + Bx 3 + Ex' 2 y + Exy 2 + By 3 . 
And so on. 

If the student find any difficulty in detecting what terms 
ought to have the same coefficient, let him remark that they are 
all derivable from each other by interchanges of the variables. 
Thus, to get all the terms that have the same coefficient as a; 3 in 
(6), putting y for x, we get y* ; putting z for x, we get z 3 ; and we 
cannot by operating in the same way upon any of these produce 
any more terms of the same type. Hence .r 3 , if, z 3 form one 
group, having the same coefficient. Next take yz~ ; the inter- 
changes x and y, x and z, y and z produce xz 2 , yx 2 , yz 2 ; applying 
these interchanges to the new terms, we get only two more new 
terms — zx 2 , xy 2 ; hence the six terms yz 2 , y 2 z, zx 2 , £x, xy 2 , x 2 y form 
another group ; xyz is evidently unique, being itself symmetrical. 

§ 23.] The rule of symmetry is exceedingly useful in abbre- 
viating algebraical work. 

Let it be required, for example, to distribute the product (a + b + c) 
(a? + b 2 + c 2 -bc- ca-ab), each of whose factors is symmetrical in a, b, c. The 
distributed product will be symmetrical in a, b, c. Now we see at once that 
the term a? occurs with the coefficient unity, hence the same must be true of 
b 3 and c 3 . Again the term b'-c has the coefficient 0, so also by the principles 
of symmetry must each of the five other terms, be 2 , c"«, cu 1 , ab 2 , a 2 b, belonging 
to the same type. Lastly, the term -abc is obtained by taking a from the 
first bracket, hence it must occur by taking b, and by taking c, that is, the 



iv INDETERMINATE COEFFICIENTS 79 

a&c-terni must have the coefficient - 3. We have therefore shown that 
(ct + b + c) (a? + b 2 + c- -bc-ca- ab) = a? + b* + c 3 - Zabc ; and the principles of 
symmetry have enahled us to abbreviate the work by about two-thirds. 

PRINCIPLE OF INDETERMINATE COEFFICIENTS. 

§24.] A still more striking use of the general principles of 
homogeneity and symmetry can be best illustrated in conjunction 
with the application of another principle, which is an immediate 
consequence of the theory of integral functions. 

We have laid down that the coefficients of ah integral function 
are independent of the variables, and therefore are not altered by 
giving any special values to the variables. If, therefore, on cither 
side of any algebraic identity involving integral functions we determine 
the coefficients, either by general considerations regarding the forms of 
the functions involved, or by considering particular cases of the identity, 
then these coefficients are determined once for all. This has (not very 
happily, it must be confessed) been called the principle of inde- 
terminate coefficients. As applied to integral functions it results 
from the most elementary principles, as we have seen ; when 
infinite series are concerned, its use requires further examination 
(see the chapter on Series in the second part of this work). 

The following are examples : — 

(x + y) 2 = (x + y)(x + y), being the product of two homogeneous 
symmetrical functions of x and y of the 1st degree, will be a 
homogeneous symmetrical integral function of the 2nd degree ; 
therefore (x + yf = Ax 2 + Bxy + Ay 2 (1). 

"We have to determine the coefficients A and B. 

Since the identity holds for all values of x and y, it must 
hold when x = 1 and y = 0, therefore 

(1 +0) 2 = A1 2 + B1 xO + AO 2 , 
1=A. 
We now have (x + y) 2 = x 2 + Bxy + y 2 ; 
this must hold when x = 1 and y - - 1, 
therefore (1 - 1) 8 = 1 +B.1.(- 1) + 1, 

that is, = 2 - B, 

whence B = 2. 

Thus finally (x + y) 2 = x 2 + 2xy + y 2 . 



80 INDETERMINATE COEFFICIENTS CHAr. 

This method of working may seem at first sight somewhat startling, but 
a little reflection will convince the learner of its soundness. We know, by 
the principles of homogeneity and symmetry, that a general identity of the 
form (1) exists, and we determine the coefficients l>y the consideration that 
the identity must hold in any particular case. The student will naturally ask 
how he is to be guided in selecting the particular cases in question, and 
whether it is material what cases he selects. The answer to the latter part of 
this question is that, except as to the labour involved in the calculation, the 
choice of cases is immaterial, provided enough are taken to determine all the 
coefficients. This determination will in general depend upon the solution of 
a system of simultaneous equations of the 1st degree, whose number is the 
number of the coefficients to be determined. (See below, chap, xvi.) So fat- 
as possible, the particular cases should be chosen so as to give equations each 
of which contains only one of the coefficients, so that we can determine them 
one at a time as was done above. 

The student who is already familiar with the solution of simultaneous 
equations of the 1st degree may work out the values of the coefficients by 
means of particular cases taken at random. Thus, for example, putting x=2, 
y = 3, and x = l, y = i successively in (1) above, we get the equations 

25 = 13A + 6B, 
25 = 17A + 4B, 

which, when solved in the usual way, give A = 1 and B = 2, as before. 

We give one more example of this important process : — 
By the principles of homogeneity and symmetry we must have 
(x + y + z) (x 2 + y 2 + z- - yz - zx - xy) 
= A(.r> + y 3 + z 3 ) + B(yz 2 + y 2 z + zx 2 + z 2 x + xy 2 + x*y) + Cyxz. 
Putting sc=l, y = 0, z — 0, we get 1=A. 
Using this value of A, and putting x=\, y = l, z = 0, we get 

2xl = 2 + Bx2, 
that is, 2 = 2 + Bx2, 

therefore 2B = 0, 

and therefore B = 0. 

Using these values of A and B, and putting x=1, y=l, 2 = 1, we get 

3xO=3 + C, 
that is, = 3 + C, 

therefore C = - 3 ; 

and we get finally 

(x + y + z) (x 2 + y 2 + z 2 -yz-zx-xij) = x 3 + if + z z -3xyz (2), 

as in § 23. 

§ 25.] Reference Table of Identities. — Most of the results given 
below will be found useful by the student in his occasional calcu- 
lations of algebraical identities. Some examples of their use 



TABLE OF IDENTITIES 



81 



have already been given, and others will be found among the 

Exercises in this chapter. Such of the results as have not 

already been demonstrated above may be established by the 
student himself as an exercise. 



(x + a) (x + b) = x" + (a + b)x + ab ; 
(x + a) (x + b) (x + c) = x 3 + (a + b + c)x* 

+ (be + ca + ab)x + abc ; 
and generally 
(x + «,) {x + a 2 ) ...(/• + a n ) = x n + P,.^ 1 " 1 + P^" 2 

+ . . . + P n _^ + P w (see §9). 

(x ± yf - x 2 ± 2xy + if ; 
(x ± yf = x 3 ± 3x 2 y + Zxif ± if j 
&c. ; 

the numerical coefficients being taken from the following 
table of binomial coefficients : — 



> 



Hi-) 



Tower. 










Coefficients. 




1 


1 


1 










o 


1 


2 1 










3 


1 


3 3 


1 








4 


1 


4 6 


4 


1 






5 


1 


5 10 


10 


5 


1 




6 


1 


6 15 


20 


15 


6 1 






1 


7 21 


35 


35 


21 7 1 




8 


1 


8 28 


56 


70 


56 28 8 1 




9 


1 


9 36 


84 126 126 84 36 9 


1 


10 


1 


10 45 120 


210 


252 210 120 15 


10 1 


11 


1 


11 55 


165 


330 462 462 330 165 


55 11 1 


12 


1 12 G6 


220 495 


792 924 792 495 


220 66 12 1 












&c. 





Mil.) 



* This table first occurs in the Arithmctica Integra of Stifel (1544), in 
connection with the extraction of roots. It does not appear tli.it he was 
aware of the application to the expansion of a binomial The table was dis- 
cussed and much used by Pascal, and now goes by the name of Pascal's 
Arithmetical Triangle. The factorial formula' for the binomial coefficients (see 
the second part of this work) were discovered by Newton. 

VOL. I G 






82 



TABLE OF IDENTITIES 



- 1i n 



(x ± yf t % = (a t y) 2 - 
(x + y) (x -y) = x 2 -y 2 ; 
(x ± y) (x 2 T sqf + y 2 ) = x 3 ± / ; 
and generally 

(x _ y) ( x « -i + x n ~ h) + . . . + «y n ~ 2 + f ~ l ) = a* 
(z + y) (sc n - 1 - » n - 2 y + . . . t -™/ n " 2 ± y n - 1 ) = aJ« ± y» 
upper or lower sign according as w is odd or even. 

(x* + f)<f + y' 2 ) = {xx' T yy') 3 + (^ ± FT; 

(.<" - *■)(*" - 2/' 2 ) = (a* ± ^7 ~ (*y ± 3*Ti 

(/ + y 2 + «■)(*/■ + y" + *' 2 ) = ixx' + yy' + zz') 2 + (yz' - y'z) 2 

+ (zx'-z'x) 2 + (xy'-x'y) 2 ; 

(x 2 + y 2 + z 2 + u 2 )(x' 2 + y' 2 + z" + u' 2 ) = (xx' + yy' + zz' + uu') 2 

+ (xy - yx' + zu' - uz') 2 
+ (xz' - yu' - zx' + uy') 2 
+ (xu' + yz' - zy' - ux') 2 . 

(x 2 + xy + f) (x 2 - xy + y 2 ) = x* + x 2 y 2 + y\ 

( a + l ) + c + d) 2 = a 2 + b 2 + c 2 + d 2 + 2ab + 2ac + 2ad 
+ 2bc + 2bd + 2cd ; 
and generally 
(a l + a 2 + . . . + a n ) 2 = sixm of squares of a ti a 2 , . . ., a, 

+ twice sum of all partial products two and two. 

(a + b + c) 3 = a 3 + b 3 + c 3 + U 2 c + 3bc 2 + Zc'a + Sea* + 3a b 
+ Sab 2 + Gabc 
= a 3 + b 3 + c 3 + 36c (6 + c) + 3ca(c + a) 
+ Sab (a + b) + Saba 

( a + b + c) (a 2 + b 2 + c 2 -bc- ca - ab) - a 3 + b 3 + c 3 - 3abc 

(b - c) (c - a) (a-b)= - a\b - c) - b\c - a) - c\a - b), 
= a(b 2 - c 2 ) + b(c 2 - a 2 ) + c(a 2 - b% 
- - bc(b -c)- ca(c - a) - ab(a - b), 
= + be 2 - b 2 c + ca 2 - c 2 a + ab 2 - a 2 b. 



CHAP. 



(III.) 



MIV.) 



Uy.y 



(VI.) 



I (VII.) 



(VIII.) 
(IX.) 



* These identities furnish, inter alia, proofs of a series of propositions in 
the theory of numbers, of which the following is typical :— If each of two 
integers he the ram of two squares, their product can he exhibited in two ways 
as the sum of two integral squares. 



iv EXERCISES VIII 83 

(b + c)(c + a)(a + b) = a s (b + c) + b 2 (c + a) + c 2 (a + b) + 2abc, \ 

= bc(b + c) + ca(c + a) + ab(a + b) + 2abc, I (XI.) 
= be 2 + b 2 c + ca 2 + c 2 a + ab 2 + ab + 2dbc. J 

(a + b + c) (a 2 + b 2 + c 2 ) = bc(b + c) + ca(c + a) + ab(a + b) \ , 

+ a 3 + b 3 + c\ j (XIL) 

(a + b + c) (Lc + ca + ab) = a 2 (b + c) + b 2 (c + a) + c 2 (a + b) \ /VTTT 

+ 3abc, J (X111) 

(b + c- a) (c + a- b) (a + b - c) = a 2 (b + c) + b 2 (c +•«) ( 

+ c\a + b)-a 3 -b 3 -c 3 - 2abc, I ( ' 

(a + 6 + c)( - a + 6 + c)(a - b + c)(a + b - c) = 2b 2 c 2 + 2c 2 a 2 ) „ 

+ 2a*b*-a t -b i -c*. / (XV.)- 

(6 - c) + (c - a) + (a - b) = ; \ 

a(b -c)+ b(c -a) + c(a - b) = ; V (XVI.) 

(6 + c)(fi - c) + (c + a)(c -a) + (a + b) (a -b) = 0. ) 

Exercises VIII. 

(1.) Write down the most general rational integral symmetrical function 
of x, y, z, u of the 3rd degree. 

(2.) Distribute the product {x"y + y"z + z"x) (xy' + yz^ + zx 2 ). Show that 
it is symmetrical ; count the number of types into which its terms fall ; and 
state how many of the types corresponding to its degree are missing. 

(3.) Construct a homogeneous integral function of x and y of the 1st 
degree which shall vanish when x = y, and become 1 when x =5 and ?/ = 2. 

(4.) Construct an integral function of x and y of the 1st degree which 
shall vanish when x = x', y = y', and also when x—x", y = y". 

(5.) Construct a homogeneous integral function of x and y of the 2nd 
degree which shall vanish when x = x', y = y', and also when x — x", y = y", and 
fchall become 1 when aj=l, y = l. 

(6.) If A(a:-3)(iB-5) + B(aJ-5)(aj-7) + C(a!-7)(a!-3)=8iB-120 for all 
values of x, determine the coefficients A, B, C. 

(7.) Show that 5.r 2 + 19a;+18 can be put into the form 

l{x - 2) {x - 3) + m(x - 3) (x - 1) + n{x - 1 ) (x - 2) ; 
and find I, m, n. 

(8.) Assuming that {x - 1) (x - 2) (a;- 3) can be put into the form 
l(x-l) {x +2){x + 3) + vi{x-2)(x + B)(x + l) + n(x - 3) (*+ 1) (»+ 2), 
determine the numbers 7, m, n. 

* Important in connection with Hero's formula for the area of a plane] 
triangle. 



84 EXERCISES VIII chap, iv 

(9.) Find a rational integral function of x of the 3rd degree which shall 
have the values P, Q, R, S when x = a, x=b, x = c, x = d respectively. 
(10.) Find the coefficients of y-z and yz- in the expansion of 

(ax + by + cz) (a-x 4 b-y 4 c 2 z) (a 3 x 4 b 3 y 4 <?z). 
(11.) Expand and simplify 2(</ 2 + z 2 - x 2 ) (y + z - x). 

Trove the following identities : — 

(12.) (ad + bc) 2 + (a + b + c-d)(a + b-c + d)(b + d)(b-d) = (b 2 -d 2 + ab + cd) 2 . 
*(13.) 2(b 2 + c 2 -a 2 + bc + ca + ab) 2 (c 2 -b 2 ) = 4(b 2 -c 2 )(c 2 -a 2 )(a 2 -b 2 ). 
(14. ) 2(ca - b 2 ) (ab - c 2 ) = (26c) (26c - 2a 2 ). 
(15.) 2(&c' - b'c) (be" - b"c) = 2a 2 2rt'«" - 2aa'2W. 
(16.) 31% 4 z) - 62?/~ = 2a-(2.c- 1) (2jj- 2) - Saj(a-1) (a!-2). 
(17.) Z(b 2 + c 2 -a"-)/2bc=(ip 1 2>-2-lh 3 -fy3)/ty3, where Pl = - 2a, p, = Zbc, 
p 3 - -abc. 

(18. ) 1% 4 ;) 2 4 2x 2 y 2 z 2 - 2x*(y 4 z) 2 = 2(2 r ) 3 . 

(19.) S(a: +y-z){(y- z) 2 -(z-x)(x- y)\ = 2a*- Zxyz. 

(20. ) n(a± b±c±d) = 2a 8 - 42a 6 6 2 4 62« 4 i 4 4 42aW - 40a 2 6 2 c 2 rf2. 

(21.) Show that 

(x 3 4 y 3 4 s 3 - 3ays) a = X 3 4 Y 3 4 Z 3 - 3X YZ, where X = x 2 4 2y;, &c. ; 
also that 

CZx 3 - 3xyz) (2a;' 3 - 2>x'y'z) = S(a»' 4 ysf* 4 ?/~~) 3 - 3II(»/ 4 yz' 4 y'z). 
(These identities have an important meaning in the theory of numbers. ) 
(22.) Show that, if n be a positive integer, then 

l-h+h~. ■ .-!(»even) = 2('^L + -L + . . .+!-)■ 

1-i+J-. . . + l(»odd)=2/-l- + _l_ + . . .+1 

n \n + l n + 3 2n 

(Blissard). 

* In this example, and in others of a similar kind, 2 is not used in its 
strict sense, but refers only to cyclical interchanges of a, b, c; that is, to 
interchanges in which a, b, c pass into b, c, a respectively, or into c, a, b 
respectively. Thus, 2cr(&-c) is, strictly speaking, =0; but, if 2 be used in 
the present sense, it is a 2 (b -c)+ b 2 (c -a)+ c 2 (a - b). 






CHAPTER V. 

Division of Integral Functions— Transformation of 

Quotients. 

§ 1.] The operations of this chapter are for the most part 
inverse to those of last. Thus, A and D being any integral 
functions of one variable x* and Q a function such that 
D x Q = A, then Q is called the quotient of A by D ; A is called 
the dividend and D the divisor. We symbolise Q by the nota- 

tion A -7- D, A/D, or =-, as explained in chap. i. 

The operation of finding Q is called division, but we prefer 
that the student should class the operations of this chapter under 
the title of transformation of quotients. 

A and D being both integral functions, Q will be a rational 
function of x, but will not necessarily be an integral function. 

When the quotient can he transformed so as to become integral, A 
is said to be exactly divisible by D. 

IFhen the quotient cannot be so transformed, the quotient is said 
to be fractional or essentially fractional. 

It is of course obvious that an essentially integral function cannot 
be equal, in the identical sense, to an essentially fractional function. 

§ 2.] When the quotient is integral, its degree is the excess of the 
degree of the dividend over tlie degree of the divisor. For, denoting 

* For reasons partly explained below, the student must be cautious in 
applying many of the propositions of this chapter to functions of more vari- 
ables than one ; or at least in such cases he must select one of the variables 
at a time, and think of it as the variable for the purposes of this chapter. 



86 THEOREM REGARDING DIVISIBILITY chap. 

the degrees of the functions represented bj 7 the various letters 
by suffixes, we have 

therefore, by chap, iv., § 7, m =p + n, that is, p = m - n. 

§ 3.] If the degree of the dividend be less than that of the divisor, 
the quotient is essentially fractional. For, m being <??, suppose, if 
possible, that the cpiotient is integral, of degree p say, then 

therefore m=p + n; but p cannot be less than by our hypo- 
thesis, and m is already less than n, hence the cpiotient cannot 
be integral, that is, it must be fractional. 

§ 4.] If A, D, Q, R be all integral functions, and if A = 
QD + R, then R will be exactly divisible by D or not according as A 
is exactly divisible by D or not. 

For, since A = QD + R, 

A _ QD + R R 

therefore ~ = j- - Q. 

Now, if A be exactly divisible by D, A/D will be integral, and 
A/D - Q will be integral, that is, R/D will be integral, that is, R 
will be exactly divisible by D. 

Again, if A be not exactly divisible by D, A/D will be 
fractional. Hence R/D must be fractional, for, if it were 
integral, Q + R/D would be integral, that is, A/D would be in- 
tegral, which is contrary to hypothesis. 

INTEGRAL QUOTIENT AND REMAINDER. 

§ 5.] The following is the fundamental theorem in the 
transformation of quotients. 

A^ and D n being integral functions of the degrees m and n respect- 
ively, we can always transform the quotient A 7n /D„ as follows . — 

A ffl _ p ^i 

XJ n XJ n 



V INTEGRAL QUOTIENT AND REMAINDER 87 

where P TO _ n is an integral function of degree m - n, and R (if it do 
not vanish) an integral function whose degree is at most n- 1. 

This transformation is effected by a series of steps. "We shall 
first work out a particular case, and then give the general proof. 

Let Ah = 8a* + 8*- 5 - 20a- 4 + 40* 3 - 50* 2 + 30* -10, 

D 4 = 2* 4 + 3* 3 -4* 2 + 6*-8, 

multiply the divisor D 4 by the quotient of the highest term of the dividend 
by the highest term of the divisor (that is, multiply D4 by 8* 6 /2* 4 = 4* 2 ), and 
subtract the result from the dividend A 6 . "We have 

A 6 = 8* fi + 8* 5 - 20k 4 + 40* 3 - 5 Ox- + 30* - 1 
4* 2 D 4 = Sx e + 1 2* 5 - 1 6* 4 + 24.C 3 - 32a? 



A 6 -4*'-D 4 = - 4ar»- 4* 4 + 16* 3 - 18a; 3 + 30*- 10 
= A 3 say ; 
therefore A 6 = 4* 2 D 4 + A 5 (1). 

Repeat the same process with the residue A 5 in place of Ae, and we have 

As= - 4a* - is* + 16* 3 - 18* 2 + 30* - 10 
- 2*D 4 = - 4* 5 - 6* 4 + 8*' - 1 2.7r + 1 Qx 



A 3 + 2*D 4 = 2x*+ 8X 3 - Qx- + 14* -10 

= A 4 say; 
therefore A 5 = - 2*D 4 + A 4 (2). 

And again with A 4 , 

A 4 = 2x* + 8x* - 6* 2 + 14* - 10 
1 x D 4 = 2** + 3*" - 4* 2 + 6* - 8 



A 4 -D 4 = 5* 3 -2* 2 + 8*- 2 

= A 3 say ; 
therefore A 4 =D 4 + A 3 (3). 

Here the process must stop, unless we agree to admit fractional multi- 
pliers of D 4 ; for the quotient of the highest term of A :! by the highest term of 
D 4 is 5* 3 /2* 4 , that is, f/*, which is a fractional function of *. Such a con- 
tinuation of the process does not concern us now, but will be considered below. 

Meantime, from (1) we have 

A 6 =4*-D 4 + A 5 (4) • 

and, using (2) to replace A 3 , 

A 6 = 4*-D 4 - 2*D 4 + A 4 (5 ) ; 

and finally, using (3), 

A 6 = 4* 2 D 4 -2*D 4 + D 4 + A 3 , 

= (4* 2 -2* + l)D 4 + A 3 (6). 

Hence A 6= J4* 2 -2a; + l)D 4 + A, 

D 4 D 4 

= 4* 2 -2* + l + 1 4>' ; 
•U4 



88 INTEGRAL QUOTIENT AND REMAINDER CHAP. 

or, replacing the capital letters by the functions they represent, 

8 ,> ; 6 + 8a; 5 - 20.r 4 + 40.T 3 - 50.r 2 + 30.r - 10 

2x* + 3x :i - ix- + 6x - 8 

. , 5,r 3 -2x 2 + 8a;-2 ... 

= ^-2*+l+ ^ + 8a ,_ 4a . + fla ,_ 3 (7). 

Since 6-4 = 2, it will be seen that we hare established the above theorem 
for this special case. It so happens that the degrees of the residues A 5 , A 4 , 
A 3 diminish at each operation by unity only ; but the student will easily see 
that the diminution might happen to be more rapid ; and, in particular, that 
the degree of the first residue whose degree falls Tinder that of the divisor 
might happen to be less than the degree of the divisor by more than unity. 
But none of these possibilities will affect the proof in any way. 

We shall return to the present case immediately, but in the first place we 
may give a general form to the proof of the important proposition which we 
are illustrating. 

§ 6.] Let A m = 2V m +ja l x m ~ 1 + p 2 x m ~ 2 + &c - 1 
D n —-q v x n + q^ 1 ' 1 + q a x n ~ 2 + &c. 

Multiplying D n by the quotient p$? n jq x n , that is, by (p /qo) xm ~ n > 
and subtracting the result from A,„,, we get 

= A m _, say, 
whence, denoting p /q by r for shortness, we get 

-A-m = tx D n + A m _ ! ( I ). 

Treating A m _t in the same way, we get 

A m _, = SX D n + A m _ 2 {-')• 

And so on, so long as the degree of the residue is not less 
than n, the last such equation obtained being — 

A„ = m>D„ + K (3), 

where E is of degree n - 1 at the utmost. Using all these 
equations in succession we get 

_ (rtfn-n + stfm-n-1 + . . . + w )J) n + R ; 

whence, dividing both sides by D n , and distributing on the 
right, 



A _ -?- r m - » T) 



v THE ORDINARY DIVISION TRANSFORMATION UNIQUE 89 

A R 

-^ = rs m - " + sx m - n ~ 1 + . . .+«;+'=-, 

which, if we bear in mind the character of R, gives a general 
proof of the proposition in cpiestion. 

§ 7.] We have shown that the transformation of § 5 can 
always be effected in a particular way, but this gives no assur- 
ance that the final result will always be the same. The proof 
that this really is so is furnished by the following proposition: — 

The quotient A/D of two integral functions can be put into the 
form P + R/D, where P and R are integral functions and the degree 
of R is less than that of D, in one way only. 

If possible let 

1 A TV R ' 

and D = P+ D' 

where P, R and P', R' both satisfy the above requirements ; 

then p + ? = P' + ^ ; 

D D ' 

subtracting P' + =r from both sides, we have 

R' R m 
r r ~ 1 ) D ' 

whence — - = P - P'. 

D 

Now, since the degrees R and R are both less than the degree 
of D, it follows that the degree of R' - R is less than that of D. 
Therefore, by § 3, the left-hand side, (R' - R)/D, is essentially 
fractional, and cannot be equal to the right, which is integral, 
unless R' - R - 0, in which case we must also have P - P' = 0, 
that is, R = R', and P = P\ 

§ 8.] The two propositions of §§ 5, 7 give a peculiar import- 
ance to the functions P and R, of which the following definition 
may now legitimately be given : — 

If the quotient A/D be transformed into V + R/D, P and R being 



90 CONDITIONS FOR EXACT DIVISIBILITY chap. 

integral and R of degree less than D, P is called the integral quotient, 
and R the remainder of A when divided by D. 

§ 9.] We can now express the condition that one integral 
function A may be exactly divisible by another D. For, if E be 
the remainder, as above denned, we have, P being an integral 
function, — 

^-P + ? 

whence, subtracting P from both sides, 



- Ax* - 

- Ax* - 


Ax* + 16a; 3 - 18a; 2 + 30a; -10 
6a; 4 + 8a; 3 - 12a; 2 + 16a; 




2x* + 8/- 6a; 2 + I Ax- 10 
2»* + 3a; 3 - 4a- 2 + 63- 8 



2x 4 + Sx 3 - 4/ + 6.r - 8 



4a: 2 - 2x + 1 






A B 

D D 

Now, if A be exactly divisible by D, A/D will be integral, and 
therefore A/D - P will be integral. Hence R/D must be integral ; 
but, since the degree of R is less than that of D, this cannot be 
the case unless R vanish identically. 

The necessary and sufficient condition for exact divisibility is there- 
fore that the remainder shall vanish. 

When the divisor is of the nth degree, the remainder will in 
general be of the (n - l)th degree, and will contain n coefficients, 
every one of which must vanish if the remainder vanish. In 
general, therefore, when the divisor is of the nth degree, n conditions 
are necessary to secure exact divisibility. 

§ 10.] Having examined the exact meaning and use of the 
integral quotient and remainder, we proceed to explain a con- 
venient method for calculating them. The process is simply a 
succinct arrangement of the calculation of §§ 5, G. It will be 
sufficient to take the particular case of § 5. 

The work may be arranged as follows : — 
8x° + 8x* - 20a; 4 + 40a; 3 - 5 Ox 2 + 3 Ox - 10 
8a; 6 + 1 2/- 16a: 4 + 24a; 3 -32a' 2 



5x 3 - 2x 2 + 8x- 2 



v DETACHED COEFFICIENTS 91 

Or, observing that the term - 10 is not waited till the last 
operation, and therefore need not be taken down from the upper 
line until that stage is reached, and observing further that the 
method of detached coefficients is clearly applicable here just as 
in multiplication, we may arrange the whole thus : — 

8+ 8 - 20 + 40 - 50 + 30 - 10 | 2 + 3 - 4 + 6 - 6 

4-2 + 1 



8 + 12-16 + 24-32 


I 


- 4- 4 + 16-18 + 30 




- 4- 6 + 8-12 + 16 




2+ 8- 6 + 14- 


10 


2+ 3- 4+ 6- 


8 



5- 2+8- 2 
Therefore, Integral quotient = 4/ - 2x + 1 \ 

Remainder = 5x 3 - 2x 2 + 8x - 2. 

The process may be verbally described as follows : — 

Arrange both dividend and divisor according to descending powers 
of x, filling in missing powers with zero coefficients. Find the quotient 
of the highest term of the dividend by the highest term of the divisor ; 
the result is the highest term of the " integral quotient." 

Multiply the divisor by the term thus obtained, and subtract the 
result from the dividend, taking down only one term to the right beyond 
those affected by the subtraction ; the result thus obtained will be less in 
degree than the dividend by one at least. Divide the highest term of 
this residt by the highest of the divisor ; the result is the second term 
of the " integral quotient." 

Multiply the divisor by the new term just obtained, and subtract, 
&c, as before. 

The process continues until the result after the last subtraction is 
less in degree than the divisor; this last result is the remainder as 
above defined. 

§ 11.] The following are some examples of the use of the 
" long rule " for division. 



92 



Example 1. 



EXAMPLES 



i. 19 „ 5 l_ 

I 36 36 r + 36 

1 _ 5 i 19 _ 6 i 1 



,x--- a; + 4 



1-i + l 





5 

3ff 
"A 


i 

•B 

1 
V 


— 1-4-1 



J 1 "} 



CHAP. 



+ 

The remainder vanishes, therefore the division is exact, and the quotient is 






Example 2. 

l+p +q 
I -a 



{x 3 +px 2 + qx + r)-i-[x-a). 
+r 1-a 



1 + (a +p) + {a 2 + ap + q) 



{a+p) + q 
(a+p)-{a 2 + ap) 

{a? + ap + q) + r 

(a 2 + ap + q)- (a 3 + a 2 p + aq) 

(a 3 + a 2 p + aq + r) 
Hence the integral quotient is 

x 2 + (a+p)x + (a 2 + ap + q) ; 
and the remainder is 

a 3 + a 2 p + aq + r. 

The student should observe the use of brackets throughout to preserve the 
identity of the coefficients. 

Example 3. 

(re 4 - 2>a 3 b + 6a 2 b- - Zab 3 + 1*) 4- (a 2 -ab + b 2 ). 

1st. Let us consider a as the variable. Since the expressions are homo- 
geneous, we may omit the powers of b in the coefficients, and use the numbers 
merely. 

1-3 + 6-3 + 1 I 1-1+1 



1-1 + 1 



-2+5-3 

-2+2-2 



1-2 + 3 
a 2 -2ab + Sb 2 



3-1 + 1 
3-3 + 3 

" 2-2 
2ab 3 - 2b* 



BINOMIAL DIVISOR 



93 



whence 



i i -3a*b + 6cr-b--Sab* + b i 



= a?-2ab + M- + 



2a't? - 2b* 



a 2 -ab + b 2 ' 

We must then arrange according 



a 2 -ab + b'- 

'2nd. Let us consider b as the variable, 
to descending powers of b, thus— 

(6 4 - 3ab 3 + 6a-b°- - 3a?b + a 4 ) +- {b- -ab + a-). 
Detach the coefficients, and proceed as before. It happens in this particular 
case that the mere numerical part of the work is exactly the same as before ; 
the only difference is in the insertion of the powers of a and b at the end. 
Thus the integral quotient is b 2 -2ba + 3a", and the remainder is 26a 3 -2a 4 , 
whence 

— , , ,., =Za--2ab + b- + — — r . 

a 2 ~ab + b 1 a- - ab + b 2 

§ 12.] The process of long division may be still further 
abbreviated (after expertness and accuracy have been acquired) 
by combining the operations of multiplying the divisor and sub- 
tracting. Then only the successive residues need be written. 
Thus contracted, the numerical part of the operations of Example 
3 in last paragraph would run thus : — 

1-3+6-3+1 1 1-1+1 
-2+5-3+i;i-2+3 
3-1 + 1 
2-2 



BINOMIAL DIVISOR — REMAINDER THEOREM. 

§ 13.] The case of a binomial divisor of the 1st degree is of 
special importance. Let the divisor be x - a, and the dividend 



p^ 1 + p x x 



n -i 



+ Pr 



,,«-2 + 



+ Pn-&+Pn- 



Then, if we employ the method of detached coefficients, the 
calculation runs as follows : — 



P0+P1 

Po-Po«- 



+ Pa + 



+ Pn-i+Pr. 



(P&+Pi)+F» 
(P&+Pi)-(Poa+Pi)a 



! 



+ (p<fl.+p 1 ) 

+ (Po°- 2 +Pia+pJ 



(j><P+Pi*+P*)+pa 

(j><P+Pi* + Pi) ~ (Po* +p x o.+ p a )a 

(p a + P,a + p,a + p 3 ) 



94 RULES FOR COEFFICIENTS OF chap. 

The integral quotient is therefore 

pjF-l + {p o a + Pl )x n ~* + {p/ + Pl a +pX' 2 + ••' 

The law of formation of the coefficients is evidently as follows: — 

The first is the first coefficient of the dividend ; 

The second is obtained by multiplying its predecessor by a and 
adding the second coefficient of the dividend ; 

The third by multiplying the second just obtained by a and adding 
the third coefficient of the dividend ; and so on. 

It is also obvious that the remainder, xchich in the present case is 
of zero degree in x (that is, does not contain x), is obtained from the 
last coefficient of the integral quotient by multiplying that coefficient by 
a and adding the last coefficient of the dividend. 

The operations in any numerical instance may be con- 
veniently arranged as follows : — * 



Example 1. 



(2x i -Sx 2 + 6x-4) + (x-2). 
2+0-3+ 6- 4 
+ 4 + 8 + 10 + 32 



2 + 4 + 5 + 16 + 28 

Integral quotient = 2a? + 4a; 2 + 5x + 1 6 ; 
Remainder = 28. 

The figures in the first line are the coefficients of the dividend. 

The first coefficient in the second line is 0. 

The first coefficient in the third line results from the addition of the two 
ahove it. 

The second figure in the second line is obtained by multiplying the first 
coefficient in the third line by 2. 

The second figure in the third line by adding the two over it. 

And so on. 

Example 2. 

If the divisor be x + 2, we have only to observe that this is the same as 



* The student should observe that this arrangement of the calculation of 
the remainder is virtually a handy method for calculating the value of an 
integral function of x for any particular value of x, for 28 is 2 x 2 4 - 3 x 2 3 
+ 6x2-4, that is to say, the value of 2x* - Zx* + 6x - 4 when x-2 (see § 14). 
This method is often used, and always saves arithmetic when some of the 
coefficients are negative and others positive. It was employed by Newton ; 
see Horsley's edition, vol. i. p. 270. 



v DIVISOB, AND FOR REMAINDER 95 

x-{-2); and we see that the proper result will be obtained by operating 
throughout as before, using - 2 for our multiplier instead of + 2. 

(2a* - 3a? + 6b -4) -Ha; +2) 

= (2a,- 4 - 3ar + 6» - 4) -=- (x - ( - 2)). 

2+0-3+ 6-4 

0-4+8-10+8 



2-4 + 5- 4 + 4. 



Integral quotient = 1)? - Ax- + 5x - 4 ; 
Remainder =4. 

Example 3. 

The following example will show the student how to bring the case of any 
binomial divisor of the 1st degree under the case of a;- a. 

3-f ' - 2X 3 + 3x 2 - 2x + 3 _ 3.x- 4 -2j? + 3 x 2 - 2.v + 3 
3a: + 2 ~ 3(x + f) 

( 3b 4 -2a; 3 + 33? -2a; + 3 "1 

Transforming now the quotient inside the bracket { } , we have 

3-2+ 3-2+3 





— 0-1- s _ 34 i 1 04 




S-iJ.n_58i.lS5 




I ntegral quotient = Sx 3 - 4a? + ^-x - Af . 


Whence 


Remainder — y 7 5 -. 


3a: 4 - 


-2 S5 »+8a?-2a!+3 f A-.it, - , W 




_te + 2 = *t 3a! 4x + ^ x ¥+ «s-(-l 




185 

— ar* ^x +-5 -a. #?+•> , .>• 



Hence, for the division originally proposed, we have — 

Integral quotient =a?-|a? + -^aj- :]; ; 
Remainder = V\ 5 -. 

The process employed in Examples 2 and 3 above is clearly 
applicable in general, and the student should study it attentively 
as an instance of the use of a little transformation in bringing 
cases apparently distinct under a common treatment. 

§ 14.] Reverting to the general result of last section, we see 
that the remainder, when written out in full, is 

p a n +_p 1 a"- 1 + . . . + £>„_!<_ +p n . 



96 REMAINDER THEOREM CHAP. 

Comparing this with the dividend 

IV'" + Pi*" " + • • • + Pn - i x + Pn , 

we have the following " remainder theorem " : — 

When an integral function of x is divided by x - a, the remainder 
is obtained by substituting a for x in the function in question. 

In other words, the remainder is the same function of a as 
the dividend is of x. 

Partly on account of the great importance of this theorem, 
partly as an exercise in general algebraical reasoning, we give 
another proof of it. 

Let us, for shortness, denote 

jj x 11 + p t x n - l + . . ■+p n - 1 x+ p n by f(x), 

/'(a) will then, naturally, denote the result of substituting a for 
x in f(x), that is, 

p o n +p l a 11 - 1 + . . . +!>„_, a +p n . 

Let x(-') denote the integral quotient, and R the remainder, 
when f(x) is divided by x - a. Then x( x ) i s an integral function 
of x of degree n-l, and R is a constant (that is, is independent 
of ,»■), and we have 

X- a ^ X- a 

whence, multiplying by x - a, we get the identity 

f( X ) = (X - a) X (x) + R. 

Since this holds for all values of x, we get, putting x = a 
throughout, 

/(a) = (a - a) X (a) + R, 

where R remains the same as before, since it does not depend on 
x, and therefore is not altered by giving any particular value 

to .''. 

Since x(«) i s finite if a be finite, (a - a)x(a) = x x («) = ; 
and we get finally 

/(a) = R, 

which, if we remember the meaning of /(a), proves the "re- 
mainder theorem." 



v FACTORISATION BY REMAINDER THEOREM 97 

Cor. 1. Since x + a - x - ( - a), it follows that 
The remainder, when an integral function f(x) is divided by 
X + a, isf(-a). 

For example, the remainder, when v 4 - 2>x* + 2a? - 5x + 6 is divided by 
as+10, is (-10) 4 -3(-10) 3 + 2(-10)--5(-10) + 6 = 13256. 

Cor. 2. The remainder, when an integral function of x, f(x), is 
divided by ax + b, is /( - bja). 

This is simply the generalisation of Example 3, § 13, above. 

By substitution we may considerably extend the application 
of the remainder theorem, as the following example will show: — 

Consider p m (x") m +p m - 1 (x» J" 1 - 1 + . . . +Pi(x")+p and x n - a". Writing 
for a moment £ in place of x n , and a in place of a", we have to deal with 
Pm$ m +p m -iZ m ~ 1 + • • ■ +]h^+Po and £-<x. Now the remainder, when the 
former of these is divided by the latter, is p m a" l +p m -ia m - 1 + . . . +Pia + p . 
Hence the remainder, when p m (x") m +p m -i{x n ) m - 1 + . . . +piaf*+po is divided 
by x n -a n , is 2} m (a") m + p m -i(a») m - 1 + . . . +p x a n +2)o- 



APPLICATION OF REMAINDER THEOREM TO THE DECOMPOSITION 
OF AN INTEGRAL FUNCTION INTO LINEAR FACTORS. 

§ 15.] If a„ a 2 , . . . , a r be r different values of x, for which the 
integral function of the nth degree f(x) vanishes, where r < n, then 
f{x) - (x - c^) (x - Og) . . . (x - a. r )cj> n _ t (.r), cf) n _ r (x) being an integral 
function of x of the (n - r)th degree. 

For, since the remainder, /(a,), when f(x) is divided by x - a„ 
vanishes, therefore f(x) is exactly divisible by x - a„ and we have 

where <£ n _i(.>') is an integral function of x of the (n - l)th degree. 
Since this equation subsists for all values of x, we have 

/(a,) = (a a - a!)4>n-i(<h), 
that is, by hypothesis, = (a. 2 - <*,)<£„_ ,(a 2 ). 

Now, since a! and a 2 are different by hypothesis, a., - a t 4= ; 
therefore <f> n -i(a 2 ) = 0. 

Hence, <f> n -\(- 1 ') is divisible by (x - a,), 
that is, </,„ _ v (x) = (x - a 2 )<f> n _ . 2 (x) ; 

whence f(x) = (x - a t ) (x - a 2 )<j> n _ 2 (x). 

VOL. I H 



98 FACTOKISATION BY MEANS OF chap. 

From this again, 

=/( a 3) = ( a 3 ~ a ( a 3 ~ a z )<£n_ a (a3), 
which gives, since a n a. 2 , a 3 are all unequal, <$> n - 2 { a 3) = ; whence 
<£»-s(a) = (■'-' - a 3 )<£m- 3 (a) J so that 

f(x) = (x - a,) (x - o s ) (a; - a 3 )<£ w _ 3 (.r). 

Proceeding in this way step by step, we finally establish the 
theorem for any number of factors not exceeding n. 

Cor. 1 . If an integral function be divisible by the factors x - a„ 
x - a 2 , . . . , x - a n all of the 1st degree, and all different, it is 
divisible by their product ; and, conversely, if it is divisible by the 
product of any number of such factors, all of the 1st degree and all 
different, it is divisible by each of them separately. The proof of 
this will form a good exercise in algebraical logic. 

Cor. 2. The particular case of the above theorem where the 
number of factors is equal to the degree of the function is of 
special interest. We have then 

f(x) = (x - a,) (X - a 3 ) . . . (X - a ?l )P. 

Here P is of zero degree, that is, is a constant. To determine it 
we have only to compare the coefficients of x n on the left and 
right hand sides, which must be equal by chap, iv., § 24. Now 
f(x) stands for p^ 1 + p^ 1 ' 1 + . . . +p n -i x +Pw Hence V-p , 
and we have 

In other words — If n different values of x can be found for which 
the integral function fix) vanishes, then f(x) can be resolved into n 
factors of the 1st degree, all different. 

The student must observe the "if" here. We have not 
shown that n such particular values of x can always be found, 
or how they can be found, but only that if they can be found the 
factorisation may be effected. The question as to the finding of 
aj, a 2 , , . . , &c, belongs to the Theory of Equations, into which 
we are not yet prepared to enter. 

§ 16.] The student who has followed the above theory will 
naturally put to himself the question, " Can more than n values 



v THE REMAINDER THEOREM 99 

of x be found for which an integral function of x of the nth degree 
vanishes, and, if so, what then 1 " The following theorem will 
answer this question, and complete the general theory of factorisa- 
tion so far as we can now follow it. 

If an integral function of x of the nth degree vanish for more than 
n different values of x, it must vanish identically, that is, each of its 
coefficients must vanish. 

Let a„ a 3 , . . . , a,! be n of the values for which f(x) vanishes, 
then by § 15 above, if p be the coefficient of the highest power 
of x in f(x), we have 

f(x) = p (x - a,) (X - a 2 ) . . . (x - a n ) ( 1 ). 

Now let ft be another value (since there are more than n) for 
which /(.r) vanishes, then, since (1) is true for all values of x, we 
have 

=/(/?) =MP - «,) (fi " «.) ... 08 - «n) (2). 

Since, by hypothesis, a 1} a 2 , . . . , a n and /? are all different, 
none of the differences /5 - a„ /3 - a 2 , . . . , (3 - a n , can vanish, 
and therefore their product cannot vanish. Hence (2) gives p = 0. 

This being so, f(x) reduces to p^x 11-1 +|) 2 a;' l ~ 2 + . . . +p n -jX 
+ p n . We have now, therefore, a function of the {n - l)th degree 
which vanishes for more than n, therefore for more than (n - 1), 
values of its variable. We can, by a repetition of the above 
reasoning, prove that the highest coefficient J9, of this function 
vanishes. Proceeding in this way we can show, step by step, 
that all the coefficients of /(./•) vanish. 

As an example of this case the student may take the following : — 

The integral function 

Q3 - 7 ) (3 - 0) (3 -y) + (y-a) (x - y) (x - a) + (a - /3) (x - a) (x - /3) 
+ (/3- 7 )( 7 - )(a-/3) 

vanishes when x — a, when x = p, and when x = y ; but it is only of the 2nd 
degree in x. We therefore infer that the function vanishes for all values of x, 
that is to say, that we have identically 

{P-y){3B-p){x-y) + (y-a.){x-y){x-a) + (a-fl(x-a)(x-p) 

+ (/*-7)(7-a)(ft-/9) =0. 

That this is so the reader may readily verify by expanding and arranging the 
left-hand side. 



100 INDETERMINATE COEFFICIENTS CHAP; 

Cor. If two integral functions of x, whose degrees are m and n 
respectively, m being > n, be equal in value for more than m different 
values of x, a fortiori, if they be equal for all values of x, that is to 
say, identically equal, then the coefficients of like powers of x in the 
two must be equal. 

We may, without loss of generality, suppose the two functions 
to be each of degree m, for, if they be not equal in degree, this 
simply means that the coefficients of x n+1 , x 1l + 2 , . . . , x m in one 
of them are zero. We have therefore, by hypothesis, 

p ( & m +p l X m ^ 1 + . . . +p m = q<pi m + qiZ m ~ 1 + • • . +2m» 

and therefore 

(p ~q )x m +(p 1 -q 1 )x m - 1 + . . . +(p m -qm) = 0, 

for more than m values of x. 
Hence we must have 

Po-?o = 0, Pi-fr^O, . . ., p m -q m = 0; 

that is, 

Po = % lh = So • • • » Pm = 2W 

This is of course merely another form of the principle of 
indeterminate coefficients. The present view of it is, however, 
important and instructive, for we can now say that, if any 
function of x can be transformed into an integral function of x, then 
this transformation is unique. 

§ 17.] The danger with the theory we have just been ex- 
pounding is not so much that the student may refuse his assent 
to the demonstrations given, as that he may fail to apprehend 
fully the scope and generality of the conclusions. We proceed, 
therefore, before leaving the subject, to illustrate very fully the 
use of the remainder theorem in various particular cases. 

To help the student, we shall distinguish, in the following 
examples, between identical and conditional equations by using 
the sign '* = " for the former and the sign " = " for the latter. 

Example 1. To determine the value of the constant k in order that 

may be exactly divisible by a; + 2. 

The remainder, after dividing by x + 2, that is, by x- ( - 2), is (-2) s 






EXAMPLES 



101 



+ 6( - 2) 2 + 4( - 2) + /-, that is, 8 + k. The condition for exact divisibility is 
therefore 8 + & = 0, that is, k= -8. 
Example 2. To determine whether 

80?- 2a- 3 -7a -2 (1) 

is divisible by (x + 1) (x- 2). 

If we put x— - 1 in the function (1) we get 

-3-2 + 7-2 = 0, 

hence it is divisible by x+1. 
If we put x = 2 we get 

24-8-14-2 = 0, 

hence it is divisible by X - 2. 

Hence by § 15 it is divisible by (x + l)(x-2). The quotient in this case 
is easily obtained, for, since the degree of (1) is the 3rd, we must have 

Bx"-2a?-7x-2 = (x+l){x-2){ax + b) (2), 

where a and b are numbers to be determined. 

If we observe that the highest term 3a- 3 on the left must be equal to the 
product x x x x ax of the three highest terms of the factors on the right, we 
see that SxP^ax 3 , hence a=3. And, since the product of the three lowest 
terms of the factors on the right must be equal to - 2, the lowest term on the 
left, we get - 2&= - 2, that is, b = l. Hence finally 

3a 3 - 2x- - 7x-2=(x+l) (x - 2) (Bx+1). 



Example 3 


If n be a positive integer, 




when 


is divided by 


the rem. is 


that is 


x n - a' 1 


X -a 


a" - a" 


always. 


x n - a" 


x + a 


{-a)"- a" 


if n be even, - 2a n if n be odd. 


x" + «" 


x-a 


a" + a' 1 


2a" always. 


x" + a" 


x + a 


( - a)" + a" 


if n be odd, 2a" if n be even. 



Hence x n -a" is exactly divisible by x-a for all integral values of n, and by 
x + a if n be even. x n + a" is exactly divisible by x + a if n be odd, but is 
never exactly divisible by z — a (so long as «=#0). These results agree with 
those given above, chap, iv., § 16. 

Example 4. To prove that 

a"'{b - c) + b 3 {c -a) + c^{a-b)= -(a + b + c) (b - c) (c - a) (a - b). 
First of all regard F^a^b - c) + b 3 {c-a)+c s (a-b) as a function of a. P 
is an integral function of a of the 3rd degree ; and, if we put a = b, 

Y = b 3 (b -c) + ¥(c -b) + c s (b - b) 
= 0; 



102 CONTINUED DIVISION chap. 

and similarly, if we put a = c, P = ; therefore P is exactly divisible both by 
a - b and by a - c. 

Again, regard P as a function of b alone. It is an integral function of b, 
and vanishes when b — c, hence it is exactly divisible by b-c. We have, 
therefore, 

T = Q{a-b){a-c)(b-c). 

Since P, regarded as a function of a, b, and c, is of the 4th degree, it 
follows that Q must be an integral function of a, b, c of the 1st degree. 
Hence, I, m, n being mere numbers which we have still to determine, we 
have 

w\b - c) + b 3 (c - a) + c 3 (a - b) = (la + mb + nc) {b-c) (a-c) (a-b) 

= —{la + mb + nc) (b -c){c- a) (a - b). 
To determine I we have merely to compare the coefficients of a 3 b on both 
sides. It thus results by inspection that 1 = 1; and similarly m = l, n = l; 
the last two inferences being also obvious by the law of symmetry. We have 
therefore finally 

a s (b - c) + b\c -a) + c 3 (a -b)= - (a + b + c)(b-c) (c-a)(a- b). 
Example 5. To show that 

Pee 2b 2 c 2 + 2c 2 a 2 + 2a 2 b 2 - a* - b* - c 4 
= (a + b + c) ( - a + b + c) (a - b + c) (a + b - c). 
First, regarding P as an integral function of a, and dividing it hy a + b + c, 
that is, by a - ( - b - c), we have for the remainder 
2J 2 c 2 +2c 2 ( - b - cf + 2b\ - b- cf- ( - 4-c) 4 - 6*-c* 

== 2b 2 c 2 + 2b 2 <? + 4.bc s + 2c 4 + 2¥ + ib 3 c + 2b 2 c 2 
- 6 4 - ib 3 c - 6b 2 c 2 - Abe 3 -c^b^-c 4 - 
=0. 
Hence P is exactly divisible by a + b + c. 

Observing that the change of a into - a, or of b into - b, or of c into - c 
does not alter P, all the powers of these letters therein occurring being even, 
we see that P must also be divisible by -a + b + c, a-b + c, and a + b-c. We 
have thus found four factors of the 1st degree in a, b, c, and there can be no 
more, since P itself is of the 4th degree in these letters. This being estab- 
lished, it is easy to prove, as in Example 4, that the constant multiplier is 
+ 1 : and thus the result is established. 



EXPANSION OF RATIONAL FRACTIONS IN SERIES BY MEANS OF 

CONTINUED DIVISION. 

§ 18.] If we refer back to §§ 5 and 6, and consider the analysis 
there given, we shall see that every step in the process of long 
division gives us an algebraical identity of the form 















V DESCENDING CONTINUED DIVISION 103 

where Q' is the part of the quotient already round, and R' the 
residue at the point where we suppose the operation arrested. 

For example, if we stop at the end of the second operation, 
8/ + 8/ - 20/ + 40/ - 50/ + 30a - 10 
2/ + 3/ - 4/ + 6.c - 8 

*_ 2/ + 8/ - 6/ + 14s- 10 

~~ X+ 2/+3/-4/+6*-8 ' 
Again, instead of confining ourselves to integral terms in x, and 
therefore arresting the process when the remainder, strictly so 
called, is reached, we may continue the operation to any extent. 
In this case, if we stop after any step we still get an identity of 
the form 

where 

Q' = AxP + BxP- 1 + . . . + Kx + L + M/:c + . . .+T/& (2), 
and 

R' = U^ 1 - 1 + Vx n ~ 2 + . . . + Z (3). 

This process may be called Descending Continued Division. 

For example, consider 

^+2.r 2 + 3a;+4 
x 2 + x + 1 
and let us conduct the division in the contracted manner of § 12, but insert 
the powers of x for greater clearness. 

a- 2 + a;+l 



x 3 + 2x 2 + Sx + i 

+ x 2 + 2x + 4 

x + d 



.2 3 1_ 2_ 

x i + x 5 



X + 1+- + ---. + -. + - 
X iV" ft, 



X 

3 2 



x ar 

L I. 

x 2 x 3 



2 1_ 

x* 
3 2 



+ ar* a* 



whence ^±g±|£±i = 1 2 3 1 2 ^fa-a^ 

ar+sc+l a; a; 2 a,- 5 ar 4 a^ 5 a^+a+l 

an identity which the student should verify by multiplying both sides by 
ar + a-+l. ' 



104: ASCENDING CONTINUED DIVISION chap. 

§ 19.] When we prolong the operation of division indefinitely 
in this way we may of course arrange either dividend or divisor, 
or both, according to ascending powers of x. Taking the latter 
arrangement we get a new kind of result, which may be illus- 
trated with the fraction used above. 
We now have 



i-x-x- + 3x 3 -2x*-x 5 



i + 3x + 2x-+ x 3 

- x-2x 2 + x 3 

- x- + 2z 3 

+ Bx 3 + x i 

- 2x i - 3ar> 

- x r ' + 2x 6 

+ 3x 6 + x 7 , 

wh ence 5 = 4 - x - x- + Bx 3 - 2x* - x 5 + , , ( o )• 

x 2 + x+l 1+x + x 2 

And, in general, proceeding in a similar way with any two 
integral functions, A^ and D n , we get 

where Q" = A + Bx + . . . + Kafl (7), 

K" = L + M.c + . . . + T.:"- 1 (8). 

This process may be called Ascending Continued Division. 

§ 20.] The results of §§ 18, 19 show us that we can, by the 
ordinary process of continued long division, expand any rational fraction 
as a "series" either of descending or ascending powers of x, continu- 
ing as many terms as ice please, plus a " residue," which is itself a 
rationed fraction. 

And there is no difficulty in showing that, when the integer q is 
given, each of the expansions (2) and (6) o/§§ 18 and 19 is unique. 

The proof depends on the theory of degree, and may be left 
as an exercise for the reader. 

These series (the Q' or Q" parts above) belong, as we shall 
see hereafter, to the general class of " Recurring Series. " 

The following are simple examples of the processes we have been describ- 
ing:— 

■ l+x + x 2 +. . .+.<" f- n — - (9). 



1 -x \-x 

111 1 l'.<" 

— -5-3-?-- • "x^^x (10) - 



EXPANSION THEOREMS 105 

-l-s+a?-. . . +(-l)»x»- [ ?\ (11). 



1 + a; ' 1 + a: 

a; a' 2 a: 3 a!" 1 + x 



•i" r -.3 • • • ,,„ t , .. 



l + ar + . . . +x n 



= 1 -x+ x"** - a;»+ 2 + a?»+ 2 - ai 2 "+ 3 + . . . 

a /r+])(n+-l) 
i r iir+r _ ^.nr+r+l . . (13). 

1 + x + . . . + a;" 



1 , „ „ , , ,, (/i + 2)a;»+ 1 -(n + l)a.'»+ 2 



(1 -a-) 2- ' v ' (1 -a-) 2 

(14). 



EXPRESSION OF ONE INTEGRAL FUNCTION IN POWERS OF 

ANOTHER. 

§ 21.] "We shall Lave occasion in a later chapter to use two 
particular cases of the following theorem. 

If P and Q be integral functions of the rnth and nth degrees 
respectively (m > n), then P may always he put into the form 

P = R + R,Q + R 2 Q 2 + . . . + iy^ (1), 

where R , E,, . . . , R^ are integral functions, the degree of each of 
which is re — 1 at most, and p is a positive integer, which cannot 
exceed mjn. 

Proof. — Divide P by Q, and let the quotient be Q and the 
remainder R . 

If the degree of Q be greater than that of Q, divide Q by 
Q, and let the quotient be Q, and the remainder Rj. 

Next divide Qi by Q, and let the quotient be Q., and the 
remainder R 2 , and so on, until a quotient Qp_i is reached 
whose degree is less than the degree of Q. Q^-,, for con- 
venience, we call also R^. "We thus have 

P = Q Q + R 
Qo = Q,Q + R, 

Hp-i = K-p- 



106 EXPANSION THEOREMS CHAP. 

Now, using in the first of these the value of Q given by the 

second, we obtain 

P = (Q,Q + R,)Q + R„, 

= R + R,Q + Q,Q 2 . 

Using the value of Q, given by the third, we obtain 

P = R + R 1 Q + R,Q 2 + Q 2 Q 3 (3). 

And so on. 

We thus obtain finally the required result ; for, R , R,, . . ., 
Rp being remainders after divisions by Q (whose degree is n), 
none of these can be of higher degree than n-1; moreover, 
since the degrees of Q , Q„ Q., . . ., Qp-i are m - n, m - 2n, 
m - 3n, . . ., m- np, p cannot exceed m/n. 

The two most important particular cases are those in which Q = a:-a and 
Q = x 2 + (3x + y. We then have 

T = a + a 1 (x-a) + . . .+a„{x-a)", 

where a , «i, • • • , «« ai ' e constants ; 

P = (oo + b&) + (rti + hx) (x°- + (3x + y) + . . . +(a p + b p x)(x" + fix + y) p, 
where a , a u . . ., a p ,bo,h, • • •, ^ are constants, a,ndj»mj2. 

Example 1. 

Let P=5« s -ll£c 2 +l0aj-2, 

Q = x-1. 

The calculation of the successive remainders proceeds as follows (see 

§ 13):— 

5 -11 +10 -2 

+ 5 - 6+4 

5 - 6 + 4| + 2 
0+5-1 



5 - l|+ 3 

0+5 



6|+ 4 


1 + 5; 

and we find 

5... 3 -1U" + 10a:-2 = 2 + 3(o;- 1) + 4(3-- l) 2 +5{x l) 3 . 

Example 2. 

P = x« + 3x 7 + 4z 6 + 4a; 2 + Sx + 1, 

Q = x 2 -x + l. 

The student will find 

R = 11 x, • Ri = - 22a! + 7, R; = 1 9a- - 22, R 3 = 7x + 15, R 4 = 1 



v EXPANSION THEOREMS 107 

so that 

Y = n.r + {-22x+7)(x 2 -x+l) + {19x-22)(x i -x+l) ,i 
+ (7x + 15) (.c 2 - x + 1 ) 8 + (k 2 - x + 1 ) 4 . 

§ 22.] If a,, a 9 , . . ., a rt &e w constants, any two or more of 
which may be equal, then any integral function of x of the nth degree 
may be put into the form 

A + A,(.c - a,) + A 2 (x - a l )(x - a, 2 ) + A 3 (x - a x )(x - a 2 )(x - a 3 ) + . . . 

+ A H (x - a,) (a; - a 2 ) . . . (x - a n ) (1 ), 

where A , A,, A 2 , . . . , A n are constants, any one of which except A n 
may be zero. 

Let P ?i , be the given integral function, then, dividing P n by 
x - a, , we have 

P» = P»-i(s- «,) + Ao (2), 

where A is the constant remainder, which may of course in any 
particular case be zero. 

Next, dividing P n _! by (x — a.,), we have 

Pn-.^Pn-^-O + A, (3); 

and so on. 

Finally, P : = A n (.?; - a n ) + A n _ , (n + 1 ). 

Using these equations, we get successively 

P n = A + A,(x - a,) + (x - a,)(x - a.,)? n _ 2 , 
= A + A,(.?; - «,) + A 2 (x - a t ) (x - «,) 

+ (x-a 1 )(x-a s )(x-a a )'P n - a , 

P n = A + A,(.?; - «,) + A,,(r - <?,) (a- - a,) + A 3 (x - a,) (x - a 2 ) (x - a 3 ) 
+ . . . + A n (x - «,) (./• - a, 2 ) . . . (x - a n ). 

This kind of expression for an integral function is often 
useful in practice. 

Knowing a priori that the expansion is possible, we can, if we choose, 
determine the coefficients by giving particular values to x. But the most 
rapid process in general is simply to carry out the divisions indicated in the 
proof, exactly as in Example 1 of last paragraph. 

Thus, to express a- 3 -! in the form A„ + A\(x- 1)4- A-..(.r- 1) {x- 2) 
+ A 3 {x - 1) (,r - 2) (x - 3), we calculate as follows : — 



108 



EXERCISES IX 



CHAl'. 






1 +0 +0 -1 

+1 +1 +1 

1 +1 +l|0 
+2 +6 



1 + 3| + 7 
+3 



1+6 



Henceo.- 3 -l = + 7(a--l) + 6(a:-l)(a:-2) + (x-l)(a.'-2)(a;-3). 



Exercises IX. 

Transform the following quotients, finding both integral quotient and 
remainder where the quotient is fractional. 

(1.) (sc 5 -5a 8 +5ar ! -l)/(ar ! +3a;+l). 

(2. ) (6a: 6 + 2a; 5 - 19a 4 + 3lar 5 - 37a; 2 + 27a: - 7)/(2a- 2 - 3a- + 1 ). 

(3. ) (4a; 5 - 2x* + Zx 3 -aj+1 )/(a; 2 - 2a; + 1). 

(4. ) (as 5 - 8aj+ 15) (x" + Sx + 15)/(x- - 25). 

(5. ) {(as- 1) {x- 2) (x - 3) (as - 4) (x - 5) - 760(.i- - 6) + 120(a: - 7)} + (a! - C) 
(x-7). 

(6.) (x 6 + 4a: 5 - 3a- 4 - 16x 3 + 2x 2 + x + 3)/{x 3 +4x 2 + 2x+l). 

(7.) (27a* + 10as s +l)/(3a?-2a: + l). 

(8. ) (x? - 9a? + 23a; - 15) (a: - 7)/(as s - Sx + 7). 

(9. ) (a; 3 + fix* + && + t& + i)/(x" ~ ix + 1 ). 
(10.) (a; 4 + fce 3 + ia- 2 + Aa: + i)/(x 2 + 2aj + 1 ). 
(11.) (a/ + T ^)/(2a:+l). 
(12. ) (a; 2 - a; + 1 ) (x s - 1 )/{x* + ar» + 1 ). 
(13.) (ai l V-flty u )/(a:-y). 
(14. ) (9« 4 + 2o 2 fi s + &*)/(3a a + 2«fe + 6 2 ). 
(15.) (a 7 + 6 7 )/(a + 6). 
(16.) (a; 4 + y 4 - 7x-y°-)l{x> + Zxy + y"). 
(17.) (x 5 - 2afy + 4.r 3 2/ 2 - 8a*Y + 1 6a-*/ 4 - 32y s )ftx* - 8)/). 
(18. ) (a; 4 + 5arfy + 7a?V + 15xif + 12y 4 )/(a- + Ay). 
(19.) (l+.r + x 2 + a? + a^ + x 6 + x 7 + a* + a; 9 + a; 15 )/(l- 
(20.) (a; 6 -5a^ + 8)/(a^ + a; + 2). 
(21 . ) {abx 3 + {ac - W )x 2 - (a/+ cd)x + <//} /(aa: - d). 
(22. ) {a 2 6 2 + uV + fe 2 c 2 + 2a?bc - 2ab~c - 2abc" } 4- { « 2 ■ 
(23. ) (1 + b+c - be - Ve '- bc n -)/(l - be). 
(24.) {(ax + by) 3 + (ax-by)*-(ay-bxf+(ay + bx) 3 }/{(a + b)hr-3ab(x>-y-)}. 
(25. ) { (a 2 + b-) 3 + b 3 a 3 }j { (a + b) n - -ba}. 
(26.) {(x 2 + xy + y y-(x i -xy + i/-) i }l{x i + Zxhj- + y i }. 
(27. ) { (x + yY -x>- /}/(a? + *>/ + 2/ 2 ) 2 . 
(28.) {(x+l) 6 -.r 6 -l}/{a? + x + l). 
(29.) {o6(a? + i/)+arj/(o s + 6 2 )}/{o6(ar'-y s )-a^(a 2 - 5 2 )}. 
(30.) (a !i + 2a 3 b- + 2a 2 b 3 -3b' i )/(ar-2ab + b-). 



t' 5 + X 6 ). 



(«-S)(a-e)}. 



v EXERCISES IX 109 

(31.) (x i -Zx 2 -2;r + 4)/(.r + 2). 

(32.) (jc*- 4a?-Zia?+76x+V)5)f(x-7). 

(33. ) Find the remainder when' a? - 6x 2 + 8a: - 9 is divided by 2x + 3. 

(34.) Find the remainder when pa? + qx 2 + qx+p is divided by x- 1 ; and 
find the condition that the function in question be exactly divisible by x 2 - 1. 

(35.) Find the condition that Ax 2m + Bx">y n + Cy 2 " be exactly divisible by 
]\v m + Qy". 

(36.) Find the conditions that x 3 + ax" + bx + c be exactly divisible by 
x- +px + q. 

(37. ) If x - a be a factor of x 2 + 2ax - 36'-, then a . = ± b. 

(38.) Determine X, /u, v, in order that x 4 + Sx 3 + \x 2 + fix + v be exactly 
divisible by (x 2 -l)(x + 2). 

(39.) If x 4 + Ax 3 + Qpx 2 + 4 qx + r be exactly divisible by x 3 + Bx 2 + 9x + S, 
find p, q, r. 

(40.) Show that px 3 + (p 2 + q)x 2 + {2pq + r)x + q 2 + s and px 3 + (p 2 - q)x 2 
+ rx-q 2 + s either both are, or both are not, exactly divisible by x 2 +p>x + q. 

(41.) Find the condition that (x m + x m ~ l + . . . +l)/(a,-" + a: n - 1 + . . . +1) 
be integral. 

(42.) Expand l/(3.c + l) in a series of ascending, and also in a series of 
descending, powers of x ; and find in each case the residue after n + 1 terms. 

(43.) Express l/iar-ax + x 2 ) in the form A + Bz + Cx 2 + Dx 3 + R, where A, 
B, C, D are constants and R a certain rational function of x. 

(44.) Divide 1+ x + ^ + 1^-3 + • • ■ t>yl-sc 

(45.) Show that, if y<\, then approximately 1/(1 +y) = l -y, 1/(1- y) 
= 1 + ?/, the error in each case being 100;/ 2 per cent. 

Find similar approximations for 1/(1+?/)" and 1/(1 -y) n , where n is a 
positive integer. 

(46.) If a>l, show that «">1 +n{a- 1), n being a positive integer. 
Hence, show that when n is increased without limit a n becomes infinitely 
great or infinitely small according as«> or <1. 

(47.) Show that when an integral function f(x) is divided by (x-ai) 
(x — ao) the remainder is {/(a-j)(a;-ai) -/(ax)(a;-ao)}/(a2- aj). Generalise 
this theorem. 

(48.) Show that f(x)-/(a) is exactly divisible by x-a; and that, if 
f(x)=p x n +piX n - 1 +p&c n - 2 + . . . +p n , then the quotient is x( x ) -pox n ~ l 
+ (p a+pi)x n - 2 + (p l) a 2 +p l a.+2h2)x n - 3 +. . . +(p a n ~ l +p x a n ~ 2 + . . .+p n - l ). 

Hence show that when f(x) is divided by (x-a) 2 the remainder is 
X(o) (z-a)+/(a), 
where f(a)=p a n +pia n - 1 +. . . +p„, 

x(a)-np a n - 1 + (n-l)2ha n - 2 +. . . +p n - 1 . 

(49.) If x"+2hx"~ 1 +. • . +p n and x"- 1 + q x x n - 2 + . . . +q»-i have the 
same linear factors with the exception of a: -a, which is a factor in the first 
only, find the relations connecting the coefficients of the two functions. 

(50.) If, when y + c is substituted for x in x n + aix n ~ 1 + . . . +a„, the 



110 EXERCISES IX chap. V 

result is y n + &i2/ n_1 + . . . +b n , show that b n , £>„_i, . . . , b x ai-e the remainders 
when the original function is divided by x - c, and the successive quotients 
by x-c. Hence obtain the result of substituting y + 3 for x in a^-l&B* 
+ 20x 3 -17x 2 -x + B. 

(51.) Express (x 2 + 3x + l)* in the form A + B(a: + 2) + C(a: + 2) 2 + &c., and 
also in the form Ax + B + {Cx + B)(x 2 + x + l) + ('Ex + F)(x- + x + lf + kc. 

(52.) Express x A + x? + x 2 + x + l in the form A + Ai(x + 1) + A- 2 (x+l)(x+3) 
+ A 3 (a: + l)(a; + 3)(a; + 5)+A4(a: + l)(a; + 3)(a; + 5)(.r + 7). 

(53.) If, when P and P' are divided by D, the remainders are R and R', 
show that, when PP' and RR' are divided by D, the remainders are identical. 

(54.) When P is divided by D the remainder is R ; and when the integral 
quotient obtained in this division is divided by D' the remainder is S and the 
integral c|uotient Q. R', S', Q' are the corresponding functions obtained by 
first dividing by D' and then by D. Show that Q = Q', and that each is the 
integral quotient when P is divided by DD'; also that SD + R = S'D' + R', 
and that each of these is the remainder when P is divided by DD'. 






CHAPTEE VI. 

Greatest Common Measure and Least Common 

Multiple. 

§ 1.] Having seen how to test whether one given integral 
function is exactly divisible by another, and seen how in certain 
cases to find the divisors of a given integral function, we are 
naturally led to consider the problem — Given two integral 
functions, to find whether they have any common divisor or not. 

We are thus led to lay down the following definitions : — 

Any integral function of x which divides exactly two or more given 
integral functions of x is called a common measure of these functions. 

The integral function of highest degree in x zchich divides exactly 
each of two or more given integral functions of x is called the greatest 
common measure (G.C.3I.) of these functions. 

§ 2.] A more general definition might be given by suppos- 
ing that there are any number of variables, x, y, z, u, &c. ; in 
that case the functions must all be integral in x, y, z, u, &c, and 
the degree must be reckoned by taking all these variables into 
account. This definition is, however, of comparatively little 
importance, as it has been applied in practice only to the case of 
monomial functions, and even there it is not indispensable. As 
it has been mentioned, however, we may as well exemplify its 
use before dismissing it altogether. 

Let the monomials be 432aWy 4 2, 270aWyV, 90a7v; 3 yV, 
the variables being x, y, z, then the G.C.M. is x 2 ifz, or C'V':, 
where C is a constant coefficient (that is, does not depend on the 
variables x, y, z). 

The general rule, of which the above is a particular case, is 
as follows : — 



112 G.C.M. BY INSPECTION 



CHAP. 



The G.C.M. of any number of monomials is the product of the 
variables, each raised to the lowest power * in which it occurs in any 
one of the given functions. 

This product may of course be multiplied by any constant 
coefficient. 

G.C.M. OBTAINED BY INSPECTION. 

§ 3.] Returning to the practically important case of integral 
functions of one variable x, let us consider the case of a number 
of integral functions, P, P', P", &c, each of which has been re- 
solved into a product of positive integral powers of certain factors 
of the 1st degree, say x — a, x - (3, x - y, &c. ; so that 

P =p(x-a)«(x-P)\x- y y..., 

P' =p'(x - a) a '(x - (3) b '(x - yY' ..., 
V"=p"{;x-ar'{x-pr{x-yr ..., 

By § 15 of chap, v., we know that every measure of P 
can contain only powers of those factors of the 1st degree that 
occur in P, and can contain none of those factors in a higher 
power than that in which it occurs in P, and the same is true for 
P', P", &c. Hence every common measure of P, P', P", &c, can 
contain only such factors as are common to P, P', P", &c. Hence 
the greatest common measure of P, P', P", &c, contains simply all the 
factors that are common to P, P', P", &c, each raised to the lowest 
power in which it occurs in any one of these functions. 

Since mere numbers or constant letters have nothing to do 
with questions relating to the integrality or degree of algebraical 
functions, the G.C.M. given by the above rule may of course be 
multiplied by any numerical or constant coefficient. 

Example 1. 

P=2x 2 -6a: + 4 = 2(a--l)(a--2), 
P' = 6a: 2 - 6z - 12 = 6(x + 1 ) (a; - 2). 
Hence the G.C.M. of P and P' is a- 2. 
Example 2. 

P = a* - 5as* + 7a: 3 + x 2 - 8x + 4 = (x - 1 )-(x + l)[x- 2) 2 , 
P' = x e - 7x>+ 17a- 4 - Ux 3 - 10a: 2 + 20a;- S = (x- l)-(o. + l) (a;-2) 3 , 
P" = x 5 -Zx*- a? + 7a," - 4 = (x - 1 ) (x + l) 2 (x - 2) 2 . 
The G. C. M. is (a; - 1) (a>+ 1) (as - 2) 2 , that is, x 4 -4x 3 + 3a; 2 + 4x - 4. 

* If any variable does not occur at all in one or more of the given func- 
tions, it must of course be omitted altogether in the G.C.M. 



vi CONTRAST BETWEEN ALGEBRAICAL & ARITHMETICAL G.C.M. 113 

§ 4.] It will be well at this stage to caution the student 
against being misled by the analogy between the algebraical and 
the arithmetical G.C.M. He should notice that no mention is 
made of arithmetical magnitude in the definition of the algebraical 
G.C.M. The word " greatest " used in that definition refers 
merely to degree. It is not even true that the arithmetical 
G.C.M. of the two numerical values of two given functions of x, 
obtained by giving x any particular value, is the arithmetical 
value of the algebraical G.C.M. when that particular value of x 
is substituted therein ; nor is it possible to frame any definition 
of the algebraical G.C.M. so that this shall be true.' 1 " 

The student will best satisfy himself of the truth of this remark by study- 
ing the following example : — 

The algebraical G.C.M. of x~-2,x + 2 and x 2 -x-2 is x-2. Now put 
sc=31. The numerical values of the two functions are 870 and 928 respectively ; 
the numerical value of x- 2 is 29 ; but the arithmetical G.C.M. of 870 and 928 
is not 29 but 58. 

LONG RULE FOR G.C.M. 

§ 5.] In chap. v. we have seen that in certain cases in- 
tegral functions can be resolved into factors ; but no general 
method for accomplishing this resolution exists apart from the 
theory of equations. Accordingly the method given in § 3 
for finding the G.C.M. of two integral functions is not one of 
perfectly general application. 

The problem admits, however, of an elementary solution by 
a method which is fundamental in many branches of algebra. 
This solution rests on the following proposition : — 

If A = BQ + R, A, B, Q, R being all integral functions of x, 
then the G.C.M. of A and B is the same as the G.C.M. of B and R. 

To prove this we have to show — 1st, that every common 



* To avoid this confusion some writers on algebra have used instead of the 
words "greatest common measure" the term "highest common factor." We 
have adhered to the time-honoured nomenclature because the innovation in 
this case would only be a partial reform. The very word /actor itself is used 
in totally dilferent senses in algebra and in arithmetic ; and the same is true 
of the words fractional and integral, with regard to which confusion is no less 
common. As no one seriously proposes to alter the whole of the terminology 
of the four species in algebra, it seems scarcely worth the while to disturb an 
old friend like the G.C.M. 

VOL. I I 



114 LONG KULE FOR G.C.M. chap. 

divisor of B and R divides A and B, and, 2nd, that every common 
divisor of A and B divides B and R. 

Now, since A = BQ + R, it follows, by § 4 of chap, v., that 
every common divisor of B and R divides A, hence every common 
divisor of B and R divides A and B. 

Again, R = A - BQ, hence every common divisor of A and B 
divides R, hence every common divisor of A and B divides B and R. 

Let now A and B be two integral functions whose G.C.M. is required; 
and let B be the one whose degree is not greater than that of the other. 
Divide A by B, and let the quotient be Q,, and the remainder R,. 

Divide B by R,, and let the quotient be Q 2 , and the remainder R 2 . 

Divide Ri by R 2 , and let the quotient be Q 3 , and the remainder R 3 , 
and so on. 

Since the degree of each remainder is less by unity at least than the 
degree of the corresponding divisor, R 15 R 2 , R 3 , &c, go on diminishing in 
degree, and the process must come to an end in one or other of two ways. 

I. Either the division at a certain stage becomes exact, and the 
remainder vanishes ; t 

II. Or a stage is reached at which the remainder is reduced to a 
constant. i i 

Now we have, by the process of derivation above described, 

A = BQ X + R, 
B - R^a + R 2 
R, = R 2 Q 3 + R 3 



M 1 )- 



Hence by the fundamental proposition the pairs of functions 

A ( B | R, ( Ro 1 R w -2 \R>i-i ( ..li v, flVP fV, P eomp ci HM 

B IRJR, jR, / • * • R B _ 1 J | R„ / 

In Case I. H n = and R ;i _ 2 = Q,iR«-i. Hence the G.C.M. of 
R, t _ 2 and Rn-u that is, of Q^R^-! and R n _i, is R»_i, for this 
divides both, and no function of higher degree than itself can 
divide R, t _i. Hence R 7l _! is the G.C.M. of A and B. 

In Case II. R ;l = constant. In this case A and B have no 
G.C.M., for their G.C.M. is the G.C.M. of R ;l _, and R n> that is, 
their G.C.M. divides the constant R n . But no integral function 






VI 



MODIFICATIONS OF LONG RULE 



115 



(other than a constant) can divide a constant exactly. Hence 
A and B have no G.C.M. (other than a constant). 

If, therefore, the process ends with a zero remainder, the last divisor 
is the G.C.M. ; if it ends with a constant, there is no G.C.M. 

§ 6.] It is important to remark that it follows from the 
nature of the above process for finding the G.C.M., which con- 
sists essentially in substituting for the original pair of functions 
pair after pair of others which have the same G.C.M., that we 
may, at any stage of the process, multiply either the divisor or the 
remainder by an integral function, provided we are sure that this 
function and the remainder or divisor, as the case may be, have no 
common factor. We may similarly remove from either the divisor or 
the remainder a factor which is not common to both. We may remove 
a factor which is common to both, provided we introduce it into the G. C M. 
as ultimately found. It follows of course, a fortiori, tluit a numerical 
factor may be introduced into or removed from divisor or remainder at any 
stage of the process. This last remark is of great use in enabling us 
to avoid fractions and otherwise simplify the arithmetic of the pro- 
cess. In order to obtain the full advantage of it, the student 
should notice that, in what has been said, "remainder" may mean, 
not only the remainder properly so called at the end of each sepa- 
rate division, but also, if we please, the " remainder in the middle of 
any such division'' or "residue," as we called it in § 18, chap. v. 

Some of these remarks are illustrated in the 
examples : — 



following 



Example 1. 

To find the G.C.M. of x 5 - 2x* - 2x? + 8x 2 - 7x + 2 audz 4 - 

x 4 - ix + 3 



ix + 3. 



X*- 
X s 


-2x i -2x 3 + 8x 2 ~ 1x + 2 
- Ax 2 + Zx 


-2) 


- 2x 4 - 2r? + 12x 2 - 10a; + 2 




ar*+ X s - 6x 2 + 5x- 1 
x* - 4a: + 3 




x 3 - 6x 2 + 9x-i 

x* - 4x + 3 
x i -Qx 3 + Qx 2 - 4x 




3) 6ar»- 9a; 2 +3 




23?- 2.x 2 +1 
2.r 3 -12a: 2 -l-18a:-8 




9) 9a^-18x + 9 



X +1 



a. 3 -6x- 2 + 9a---4 
x +2 



c 2 - 2x+l 



116 



EXAMPLES 



CHAP. 



X 3 - 


- 6a; 2 + 9a; - 4 

- '2a; 2 + x 


-4) - 


-4a; 2 +8a:-4 




a; 2 -2a; + l 
a- 2 -2a:+l 



x 2 -2a;+l 



a; +1 



Hence the G.C.M. is xr-2x + l. 

It must be observed that what we have written in the place 
of quotients are not really quotients in the ordinary sense, owing 
to the rejection of the numerical factors here and there. In 
point of fact the quotients are of no importance in the process, 
and need not be written down ; neglecting them, carrying out the 
subtractions mentally, and using detached coefficients, we may 
write the whole calculation in the following compact form : — 



_ o 



1-2-2+ 8- 7 + 2 

-'2-2 + 12-10 + 2 

1 + 1- 6+ 5-1 

1- 6+ 9-4 



4 + 
1- 



8-4 
2 + 1 







1+0+0- 4+3 

6-9+ 0+3 

2-3+ 0+1 

9-18 + 9 

1- 2 + 1 



-^3 
-j-9 



zx 



G.C.M., x 2 

Example 2. 

Required the G.C.M. of 4x 4 + 26a,- 3 + 41a; 2 - 2a;- 24 and 3a!* + 20a? + 32a; 2 
- 8a; - 32. 

Bearing in mind the general principle on which the rule for finding the 
G.C.M. is founded, we may proceed as follows, in order to avoid large num- 
bers as much as possible : — 



4 + 26 + 41- 
x2 1+ 6+ 9 + 


2- 

6 + 


24 

8 


2 + 12 + 18+ 12+ 16 
7 + 44+ 68+ 16 
1+29 + 146 + 184 


-+23 23 + 138 + 184 

1+ 6+ 8 



3 + 20 + 32- 
2+ 5- 



26- 



32 
56 



-53-318-424 

1+ 6+ 8 



-53 



The G.C.M. is a 2 + 6a: + 8. 

Here the second line on the left is obtained from the first by subtracting 
the first on the right. By the general principle referred to, the function 
se 4 + 6ar 3 + 9a? + 6a; + 8 thus obtained and 3a; 4 + 203 3 + 32a; 2 - Sx - 32 have tlie 
same G.C.M. as the original pair. Similarly the fifth line on the left is the 
result of subtracting from the line above three times the second line on the 
right. 



VI 



SECOND RULE FOR G.C.M. 117 



If the student be careful to pay more attention to the prin- 
ciple underlying the rule than to the mere mechanical application 
of it, he will have little difficulty in devising other modifications 
of it to suit particular cases. 



METHOD OF ALTERNATE DESTRUCTION OF HIGHEST AND 

LOWEST TERMS. 

§ 7.] If /, m. p, q be constant quantities (such that Iq - mp is not 
zero), and if 

P = ZA + mB (1), 

Q = M + qB (2), 

where A and B, and therefore P and Q, are integral functions, then 
the G.C.M. ofP and Q is the G.C.M. of A and B. 

For it is clear from the equations as they stand that every 
divisor of A and B divides both P and Q. Again, we have 

qP - wQ = q(IA + mS) - m (pA + qB) = (Iq - mp)k (3), 
-pP + ZQ = -p(IA + nzB) + I(pA + qB) = (Iq - mp)B (4) ; 

hence (provided Iq - mp does not vanish), since I, p, m, q, and 
therefore Iq - mp, are all constant, it follows that every divisor of 
P and Q divides A and B. Thus the proposition is proved. 

In practice I and m and p and q are so chosen that the 
highest term shall disappear in I A + m*B, and the lowest in 
pA + qB. The process will be easily understood from the follow- 
ing example : — 

Example 1. 

Let A = 4ar* + 26a- 3 + 41a; 2 - 2x - 24, 
B = 3a; 4 + 20a? + 32a; 2 - 8x - 32 ; 
then -3A + 4B = 2ar> + 5a; 2 -26a;-56, 

4 A - 3B = 7x* + 44a.- 3 + 68a; 2 + 16a;. 

Rejecting now the factor x, which clearly forms no part of the G.C.M., we 
have to find the G.C.M. of 

A' = 7a;' , + 44a: 2 + 68a;+16 I 
B' = 23? + 5a; 2 -26a: -56. 
Repeating the above process — 

2A'- 7B' = 53a- 2 + 318a; +424, 
7A' + 2B'=53a- 3 + 318.v 2 +424a\ 



118 TENTATIVE PROCESSES 



CHAP. 



the G.C.M. of which is 53a; 2 + 318a; + 424. Hence this, or, what is equivalent 
so far as the present quest is concerned, x 2 + Qx + 8, is the G.C.M. of the two 
given functions. 

When the functions differ in degree, we may first destroy 
the lowest term in the function of higher degree, divide the 
result by x, and replace the function of higher degree by the 
new function thus obtained. We can proceed in this way until 
we arrive at two functions of the same degree, which can in 
general be dealt with by destroying alternately the highest and 
lowest terms. 

Detached coefficients may be used as in the following 
example : 

Example 2. 

To find the G.C.M. of % A - 3ar> + 2a; 2 + x - 1 and x 3 - x 2 - 2x + 2, we have 
the following calculation : — 



A 
B 


1-3+2+1-1 
1-1-2+2 


A' = (2A + B)/* 
B' = B 


2-5+3+0 
1-1-2+2 


A" = A'/x 
B" =(-A' +2B') 


2-5 + 3 

3-7 + 4 


A"' = (-4A" + 3B")/x 
B'"= -3A" + 2B" 


1-1 
1-1 



Hence the G.C.M. is x-1. 

The failing case of the original process, where Jq -mp = 0, may be treated 
in a similar manner, the exact details of which we leave to be worked out as 
an exercise by the learner. 

§ 8.] The following example shows how, by a semi-tentative 
process, the desired result may often be obtained very quickly : — 

Example. 

A = 2a; 4 -3a,- 3 -3a; 2 + 4, 
B = 2x i -x s -9x i + ix + i. 

Every common divisor of A and B divides A - B, that is, 

- 2x a + 6.c 2 - ix, that is, rejecting the numerical factor - 2, 

x(x 2 — 3x + 2), that is, x (x - 1) (.c - 2). We have therefore merely 

to select those factors of x(x-\)(x-2) which divide both A 









VI PROPOSITIONS REGARDING G.C.M. 119 

and B. x clearly is not a common divisor, but we see at once, 
by the remainder theorem (§ 13, chap, v.), that both x—\ and 
x - 2 are common divisors. Hence the G.C.M. is (x - 1) (a; — 2), 
or x 2 - 3a + 2. 

§ 9.] The student should observe that the process for finding 
the G.C.M. has the valuable peculiarity not only of furnishing 
the G.C.M., but also of indicating when there is none. 

Example. 

k=x--Sx + \, 
B = o; 2 -4a;+6. 

Arranging the calculation in the abridged form, we have 

1-3+1 I 1-4+6 
2+1 -1+5 



11 I 
The last remainder being 11, it follows that there is no G.C.M. 

G.C.M. OF ANY NUMBER OF INTEGRAL FUNCTIONS. 

§ 10.] It follows at once, by the method of proof given in 
§ 5, that every common divisor of two integral functions A and B is 
a divisor of their G. CM. 

This principle enables us at once to find the G.C.M. of any 
number of integral functions by successive application of the 
process for two. Consider, for example, four functions, A, B, C, D. 
Let Gi be the G.C.M. of A and B, then G v is divisible by every 
common divisor of A and B. Find now the G.C.M. of Gj and 
C, G, say. Then Gr 3 is the divisor of highest degree that will 
divide A, B, and C. Finally, find the G.C.M. of G 2 and D, 
G 3 say. Then G 3 is the G.C.M. of A, B, C, and D. 

GENERAL PROPOSITIONS REGARDING ALGEBRAICAL PRIMENESS. 

§ 11.] We now proceed to establish a number of propositions 
for integral functions analogous to those given for integral 
numbers in chap, iii., again warning the student that he must 
not confound the algebraical with the arithmetical results ; 



120 PROPOSITIONS REGARDING G.C.M. chav. 

although he should allow the analogy to lead him in seeking for 
the analogous propositions, and in devising methods for proving 
them. 

Definition. — Two integral functions are said to be prime to each 
other when they have no common divisor. 

Proposition. — A and B being any two integral functions, there 
exist always two integral functions, L and M, 'prime to each other, such 
that, if A and B have a G.C.M. , G, then 

LA + MB = G ; 

and, if A and B be prime to each other, 

LA + MB=1. 

To prove this, we show that any one of the remainders in 
the process for finding the G.C.M. of A and B may be put into 
the form PA + QB, where P and Q are integral functions of x. 

We have, from the equalities of § 5, 

E^A-Q.B (1), 

E, = B - Q 2 E, (2), 

E 3 = E x - Q 3 E 3 (3), 

Equation (1) at once establishes the result for Ei (only here 

P = l, Q= -Q,). 

From (2), using the value of E t given by (1), 

E 2 = B - Q,(A - Q t B) = ( - Q 3 )A + ( + 1 + Q.Q^B, 

which establishes the result for E 2 . 

From (3), using the results already obtained, we get 

E 3 = A - Q,B - Q, 3 {( - Q g )A + ( + 1 + Q l Q B )B} 
= (1 + QA)A + (-<&- Q 3 - Q I Q,Q 3 )B, 

which establishes the result for E 3 , since Q,, Q 2 , Q 3 are all in- 
tegral functions. Similarly we establish the result for E 4 , E 6 , 
&c. 

Now, if A and B have a G.C.M., this is the last remainder 
which does not vanish, and therefore we must have 

G = LA + MB (L), 



vi ALGEBRAIC PRIMENESS 121 

where L and M are integral functions ; and these must be prime 
to each other, for, since G divides both A and B, A/G ( = a say) 
and B/'G ( = b say) are integral functions ; we have therefore, 
dividing both sides of (I.) by G-, 

1 = La + M5 ; 

so that any common divisor of L and M would divide unity. 

If A and B have no G.C.M., the last remainder, R rt , is a 
constant ; and we have, say, R n = L' A + M'B, where L' and M' 
are integral functions. Dividing both sides by the constant R n , 
and putting L = L'/R w , M = M'/R n , so that L and M are still 
integral functions, we have 

1 = LA + MB (II.). 

Here again it is obvious that L and M have no common divisor, 
for such divisor, if it existed, would divide unity. 

The proposition just proved is of considerable importance in 
algebraical analysis. We proceed to deduce from it several con- 
clusions, the independent proof of which, by methods more 
analogous to those of chap, iii., § 10, we leave as an exercise 
to the learner. Unless the contrary is stated, all the letters 
used denote integral functions of x. 

§ 12.] If X be prime to B, then any common divisor of AH and 
B must divide H. 

For, since A is prime to B, Ave have 

LA + MB=1, 
whence 

LAH + MBH = H, 

which shows that any common divisor of AH and B divides H. 

If A and B have a G.C.M. a somewhat different proposition 
may be established by the help of equation (I.) of § 11. The 
discovery and proof of this may be left to the reader. 

Cor. 1. IfB divide AH and be prime to A, it must divide H. 

Cor. 2. If A' be prime to each of the functions A, B, 0, <&c, it 
is prime to their product ABC . . . 



122 ALGEBRAIC PPJMENESS chav. 

Cor. 3. If each of the functions A, B, C, . . . be prime to each 
of the functions A', B', C, . . . , then the product ABC . . . is prime 
to the product A'B'C . . . 

Cor. 4. If A be prime to A', then A a is prime to A' a ', a and a' 
being any positive integers. 

Cor. 5. If a given set of integral functions be each resolved into a 
product of powers of the integral factors A, B, C, . . ., which are 
prime to each other, then the G.C.M. of the set is found by uniting 
down the product of cdl the factors that are common to all the given 
functions, each raised to the lowest power in which it occurs in any of 
these functions. 

This is a generalisation of § 3 above. 

After what has been done it seems unnecessary to add de- 
tailed proofs of these corollaries. 



LEAST COMMON MULTIPLE. 

§ 13.] Closely allied to the problem of finding the G.C.M. of 
a set of integral functions is the problem of finding the integral 
function of least degree which is divisible by each of them. This 
function is called their least common multiple (L.C.M.). 

§ 14.] As in the case of the G.C.M., the degree may, if we 
please, be reckoned in terms of more variables than one ; thus 
the L.C.M. of the monomials Sx 3 yz 2 , 6.c 2 ?/V, 8xyzu, the variables 
being x, y, z, u, is x s y 3 z*u, or any constant multiple thereof. 

The general rule clearly is to write down all the variables, each 
raised to the highest power in which it occurs in any of the mono- 
mials. 

§ 15.] Confining ourselves to the case of integral functions 
of a single variable x, let us investigate what are the essential 
factors of every common multiple of two given integral functions 
A and B. Let G be the G.C.M. of A and B (if they be prime 
to each other we may put G = 1) ; then 

A = aG, B = bG, 

where a and b are two integral functions which are prime to each 



vi LEAST COMMON MULTIPLE 123 

other. Let M be any common multiple of A and B. Since M 
is divisible by A, we must have 

M = PA, 

where P is an integral function of x. 

Therefore M = PaG. 

Again, since M is divisible by B, that is, by bG, therefore 
M/bG, that is, TaG/bG, that is, ~Pa/b must be an integral function. 
Now b is prime to a; hence, by § 12, b must divide P, that is, 
P = GJ), where Q is integral. Hence finally 

M = QabG. 

This is the general form of all common multiples of A and B. 

Now a, b, G are given, and the part which is arbitrary is the 
integral function Q. Hence we get the least common multiple 
by making the degree of Q as small as possible, that is, by making 
Q any constant, unity say. The L.C.M. of A and B is therefore 
abG, or (aG)(bG)/G, that is, AB/G. In other words, the L.C.M. 
of two integral functions is their product divided by their G.C.M. 

§ 16.] The above reasoning also shows that every common 
multiple of two integral functions is a multiple of their least common 
multiple. 

The converse proposition, that every multiple of the L.C.M. 
is a common multiple of the two functions, is of course obvious. 

These principles enable us to find the L.C.M. of a set of any 
number of integral functions A, B, C, D, &c. For, if we find 
the L.C.M., L! say, of A and B; then the L.C.M., L 2 say, of L, 
and C ; then the L.C.M., L 3 say, of L 2 and D, and so on, until all 
the functions are exhausted, it follows that the last L.C.M. thus 
obtained is the L.C.M. of the set. 

§ 17.] The process of finding the L.C.M. has neither the 
theoretical nor the practical importance of that for finding the 
G.C.M. In the few cases where the student has to solve the 
problem he will probably be able to use the following more direct 
process, the foundation of which will be obvious after what has 
been already said. 

If a set of integral functions can all be exhibited as powers of a 



124 EXERCISES X CHAP. 

set of integral factors A, B, C, &e., which are either all of the 1st 
degree and all different, or else are all prime to each other, then the 
L.C.M. of the set is the product of all these factors, each being raised 
to the highest power in which it occurs in any of the given functions. 

For example, let the functions be 

(,r-l) 2 (a; 2 + 2) 3 (a: 2 + a- + l), 

(x-l) 5 {x-2) 3 (x-BY(x 2 + x + l) 3 , 
then, by the above rule, the L.C.M. is 

(x-l) 5 (x-2) 5 (x-Z) i (x 2 +2f(x 2 + x + lf(x"-x + l)\ 

Exercises X. 

Find the G.C.M. of the following, or else show that they have no CM. 

(1.) (x 2 -lf, x 6 -l. 

(2.) a: 6 -l, x i -2x 3 + Zx 2 -2x + \. 

(3.) z*-x 2 + l, a- 4 + ar + l. 

(4.) jc 9 + 1, x ll + h 

(5.) a?-x 2 -8x + \2, sP+ia?- 3x- 18. 

(6.) a: 4 -7a: 3 -22a: 2 + 139x + 105, x A -8x* -Ux 2 + \l6x + 70. 

(7.) a 4 -286a: 2 + 225, x* + 140?- 480a: 2 - 690a: - 225. 

(8.) x 6 -x*-8x 2 + 12, a; 6 + 4a; 4 -3a; 2 -18. 

(9.) x> - 2a: 4 - 2a: 3 + 4ar + x -2, x 5 + 2x i -2x"' - 8x 2 -7x-2. 

(10.) x s + 6x 6 - 8.C 4 + 1 , a: 12 + 7a: 10 - 3a? - 3a: 2 - 2. 

(11.) 12x 3 + lBx 2 + 6x + l, 16a: 3 + 16a: 2 + 7a: + l. 

(12.) 5ar 3 + 38a: 2 -195a; -600, 4a; 3 - 15a: 2 - 38a; + 65. 

(13.) 16a 4 - 56a,- 3 - 88a; 2 + 278a; +105, 16a; 4 - 64a; 3 - 44a; 2 + 232a; + 70. 

(14.) 7a 4 + 6a: 3 -8a; 2 - 6a; + 1, lla: 4 + 15a; 3 - 2a: 2 - 5;<;+ 1. 

(15.) a 4 + 64a 4 , (a; + 2a) 4 - 16« 4 . 

(16.) 9a; 4 +4a; 2 + l, Zsj2x 3 + x 2 +l. 

(17.) a; 3 + 3i?a; 2 - (1 + 3p), px 3 - 3( 1 + Zp)x + (3 + 8p). 

(18. ) x 3 - 3(a - b)x 2 + (4a 2 - Zab)x - 2a°{2a - 3b), 

x i -{3a + b)x 3 +{5a 2 + 2ab)x 2 -a 2 (5a + 3b)x + 2a 3 {a + b). 

(19. ) 7iaf+ 1 - (it + l)x n + 1, a"' - nx+{n - 1 ). 

(20.) Show that ar 3 +^a: 2 + g'a; + l, x 3 + qx 2 + px + 1 cannot have a common 
measure, unless either p = q or p + q + 2 = 0. 

(21. ) Show that, if ax 2 + bx + c, ex 2 + bx + a have a common measure of the 
1st degree, then a±b + c — 0. 

(22.) Find the value of a for which {x 3 -aa; 2 + 19a'-«- 4] /{a: 3 - (a+l)a- 2 
+ 23a;-a-7} admits of being expressed as the quotient of two integral 
functions of lower degree. 

(23.) If ax 3 + 3bx 2 + d, bx 3 + Bdx + e have a common measure, then 
(ae -4bd) 3 = 27 {ad 2 + b 2 e) 2 . 



VI EXERCISES X 125 

(24.) Ax 2 + Bxy + Cy 2 , Ba? - 2(A - C)xy - By 8 cannot have a common 
measure unless the first be a square. 

(25.) ax 3 + bx 2 + cx + d, dx 3 + cx 2 + bx + a will have a common measure of 
the 2nd degree if 

abc - a"b - b 2 d + acd ac 2 - bed - a 3 + ad 2 d(ac - b<l) 
ac-bd ab-cd a 2 -d? 

show that these conditions are equivalent to only one, namely, ac~bd = 
or - dr. 

(26.) Find two integral functions P and Q, such that 

P(^ 2 - Zx + 2) + Q(.x 2 + x + 1) = 1. 

(27.) Find two integral functions P and Q, such that 

P(2^-7a; 2 + 7^-2) + Q(2^ + a; 2 + x-l) = 2a;-l. 

Find the L.C. M. of the following : — 

(28.) a 5 -ab\ a?+a*b, a 6 + b (i + a 2 b"-(a 2 + b 2 ). 

(29.) x 3 -x 2 -Ux + 24, x 3 -2x 2 -5x +6, x 2 -4a- + 3. 

(00.) 3x 3 + x 2 -8x + 4, 3x 3 + 7x 2 -4, x 3 + 2a: 2 - x - 2, dx 3 + 2,r 2 - 3r - 2. 

(31. ) x 3 - V2x + 16, x A - i.r 3 - x 2 + 20,r - 20, x* + 3x 3 - llx 2 - 3x + 10. 

(32.) x 6 + 2ax 5 + a 2 x 4 + 5a 5 x + a 6 ) x 3 + crx - ax 2 - a 3 . 

(33.) If x 2 + ax + b, x 2 + a'x + b' have a common measure of the 1st 
degree, then their L.C.M. is 

, ab-a'V . / , [b-V\*\ , ,.,«-«' 

(34.) Show that the L.C.M. of two integral functions A and B can always 
be expressed in the form PA + QB, where P and Q are integral functions. 



CHAPTEE VII. 

On the Resolution of Integral Functions into 

Factors. 

§ 1.] Having seen how to determine whether any given 
integral function is a factor in another or not, and how to deter- 
mine the factor of highest degree which is common to two in- 
tegral functions, it is natural that we should put to ourselves 
the question, How can any given integral function be resolved 
into integral factors ? 

TENTATIVE METHODS. 

§ 2.] Confining ourselves at present to the case where 
factors of the 1st degree, whose coefficients are rational integral 
functions of the coefficients of the given function, are suspected 
or known to exist, we may arrive at these factors in various ways. 

For example, every known identity resulting from the distri- 
bution of a product of such factors, when read backwards, gives 
a factorisation. 

Thus (x + y) (x - y) — x 2 - y" tells us that of - y 2 may be re- 
solved into the product of two factors, x + y and x — y. In a 
similar way we learn that x + y + z is a factor in x 3 + y 3 + z 3 — 3xyz. 
The student should again refer to the tables of identities given 
on pp. 81-83, and study it from this point of view. 

When factors of the 1st degree with rational integral 
coefficients are known to exist, it is usually not difficult to find 
them by a tentative process, because the number of possible 
factors is limited by the nature of the case. 



chap, vii TENTATIVE FACTORISATION EXAMPLES 127 

Example 1. 

Consider a ,2 -12x + 32, and let us assume that it is resolvable into (x-a) 
{x-b). 

Then we have 

a- 2 - 12a! + 32 = x- - (a + b)x + ab, 

and we have to find a and b so that 

«6=+32, a + b- +12. 
"We remark, first, that a and b must have the same sign, since their pro- 
duct is positive ; and that that sign must be +, since their sum is positive. 
Further, the different ways of resolving 32 into a product of integers are 
1x32, 2x16, 4x8; and of these we must choose the one which gives 
a + b= +12, namely, the last, that is, a = 4, b — 8. 
So that 

a- 2 -12a- + 32 = (a--4)(a--8). 
Example 2. 

x 3 - 2a- 2 - 23a: + 60 = (x -a)(x- b) (x - c) say. 
Here - abc — + 60. 

Now the divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ; and we 
have therefore to try a-±l, a 1 ±2, x±3, &c. The theorem of remainders 
(chap. v. , § 14) at once shows that x + 1, x - 1, x + 2, x - 2, are all inadmissible. 
On the other hand, for a- -3 we have (see chap, v., § 13) 

1-2-23 + GO 
+ 3+ 3-60 



1 + 1-20+ 
that is, a- -3 is a factor ; and the other factor is a; 2 + a;-20, which we resolve 
by inspection, or as in Example 1, into {x- 4) {x + 5). 
Hence x* - 2a- 2 - 23a- + 60 = {x - 3) (a- - 4) (x + 5). 

Example 3. 

&x- - 1 9x + 1 5 = (ax + b)(cx + d). 

Here ac= +6, bd=+ 15 ; and we have more cases to consider. "We might 
have anyone of the 32 factors, a;±l, a; ±3, a' ±5, a' ±15, 2a; ±1, 2a- ±3, 
2x ± 5, 2xzk 15, &c. A glance at the middle coefficient, - 19, at once excludes 
a large number of these, and we find, after a few trials, 
6x°- - 19a- + 15 = (2a- - 3) (3a- - 5). 
§ 3.] In cases like those of last section, we can often detect 
a factor by suitably grouping the terms of the given function. 
For it follows from the general theory of integral functions 
already established that, if P can be arranged as the sum of a 
series of groups in each of which Q is a factor, then Q is a factor 
in P ; and, if P can be arranged as the sum of a series of groups 
in each of which Q is a factor, plus a group in which Q is not a 
factor, then Q is not a factor in P. 






128 



FACTORISATION OF 



CHAP. 



Example 1. 

x 3 - 2x" - 23a; + 60 

= .>•■-(./■- 2) -23(a:- 2) + 14, 

that is, x - 2 is not a factor. 

a; 3 -2« 2 -23a- + 60 

= a"(a;-3)+a: 2 -23a: + 60 

= x 2 (x-3) + x(x-Z) - 20a;+ 60 

= x 2 {x - 3) + x(x - 3) - 20(a; - 3), 

that is, a; -3 is a factor. 

Example 2. 

-])x 2 + xy+pqxy + qy 2 
= x{px + y) + qy(px + y) , 
that is, px + y is a factor, the other being x + qij. 

Example 3. 

x 3 + (m + n + 1 )a,"a + (m + ?i + mn)xa 2 + mna? 
= x 3 + a; 2 a + (w + ?i) (ara + xa 2 ) + mn(xa 2 + « 3 ) 
= ar^ai + a) + (m + n)xa(x + a) + mna 2 {x + a) 
= {a; 2 + (m + yi)a'a + mrea 2 } (x + a) 
= {x(x + via) + ?ta(a; + ma)} (x + a) 
= (x + ma) (x + ?ia) (x + a). 



GENERAL SOLUTION FOR A QUADRATIC FUNCTION. 

§ 4.] For tentative processes, such as Ave have been illustrat- 
ing, no general rule can be given; and skill in this matter is one 
of those algebraical accomplishments which the student must 
cultivate by practice. There is, however, one case of great im- 
portance, namely, that of the integral function of the 2nd degree 
in one variable, for which a systematic solution can be given. 

We remark, first of all, that every function of the form 
x'+px + q can De made a complete square, so far as x is con- 
cerned, by the addition of a constant. Let the constant in 
question be a, so that we have 

x 2 + px + q + a = (./• + /5) 2 = x" + 2 fix + f3 2 , 
(3 being by hypothesis another constant. Then we must have 

p = 2f3, q + a = ft 2 . 
The first of these equations gives (3-2>j2, the second a = /3 2 -q 
- (jpffi ~ '1- Thus our problem is solved by adding to x 2 +px + q 
the constant (jj/2) 2 - q. 






vii A QUADRATIC FUNCTION 129 

The same result is obtained for the more general form, 
ax* + bx + c, as follows : — 

2 , / 2 6 c 

ax + bx + c = a[x + - x + - 

\ a a 

Now, from the case just treated, we see that x 2 + (b/a)x + c/a is 
made a complete square in x by the addition of (b/2a) 2 — c/a, 
that is, (b 2 - 4:0c)/ia a . Hence ax 2 + bx + c will be made a com- 
plete square in x by the addition of a(b 2 — lac)/ la 2 , that is, 
(b 2 - lac)/ la. We have, in fact, 

, , b 2 -iac ( b\ 2 

ax + bx + c + — ; = a[x + -—) ■ 

la \ 2aJ 

§ 5.] The j)rocess of last article at once suggests that 
aaf + bx + c can always be put into the form a{(x + I) 2 - m 2 }, 
where I and m are constant. 

In point of fact we have 

2i f « b c ) 

ax + bx + c = a< x~ + - x + - > 
I a a ) 

..I ,t+ 9 > .+ (*)'.(>)'+:] 

{ 2a \2aJ \2aJ a ) 

In other words, our problem is solved if we make I = b/2a and 
find m, so that m 2 = (b 2 - iac)/ia 2 . 

This being done, the identity X 2 - A 2 = (X - A) (X + A) at 
once gives us the factorisation of ax 2 + bx + c ; for we have 

ax 2 + bx + c = a {(x + I) 2 - m 2 } 

= a {(x + 1) + m } {(x + /) - m }. 
Example 1. 
Consider 6x 2 - 19a; + 15 ; we have 

-{'-S-*(S)'-S*¥} 

19 / 1 \ 2 1 

Here 1= - — , and m-= ( — ) ; so that our problem is solved if we take w = — • 

VOL. I K 



130 REAL ALGEBRAICAL QUANTITY chap. 

We get, therefore, 

"— + M(-S)4}{(--S)-£} 

=<¥)(^) 

= (2as-3)(3«-5); 

the same result as we obtained above (in § 2, Example 3), by a tentative 
process. 

Example 2. 

Consider x 6 - 5x 3 + 6. We may regard this as (x 3 ) 2 - 5(,r 3 ) + 6, that is to 
say, as an integral function of x 3 of the 2nd degree. We thus see that 

x 6 -5x 3 + 6 = (x 3 ) 2 ~5(x s ) + 6, 
= (.k 3 -3)(.b 3 -2). 

INTRODUCTION INTO ALGEBRA OF SURD AND 
IMAGINARY NUMBERS. 

§ 6.] The necessities of algebraic generality have already led 
us to introduce essentially negative quantity. So far, algebraic 
quantity consists of all conceivable multiples positive or negative 
of 1. To give this scale of quantity order and coherence, we 
introduce an extended definition of the words greater than and 
less than, as follows : — a is said to be greater or less than b, according 
as a- b is positive or negative. 

Example. 

( +3) - ( + 2)= + 1 therefore + 3 > + 2 ; (-3) -(-5) =+2 therefore - 3 > - 5 ; 
( + 3) - ( - 5) = + 8 therefore +3> -5; (-7) -(-3)= - 4 therefore -7< -3. 

Hence it appears that, according to the above definition, any 
negative quantity, however great numerically, is less than any 
positive quantity, however small numerically ; and that, in the 
case of negative quantities, descending order of numerical magni- 
tude is ascending order of algebraical magnitude. 

We may therefore represent the whole ascending series of 
algebraical quantity, so far as we have yet had occasion to con- 
sider it, as follows : — 



- CO 



1 * ..-£... ...+£... +1 ...+ oo 



* 



t 

* The symbol oo is here used as an abbreviation for a real quantity as 
great as we please. 



vii RATIONAL AND SURD QUANTITY 131 

The most important part of the operations in the last para- 
graph is the finding of the quantity m, whose square shall be equal 
to a given algebraical quantity. We say algebraical, for we must 
contemplate the possibility of (b 2 - 4ac)/4a s , say k for shortness, 
assuming any value between - <x> and + qo . When m is such that 
in 2 = k, then in is called the square root of k, and we write in = \/k. 
We are thus brought face to face with the problem of finding 
the square root of any algebraical quantity ; and it behoves us 
to look at this question somewhat closely, as it leads us to a new 
extension of the field of algebraical operations, similar to that 
which took place when we generalised addition and subtraction 
and thus introduced negative quantity. 

1st. Let us suppose that k is a positive number, and either 

a square integer = + k, say, or the square of a rational number 

= + (*/A) 2 , say, where k and A are both integers, or, which is 

the same thing [since (k/X) 2 = k'/'A 2 ], the quotient of two square 

integers. Then our problem is solved if we take 



in the one case, or 



m — + k, or m = - k 

m = + k/X, or m - - k/X 



in the other ; 

for m* - ( ± «) 2 = k = k, 



in 



-(^■-©■-* 



which is the sole condition required. 

It is interesting to notice that we thus obtain two solutions 
of our problem ; and it will be afterwards shown that there are 
no more. Either of these will do, so far as the problem of 
factorisation in § 5 is concerned, for all that is there required is 
any one value of the square root. 

More to the present purpose is it to remark that this is the 
only case in which m can be rational; for if m be rational, that 
is, = ± k/X where k and A are integers, then m 2 = (k/A) 3 , that is, 
k = (k/X) 2 , that is, k must be the square of a rational number. 

2nd. Let k be positive, but not the square of a rational 



132 DEFINITION OF THE IMAGINARY UNIT i CHAP. 

number ; then everything is as before, except that no exact 
arithmetical expression can be found for m. We can, by the 
arithmetical process for finding the square root, find a rational 
value of m, say v, such that m 2 = ( ± v) 2 shall differ from k by 
less than any assigned quantity, however small ; but no such 
rational expression can be absolutely exact. In this case m 
is called a surd number. When k is positive, and not a square 
number, as in the present case, it is usual to use \/k to denote 
the mere (signless) arithmetical value of the square root, which 
has an actual existence, although it is not capable of exact arith- 
metical expression ; and to denote the two algebraical values of 
m by ± \/k. Thus, if k = + 2, we write m = ± v/2. In any 
practical application we use some rational approximation of 
sufficient accuracy ; for example, if k = + 2, and it is necessary 
to be exact to the l/10,000th, we may use m- ± 1-4142. 

A special chapter will be devoted to the discussion of surd 
numbers ; all that it is necessary in the meantime to say further 
concerning them is, that they, or the symbols representing them, 
are of course to be subject to all the laws of ordinary algebra.! 

3rd. Let k be negative = - k', say, where k' is a mere arith- 
metical number. A new difficulty here arises ; for, since the 
square of every algebraical quantity between - co and + oo (ex- 
cept 0, which, of course, is not in question unless k' = 0) is 
positive, there exists no quantity m in the range of algebraical 
quantity, as at present constituted, which is such that m 2 = - k'. 
If we are as hitherto to maintain the generality of all algebraical 
operations, the only resource is to widen the field of algebraical quantity 
still farther. This is done by introducing an ideal, so-called imaginary, 
unit commonly denoted by the letter i* whose definition is, that it is 
such that 

i 2 = -1. 

It is, of course, at once obvious that i has no arithmetical 
existence whatsoever, and does not admit of any arithmetical 
expression, approximate or other. We form multiples and sub- 
multiples of this unit, positive or negative, by combining it with 

* Occasionally also by t. t See vol. ii. chap. xxv. § 28-41. 



VII 



COMPLEX NUMBERS 133 



quantities of the ordinary algebraical, now for distinction called 

real, series, namely, 

-oo... -1...-|...0. ..+!•.. +1...+CO. 
We thus obtain a new series of purely imaginary quantity : — 
— oo *...— *.. . - \i. . . 0* . . . + \i . . . + i , . . + co i. 

These new imaginary quantities must of course, like every other 
quantity in the science, be subject to all the ordinary laws of 
algebra when combined either with real quantities or with one 
another. All that the student requires to know, so far at least as 
operations with them are concerned, beyond the laws already laid 
down, is the defining property of the new unit i, namely, i 2 — — 1. 

When purely real and purely imaginary numbers are com- 
bined by way of algebraical addition, forms arise like!? + qi, where 
P and q are real numbers positive or negative. Such forms are 
called complex numbers ; and it will appear later that every alge- 
braical function of a complex number can itself be reduced to 
a complex number. In other words, it comes out in the end 
that the field of ordinary algebraical quantity is rendered com- 
plete by this last extension. 

The further consequences of the introduction of complex 
numbers will be developed in a subsequent chapter. In the 
meantime we have to show that these ideal numbers suffice for 
our present purpose. That this is so is at once evident ; for, if 
we denote by »Jk' the square root of the arithmetical number 
k\ so that ijk' may be either rational or surd as heretofore, but 
certainly real, then m - ±i Jk' gives two solutions of the problem 
in hand, since we have 

m 2 = ( ± i Jk') 2 

= ( ± i Jk') x ( ± i Jk'), 
upper signs going together or lower together, 

= (i 2 ) x ( </V Y 
= (-l)x(*') 

= -k\ 

§ 7.] We have now to examine the bearing of the discus- 
sions of last paragraph on the problem of the factorisation of 
ax 2 + bx + c. 



134 FACTORISATION OF QUADRATIC chap. 

It will prevent some confusion in the mind of the student if 
we confine ourselves in the first place to the supposition that a, b, c 
denote positive or negative rational numbers. Then I = bj2a is in all 
cases a real rational number, and we have the following cases : — 

1st. If b 2 - 4ac is the positive square of a rational number, 
then m has a real rational value, and 

ox 2 + bx + c = a(z + I + m)(x + I - m) 
is the product of two linear factors whose coefficients are real rational 
numbers. Example 1, § 5, will serve as an illustration of this case. 

2nd. If b 2 — iac is positive, but not the square of a rational 
number, then m is real, but not rational ; and the coefficients in 
the factors are irrational. 

Example 1. 

x 2 + 2x-l=x" + 2x + l-2, 
= (a; + l) 2 -(V2) 2 , 
= (x + l + yj2)(x + l-^2). 

3rd. If b 2 - iac is negative, then m is imaginary, and the 
coefficients in the factors are complex numbers. 
Example 2. 



Example 3. 



a: 2 + 2a: + 5 = a: 2 + 2a' + l + 4, 
= (a- + l) 2 -(2i) 2 , 
= {x + l + 2i){x + l-2i). 



x- + 2x + 3 = x 2 + 2x + l + 2, 

= (x + lf-(i^2T~, 

= (x + 1 + i\j2) (x + 1 -- i V2). 

4th. There is another case, which forms the transition 
between the cases where the coefficients in the factors are real 
and the case where they are imaginary. 

If b 2 - iac = 0, then m = 0, 
and we have ax 2 + bx + c = a(x + I) 2 ; 

in other words, ax 2 + bx + c is a complete square, so far as x is 
concerned. The two factors are now x + I and x + I, that is, 
both real, but identical. 

We have, therefore, incidentally the important result that 

ax 2 + bx + c is a complete square in x ij ' tf - 4ac = 0. 

Example 4. 

3a; 2 - 3a; + 1 = 3(a; 2 - 2. \x + J), = 8(» - h) 2 - 

* I'^-iac is called the Discriminant of the quadratic function ax- + bx+c. 



vil FUNCTIONS RESUMED 135 

§ 8.] There is another point of view which, although usually 
of less importance than that of last section, is sometimes taken. 

Paying no attention to the values of a, b, c, but regarding 
them merely as functions of certain other letters which they may 
happen to contain, we may inquire under what circumstances the 
coefficients of the factors will be algebraically rational functions of 
those letters. 

In order that tins may be the case it is clearly necessary and 
sufficient that b 2 - iac be a complete square in the letters in 
question, = P 2 say. 
Then 

/ b P\ / b P 
= a ( x + — + — ) (x + — - — 

\ 2a la J \ 2a lo 

which is rational, since P is so. 

If b 2 - iac - - P 2 , where P is rational in the present sense, 
then 



/ b P.\/ b P . 
= a(x + — - + — -n (x + — - - —i 

V 2a 2a / \ 2a 2a 

where the coefficients are rational, but not real. 

Example 5. 

jtx 2 + {p + q)x- J r q 

«j»(«+i)(«+J). 

= (x + l)(j>X + q); 

a result which would, of course, be more easily obtained by the 
tentative processes of §§ 2, 3. 



136 HOMOGENEOUS QUADRATIC FUNCTIONS chap. 

§ 9.] It should be observed that the factorisation for 
ax 2 + bx + c leads at once to the factorisation of the homogeneous 
function ax 2 + bxy + cif of the 2nd degree in two variables ; for 

ax 2 + bxy + cf 

,f z b ///" - lac ) f x b lb 1 - iuc > . 

( / b lb 2 - \ac\ ") f / b lb 1 - 4ac\ ) 

= " { x+ (a; + V ~iH-" \{ r+ (s " v tt) "I • 

By operating in a similar way any homogeneous function of 
two variables may be factorised, provided a certain non-homo- 
geneous function of one variable, having the same coefficients, 
can be factorised. 

Example 1. From 

a: 2 + 2x + 3={x+ 1 + is/2) {x + 1 - z'V2), 

we deduce 

x 2 + 2xy + 3y n - = {x + (1 + i\/2)y} {x+(l - i\/2)y}. 
Example 2. From 

x f _ 2x- - 23a- + 60 = (x - 3) {x - 4) (as + 5), 
we deduce 

a? - 2a' 2 ;/ - 23a-y 2 + 60?/ 3 = (.r - Zy) [x - iy) (x + 5y). 

§ 10.] By using the principle of substitution a great many 
apparently complicated cases may be brought under the case of 
the quadratic function, or under other equally simple forms. 
The following are some examples : — 

Example 1. 
xt + xhf + y^^ + y^-ixyT-, 

= {x*+y*+xy)(x*+y*-xy), 

-{(•^'-(^^{(-W'-W}- 
-{. + G + ^>}{. + G-^>H- + (-^>} 



vil EXAMPLES 137 

Here the student should observe that, if resolution into quadratic factors only 
is required, it can be effected with real coefficients ; but, if the resolution be 
carried to linear factors, complex coefficients have to be introduced. 

Example 2. 

x 3 + y 3 ={x + y)(x 2 -.ry + ij 2 ) 



-*"»»{•+(-£+#>}{ 



-«-#>) 



Example 3. 

.< i + y i = (x 2 +y 2 ) 2 -2x 2 y 2 

= (■<•- + j/ 2 ) 2 -W-2xy? 

= (x 2 + V&ey + y 2 ) [a? - ^2xy + y 2 ). 

Again 

a? + s/2xy + y-=f x + ^fy Y + ~y 2 

V2 y /V2. 



X+ 2 1 ' -\2 iy 



x + ^(l + i)y] {*+^(l-%}. 



The similar resolution for x 2 - \/2xy + y 2 will be obtained by changing the 
sign of \J2. Hence, finally, 

= {^f(l + i)y}{^^l-i) y }{,-^lU)y}{x-^ { l-i )y ). 

Example 4. 

x^-i/^^y-iy 6 ) 2 

= (x e -y , ')(x G + y R ) 

= {(* 2 ) 3 -(2/ 2 ) 3 } {(* 2 ) 3 + (ir) 3 } 

= (x 2 - y 2 ) (x 4 + x 2 y 2 + y*) (x 2 + y 2 ) (x* - x 2 y 2 + y 4 ) 
= {x + y){x-y) (x + iy) {x - iy) (x 4 + x 2 y 2 + y*) (x* - x 2 y 2 + y*), 
where the last two factors may be treated as in Example 1. 
Example 5. 

2i 2 c 2 + 2cV + 2a 2 6 2 - « 4 - ¥ - c 4 
= 4i 2 c 2 - (a- -b 2 - c 2 ) 2 
= (2bc + a 2 - b 2 - c 2 ) (26c - d 2 + b 2 + c 2 ) 
= {a 2 -(b-c) 2 } {(b + c) 2 -a 2 } 
= (a + b - c) (a - b + c) (b + c + a) (b + c - a). 



* The student should observe that the decomposition x 2 + y 2 + xy = 
(x + y+ \Jxy) (x + y- \/xy), which is often given by beginners when they are 
asked to factorise x 2 + y 2 + xy, although it is a true algebraical identity, is no 
solution of the problem of factorisation in the ordinary sense, inasmuch as 

the two factors contain \Jxy, and are therefore not rational integral functions 
of a- and y. 



138 USE OF REMAINDER THEOREM chap. 



RESULTS OF THE APPLICATION OF THE REMAINDER THEOREM. 

§ 11.] It may be well to call the student's attention once 
more to the use of the theorem of remainders in factorisation. 
For every value a of x that we can find which causes the integral 
function f(x) to vanish we have a factor x - a off(x). 

It is needless, after what has been shown in chap, v., §§ 13-16, 
to illustrate this point further. 

It may, however, be useful, although at this stage we cannot 
prove all that we are to assert, to state what the ultimate result 
of the rule just given is as regards the factorisation of integral 
functions of one variable. If f(x) be of the wth degree, its coeffi- 
cients being any given numbers, real or imaginary, rational or 
irrational, it is shown in the chapter on Complex Numbers 
that there exist n values of x (called the roots of the equation 
f(x) = 0) for which f(x) vanishes. These values will in general 
be all different, but two or more of them may be equal, and one 
or all of them may be complex numbers. 

If, however, the coefficients of f(x) be all real, then there 
will be an even number of complex roots, and it will be possible 
to arrange them in pairs of the form X ± jxi. 

It is not said that algebraical expressions for these roots in 
terms of the coefficients of /(.<:) can always be found ; but, if 
these coefficients be numerically given, the values of the roots 
can always be approximately calculated. 

From this it follows that f(x) can in all cases be resolved into n 
linear * factors, the coefficients of which may or may not be all real. 

If the coefficients of f(x) be all real, then it can be resolved into a 
product of p linear and q quadratic factors, the coefficients in all of 
which are real numbers which may in all cases be calculated approxi- 
mately. We have, of course, p + 2q = n, and either p or q may be 
zero. 

The student will find, in §§ 1-10 above, illustrations of these 
statements in particular cases ; but he must observe that the 

* " Linear" is used here, as it often is, to mean "of the 1st degree." 



VII QUADRATIC FUNCTION WITH TWO VARIABLES 139 

general problem of factorising an integral function of the nth. 
degree is coextensive with that of completely solving an equation 
of the same degree. When either problem is solved the solution 
of the other follows. 



FACTORISATION OF FUNCTIONS OF MURE THAN ONE VARIABLE. 

§ 12.] Jflien the number of variables exceeds unity, the problem of 
factorisation of an integral function {excepting special cases, such as 
homogeneous functions of two variables) is not in general soluble, at 
least in ordinary algebra. 

To establish this it is sufficient to show the insolubility of 
the problem in a particular case. 

Let us suppose that x 2 + y 2 + 1 is resolvable into a product of factors which 
are integral in x and y, that is, that 

x" + y 2 + 1 = (px + qy + r) (p'x + q'y + r'), 
then x 2 + y 2 + l =pp'x 2 + qq"y 2 + rr' 

+ (Pi' +P'l)xy + (pr' +p'r)x 

+ (qr' + q'r)y. 

Since this is, by hypothesis, an identity, we have 

pp' = 1 
qq' = \ 
rr' = 1 

First, we observe that, on account of the equations (1) (2) (3), none of the six 
quantities p q r p' q' r' can be zero ; and further, p' = -, <?' = -, r' — -. Hence, 



(1) 


Pi' + v'i ~ o 


(4) 


(2) 


pr' + p'r = 


(5) 


(3) 


q?''+q'r = 


(6) 



as 


logical 


consequences 


of 


our 


hypothesis, we 

^ + 2 = 
q p 

r p 

o* n 


have 


from 


(4) (5) 


and 


(6V- 


(7) 
(8) 
(9); 



T q 

and, from these again, if we multiply by pq, rp, and qr respectively, we get 

p 2 + q 2 =0 (10) 

^2 + r 2 = ( H ) 

q 2 + r 2 =0 (12). 
Now from (11) and (12) by subtraction we derive 

p 2 -q 2 =0 (13); 



140 QUADRATIC FUNCTION chap. 

and from (10) and (13) by addition 

2p 3 =0; 
from this it follows that ^ = 0, which is in contradiction with the equation (1). 
Hence the resolution in this case is impossible. 

§ 13.] Nevertheless, it may happen in particular cases that 
the resolution spoken of in last article is possible, even when the 
function is not homogeneous. This is obvious from the truth 
of the inverse statement that, if we multiply together two 
integral functions, no matter of how many variables, the result 
is integral. 

One case is so important in the applications of algebra to 
geometry, that we give an investigation of the necessary and 
sufficient condition for the resolvability. 

Consider the general function of x and y of the 2nd degree, and write it 

F = ax 2 + 2hxy + by 2 + 2gx + 2/y + c. 
We observe, in the first place, that, if it be possible to resolve F into two 
linear factors, then we must have 

F = ( \Jax + ly + m) (\Jax + I'y + to'), 
= [sjax + >J(Z + Z') + h(l - I')} y + h(m + to') +£(m- »»')] 

x[sjctx+{\{l+l')-l{l-l')}y + ii{m + m')-\{m-m')], 
= { sjax + J(J + V)y + \{m + m')} 2 - {h(l - l')y + i(m - to')} 2 - 

Hence, when F is resolvable into two linear factors, it must be expressible in 
the form L 2 -M 2 , where L is a linear function of x and y, and M a linear 
function of y alone ; and, conversely, when F is expressible in this form, it 
is resolvable, namely, into (L + M) (L- M). 

Let us now seek for the relation among the coefficients of F which is 
necessary and sufficient to secure that F be expressible in the form L 2 - M 2 . 

1st. Let a 4=0, then 

F = a[x* + 2(hy + g)x/a +(by"- + 2/y + c)/a], 
= a[ {x + (hy + g)/a} 2 - {(hy + gf- - a(by* + 2/y + c)} /a"], 
= a[ {x + (hy + g)/a} 2 - {(^ 2 - a% 2 +2(gh - af)y + (f - ac)} /«'-]. 

Hence the necessary and sufficient condition that F be expressible in the form 
L 2 - M 2 is that (K 2 - ab)y" + 2(gh - af)y + (g 2 - ac) be a complete square as regards 
y. For this, by § 7, it is necessary and sufficient that 

l{gh-af)*-4(h*-db)(g*-ac) = 0; 

that is, - a {abc + 2fgh - a/ 2 - bg 2 - ch 2 ) — 0. 

Now, since a 4=0, this condition reduces to 

abc + 2/gh - a/ 2 -bg 2 -ch 2 = { 1 ). 

2nd. If a = 0, but b 4=0, we may arrive at the same result by first arranging 
F according to powers of y, and proceeding as before. 



vii OF TWO VARIABLES HI 

3rd. If a = 0, 6 = 0, and A#0, the present method fails altogether, but F 
now reduces to 

F = 2hxy + 2gx + 2fy + a, 

and it is evident, since x 2 and y 2 do not occur, that if this be resolvable into 
linear factors the result must be of the form 2h(x+p)(y + q). We must 
therefore have 2g=2kq, 

2f=Zhp, 
c = 2hpq. 

Now the first two of these give fg = h-pq, that is, 2hpq=~- ; whence 

using the third, 

ch=2fg, 
or, since h * 0, 2fgh - cK 2 = (2) ; 

but this is precisely what (1) reduces to when a = 0, 6 = 0, so that in this third 
case the condition is still the same. 

Moreover, it is easy to see that when (2) is satisfied the resolution is 
possible, being in fact 

2hxy + 2gx + 2fy + c = 2hfx + fVy + ^\ (3), 

which is obviously an identity if c = 2/g/h. 

4th. If a = 0, 6 = 0, h = 0, F reduces to 2gx + 2/y + c. In this case we may 
hold that F is resolvable, it being now in fact itself a linear factor. It is 
interesting to observe that in this case also the condition (1) is satisfied. 

Returning to the most general case, where a does not vanish, we observe 
that, when the condition (1) is satisfied, we have, provided /i' 2 -«6 + 0, 

V{(# - ab),f- + 2(gh - afjy + (g* - ac)} = VF^L+ ? p^Q, 

so that the required resolution is 

f h + \/l)?~-ab g gh-af /r „ , "| 

Y=a\x+ y + -+ 7£5 — ^-.\/h 2 -ab [ 

I a a a(h 2 - ab) ) 



( h-\/h 2 -ab q qh-af /.« "| r a\ 

I. a a a{k- - ab) ) 

To the coefficients in the factors various forms may be given by using the 
relation (1) ; but they will not be rational functions unless h 2 -ab be a com- 
plete square, and they will be imaginary unless h 2 -ab is positive. 

If h 2 -ab = 0, then (1) gives (gh-af ') 2 = 0, that is, gh-af— 0; and the 
required resolution is 

( h g \'q- -ac\ | h g \fg 2 -ac\ ,-> 

Y = a\ .T + -2/ + -+ — * M x+-y+- - h (5). 

I a a a J I a fa a ) 

The distinction between these cases is of fundamental importance in the 
analytical theory of curves of the 2nd degree. 

The function abc + 2fgh-af 2 -bg 2 -ch 2 , whose vanishing is the condition 
for the resolvability of the function of the 2nd degree, is called the Discrimi- 
nant of that function. 



142 



EXERCISES XI 



CHAP. 



It should be noticed that, if 

F = ax 2 + 2hxy + by 2 + 2gx + 2fy + c 

=(Ise+my+n)(l'x+m'y+n') (6), 

then 

ax 2 + 2hxy + by 2 =(lx + my) (Vx + m'y), 

so that the terms of the 1st degree in the factors of F are simply the factors 
of ax 2 + 2h xy + b y". We have therefore merely to find, if possible, values for 
n and n' which will make the identity (6) complete. 

Example. To factorise 3a; 2 + 2xy - y- + 2x - 2y - 1 . We have 3a; 2 + 2xy - y 2 
= (3x-y) (x + y). Hence, if the factorisation be possible, we must have 

3x 2 + 2xy-y 2 + 2x-2y-l = (3x-y + n){x + y + n') (7). 

Therefore, we must have 

»+3»'=2 (8), 

n-n'=-2 (9), 

«/;'=- 1 (10). 

Now, from (8) and (9), we get n= - 1, and n = +1. Since these values 
also satisfy (10), the factorisation is possible, and we have 

3x- 2 + 2xy - y 2 + 2x - 2y - 1 = (3x - y - 1 ) (x + y + 1 ). 

It should be noticed that the resolvability of 
F = ax 2 + 2hxy + by 2 + 2gx + 2fy + c 
carries with it the resolvability of the homogeneous function of 
three variables having the same coefficients, namely, 
F = ax 5 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy, 
as is at once seen by writing xfz, y/z, in place of x and y. 






Exercises XL 
Factorise the following functions : — 



(1. 
(3. 
(5. 

(7. 
(10 
(13. 
(16. 
(18. 
(20. 
(22. 
(24. 
(23. 
(27. 
(28. 
(29. 
(30. 



(a + b) 2 + [a + c) 2 - (c+ d) 2 -{b + d) 2 . (2. ) ia 2 b 2 - {a 2 + b 2 - c 2 ) 2 . 

(a 2 - 2¥ - c 2 ) 2 - i(b 2 - c 2 ) 2 . (4. ) (5a 2 - 11a; + 12) a - (4.C 2 - 15a; + 6) 3 . 

{x 2 - 03 + y)x + fa} 2 -(x- yT-(x - a) 2 . (6. ) a* - y 6 . 

x$-y\ (8.) x 2 + Qxy + 9y 2 -±. (9.) 2a^ + 3a;-2. 

x 2 +6x-16. (11.) a; 2 -10a; + 18. (12.) x 2 + a;-30. 

a; 2 + 14a; + 56. (14.) ar + 4a;+7. (15.) 2a; 2 + 5a;-12. 

x 2 + 2xs/( 2 J + q) + 2q. (17.) x*-2bx/(b + c) + (b-c)/(b + c). 

[x 2 +pq) 2 -(p + q) 2 x 2 . (19.) ab(x 2 -y 2 )+xy(a 2 -b 2 ). 

pq[x + y?-{p + q){x 2 -y 2 ) + {x-y) 2 . (21.) a; 3 -15a; 2 + 71a;- 105. 

a, J -14x 2 +148a;. (23.) a 3 - 13.^ + 54a; -72. 

3? - 8.c 2 + x - 8. (25. ) x 3 + Zpx 2 + (3p 2 - q 2 )x +p(p 2 - q 2 ). 

(p + q)x 3 + (p-q)x 2 -(p + q)x-(p-q). 

x 3 - (1 +p +p 2 )x 2 + (p +p 2 +p 3 )x -p 3 . 

x* - (a + byx 3 + (a 2 b + ab 2 )x - a 2 b 2 . 

x 6 + x i a + xta 2 - x 2 ^ - xa s - a\ 

(l+x) 2 (l+i/ 2 )-(l+y) 2 (l+ar ! ). (31.) x 4 + x 2 y 2 + y*. 



vii EXERCISES XI 143 

(32. ) Assuming x l + y i =(x~ +pxy + y 2 ) (x 2 + qxy + y 2 ), determine p and q. 
(33.) Factorise a 4 + y 4 -2(a- 2 + ?/ 2 ) + l. 

(34.) Determine r and s in terms of a, p, and q in order that x~- a 2 may 
be a factor in x* +px* + qx 2 + rx + s. 
Factorise 

(35. ) (x m +») 2 - {x'"a n ) 2 - (x"a m ) 2 + («'»+»)-. 
(36. ) (x 2 + a 2 ) 2 ^ + a 2 x 2 + a 4 ) - (x* + a; 4 a 4 + a 8 ). 
(37. ) xy 2 - 2xy -y 2 + x + 2y-l. (38. ) 2a- 2 + xy + 7x + 3y + 3. 
(39.) 2.)j 2 + a-2/-32/ 2 -a;-47/-l. (40.) xy + 7x + 3y + 21. 
(41.) a: 2 -2!/ 2 -3z 2 + 7yz + 2za; + a;#. 

(42.) Determine X so that (x + 6y -l){6x + y -l)+\(3x + 2y + l)(2x + Sy+l) 
may be resolvable into two linear factors. 

(43.) Find an equation to determine X so that ax 2 + by 2 + 2hxy + 2gx + 2/y 
+ c + \xy may be resolvable into two linear factors ; and find the value of X 
when c = 0. 

(44.) Find the condition that (ax + Py + yz) {a'x + p'y + y'z) - (a"x + p"y 
+ y"z) 2 break up into two linear factors. 

(45.) If (x+p) {x + 2q) + (x + 2p) (x + q) be a complete square in a', then 
9p 2 -Upq + 9q 2 =0. 

(46. ) If (x 4- b) (x + c) + (x + c) (x + a) + (x + a) (x + b) be a complete square in 
x, show that a = b = c. 

Factorise 

(47.) a 3 + b 3 + c?-3abc. (48.) x 3 + 3axy + ?/ 3 - a 3 . 

(49.) (x-x 2 ) 3 + {x 2 -l) 3 + (l-x) 3 . 

Factorise the following functions of a:, y, z: — * 

(50.) ^(y 2 + x 2 )(z 2 + x 2 )(y-z). 

(51.) 2(^ + ^)^-2/). (52.) 2,x i (y 2 -z 2 ). (53.) (2a-) 3 -Zx 3 . 

(54.) Simplify {Z(x 2 + y 2 -z 2 )(x 2 + z 2 -y 2 )} /U{x±y±z). 

(55.) Show that 2(1/"^" - y n z m ) and 2,x n (y m zt> - yPz m ) are each exactly 
divisible by (y -z)(z- x) (x - y). 

(56.) Show that nx n+1 - (n + 1 )x n + 1 is exactly divisible by (a*-!) 2 . 

(57.) Show that ~Zx 2 (y + z - a:) 3 is exactly divisible by 2a; 2 -22j/z. 

(58.) Show that (x + y + z) 2n+1 -x 2n + 1 ~y 2rl + 1 -z 2n + 1 is exactly divisible by 
{y + z){z + x)[x + y). 

(59. ) (y - z) 2 ^ 1 + {z- a-) 2 "+ 1 + (x - y) 2 '* 1 is exactly divisible by {y - z) (z - x) 

(x ~ !/)■ 

(60. ) If n be of the form 6m - 1, then (,y - z) n + (z - x) n + (x - y)» is exactly 
divisible by 2.x 2 -Zxy ; and, if n be of the form 6»i + l, the same function is 
exactly divisible by (2a: 2 - 2a*i/) 2 . 

(61.) Prove directly that xy-\ cannot be resolved into a product of two 
linear factors. 

(62.) If a and b be not zero, it is impossible so to determine p and q that 
x +py + qz shall be a factor of x 3 + ay 3 + bz 3 . 

* Regarding the meaning of 2 in (50), (51), &c, see the footnote on p. 84. 



CHAPTER VIII. 
Rational Fractions. 

§ 1.] By a rational algebraical fraction is meant simply the 
quotient of any integral function by any other integral function. 

Unless it is otherwise stated it is to be understood that we 
are dealing with functions of a single variable x. 

If in the rational fraction A/B the degree of the numerator 
is greater than or equal to the degree of the denominator, the 
fraction is called an improper fraction, if less, a proper fraction. 



GENERAL PROPOSITIONS REGARDING PROPER AND 
IMPROPER FRACTIONS. 

§ 2.] Every improper fraction can be expressed as the sum of 
an integral function and a proper fraction ; and, conversely, the sum 
of an integral function and a proper fraction may be exhibited as an 
improper fraction. 

For if in '— the degree m of A m be greater than the 

degree n of B n , then, by the division-transformation (chap, v.), 
we obtain 

A w _ pj J* 

n -Dn 

which proves the first part of our statement, since Q m . n is 
integral, and the degree of II is < n. 

Again, if F p be any integral function whatever, and A TO /B n a 
proper fraction (that is, m<n), then 



CHAP, nil PROPER AND IMPROPER FRACTIONS 1-45 

A P P, + A 

P -"Ml _ P n "I 

p K~~ "bT ' 

which is an improper fraction, since the degree of the numerator, 
namely, n + p, is > n. 

Examples of these transformations have already been given 
under division. 

It is important to remark that, if two improper fractions be 
equal, then the integral parts and the properly fractional parts must be 
equal separately. 

For let ^• = Q M + ^ 

A' / "R' 

am ' t>, — H m'-n' + T3' ' 

-t> ,i' -d n ' 

by the above transformation. 

A A' , 
I hen, it 



B„ B' tt < ' 



R ~ R' 



we have Q m _„ + =-= Q' m < _ M - + 



B„ ^" l -" BV 



'/i 



nence h»/i-« V»h'-)i' — t> tv 

*> » & ri 

Now, since the degrees of R' and R are less than n' and n 

respectively, the degree of the numerator on the right-hand 

side of this last equation is less than n + n' . Hence, unless 

Qm-n — Q'm'-ri — Q) we nave an integral function equal to a 

proper fraction, which is impossible (see chap, v., § 1). We must 

therefore have 

T> TV 

Qm-n = Q'm'-n'j and consequently ~- = ™~. 

-D/i -° ri 

X.B. — From this of course it follows that m -n= m' - it'. 
As an example, consider the improper fraction [a? + 2x- + Zx + 4)/(ar + x + 1), 
and let us multiply both numerator and denominator by ar + 2x+ 1 ; we thus 
obtain the fraction 

{a? + Ax 4 + 8x* + 12x- 2 + 11«+ 4)/(* 4 + Zx 3 + 4.? 2 + 3a?+ 1), 
which, by chap, iii., § 2, must be equal to the former fraction. Now transform 
each of these by the long-division transformation, and we obtain respectively 

£+3 



■ +1 +?+.+r 



VOL. I 



146 DIRECT OPERATIONS chap. 

x 3 + 5x 2 + 7x + 3 



and X + 1 + 



x 4 + 3x 3 + ^- + 3x + Y 



The integral parts of these are equal ; and the fractional parts are also equal 
(see next section). 

The sum of two 'proper algebraical fractions is a proper algebraical 
fraction. 

After what has been given above, the proof of this proposition will present 
no difficulty. The proposition is interesting as an instance, if any were needed, 
that fraction in the algebraical sense is a totally different conception from 
fraction in the arithmetical sense ; for it is not true in arithmetic that the 
sum of two proper fractions is always a proper fraction ; for example, f + i = |, 
which is an improper fraction. 

§ 3.] Since by chap, iii., § 2, we may divide both numerator 
and denominator of a fraction by the same divisor, if the nu- 
merator and denominator of a rational fraction have any common 
factors, we can remove them. Hence every rational fraction can 
be so simplified that its numerator and denominator are algebraically 
prime to each other ; when thus simplified the fraction is said to be at 
"its lotvest terms." 

The common factors, when they exist, may be determined by 
inspection (for example, by completely factorising both numerator 
and denominator by any of the processes described in chap, vii.) ; 
or, in the last resort, by the process for finding the G.C.M., which 
will either give us the common factor required, or prove that 
there is none. 

Example 1. 

a? +53?+ 7s +3 



se*+3a 8 +4a! 2 +3a;+l' 

By either of the processes of chap. vi. the G. C. M. will be found to be x- + 2.r + 1. 
Dividing both numerator and denominator by this factor, we get, for the 
lowest terms of the given fraction, 

a+3 
a, a +ss+r 

The simplification might have been effected thus. Observing that both 
numerator and denominator vanish when x= - 1, we see that x+ 1 is a com- 
mon factor. Removing this factor we get 

.-•--}- ■!..• + 3 
a*+2ar 1 +2a+l' 
Here numerator and denominator both vanish when x= - 1, hence there is the 
common factor as+1. Removing this we get 



vin WITH RATIONAL FRACTIONS 147 

x+3 

aP + x + l' 
It is now obvious that numerator and denominator are prime to each 
other ; for the only possible common factor is x + 3, and this does not divide 
the denominator, which does not vanish when x = - 3. 

§ 4.] The student should note the following conclusion from 
the above theory, partly on account of its practical usefulness, 
partly on account of its analogy with a similar proposition in 
arithmetic. 

If two rational fractions, P/Q, P'/Q', be equal, and P/Q be at its 
lowest terms, then P' = AP, Q' = AQ, where A is an integral function 
of x, which will reduce to a constant if P'/Q' be also at its lowest 
terms. 

To prove this, we observe that 

p p 

Q'~Q' 

whence r = — ~-, 

Q 

that is, Q'P/'Q must be integral, that is, Q'P must be divisible 

by Q ; but P is prime to Q, therefore by chap, vi., § 1 2, Q' = AQ, 

where A is an integral function of x. We now have 

Q 

so that P' = AP, Q' = AQ. 

If P'/Q' be at its lowest terms, P' and Q' can have no com- 
mon factor ; so that in this case A must be a constant, which 
may of course happen to be unity. 

DIRECT OPERATIONS WITH RATIONAL FRACTIONS. 

§ 5.] The general principles of operation with fractions 
have already been laid down ; all that the student has now to 
learn is the application of his knowledge of the properties of 
integral functions to facilitate such operation in the case of 
rational fractions. The most important of these applications is 
the use of the G.C.M. and the L.C.M., and of the dissection of 
functions by factorisation. 



148 EXAMPLES OF DIRECT OPERATIONS chap. 

No general rules can be laid down for such transformations as we proceed 
to exemplify in this paragraph. But the following pieces of general advice 
will be found useful. 

Never make a step that you cannot justify by reference to the fundamental 
laws of algebra. Subject to this restriction, make the freest use of your judg- 
ment as to the order and arrangement of steps. 

Take the earliest opportunity of getting rid of redundant members of a 
function, unless you see some direct reason to the contrary. 

Cultivate the use of brackets as a means of keeping composite parts of a 
function together, and do not expand such brackets until you see that some- 
thing is likely to be gained thereby, inasmuch as it may turn out that the 
whole bracket is a redundant member, in which case the labour of expanding 
is thrown away, and merely increases the risk of error. 

Take a good look at each part of a composite expression, and be guided in 
your treatment by its construction, for example, by the factors you can per- 
ceive it to contain, by its degree, and so on. 

Avoid the unthinking use of mere rules, such as that for long division, 
that for finding the G.C.M., &c. , as much as possible ; and use instead pro- 
cesses of inspection, such as dissection into factors ; and general principles, 
such as the theorem of remainders. In other words, use the head rather than 
the fingers. But, if you do use a rule involving mechanical calculation, be 
patient, accurate, and systematically neat in the working. It is well known 
to mathematical teachers that quite half the failures in algebraical exercises 
arise from arithmetical inaccuracy and slovenly arrangement. 

Make every use you can of general ideas, such as homogeneity and sym- 
metry, to shorten work, to foretell results without labour, and to control 
results and avoid errors of the grosser kind. 

Example 1. Express as a single fraction in its simplest form — 
2x s + 4x- + 3x + A 2ar>+4x 2 -3.c-2 „ 

— &n ^i — =F say - 

Transform each fraction by division, then 

F = (2x + 4) +-£-, ■- (2x+ 4) - --£±?, 

x + 1 a; — l 

_ x{a?-1)-(-x + 2){j?+l) 
.>- 4 -l 

_ 2x?-2xr-1 
~ x*-l ' 

_ 2(3 3 -a 8 -l) 

x A -l ' 

Example 2. Express as a single fraction 

111 1 



F= 



tf-Zxt+Zx-l x*-a~-x+l a A -2x 3 + 2x~l x i - 2x* + 2x : -~ 2x+ 1" 



vin WITH RATIONAL FRACTIONS H9 

AVe have 

x^-ofi-x+l =x 3 + 1-x(x+1) = (j: + 1)(x 2 -x + 1-x) 

= (x+l) (x-lf; 
x 4 - 2:< 3 + 2x - 1 =x* - 1 - 2x{x 2 - 1), 
= (* 2 -l)(.>'-l)-, 
= (x-l)*(x+l); 
x* - 2X 3 + 2x 2 - 2x + 1 = (a? + 1 ) 2 - 2x(x- + 1 ), 
= (x*+l)(x-l)*. 
Whence 

ill 1 



F = 



(x-1? (jc+l)(a;-l) a (x-lf(x+l) (ar + lU:c-l)-' 

_ (a;+l)-(.r~l) _ (a; 2 + l) + (a--l)(a;+l ) 
[x + 1 ) (x - 1 ) 3 [x - l) s (aj+ 1) (ar> + 1) ' 

2 2^ 

(x + l)(a;-l) 3 (a}-l) s (se+l)(a?+l)' 

x^+l-x* 
(x + l)(x 2 + l){x-lf 

2 



"(x*-l)(x--l) 2 ' 
2 



x*-2x i + x i - x 2 + 2x - 1 

Example 3. 

/ .k - y x 3 - y 3 \ / x + y x s + y 3 \ 
\x + y ar' + y 3 / \x-y x^-y 3 /' 

-( X -V\ (i x* + x*/ + y- \ (x+y\ / t | x 2 -xy + y 2 \ 
\x+yj \ xP-xy + y 2 / \x-y) \ x 2 + xy + y 2 )' 

=( ~ 2xy \ x ( 2 ^ + y z )\ 

\x 2 -xy + y 2 J \^ + xij + y-)' 

ixy(x- + y") 
x 4 + x 2 y 2 + y f 

Example 4. 

^2 b-c 2 c-a 2 a-b 

* =T~:+ i \t~ m + r— : + r— 5T7T — -, + — ; + 



b-c (c-a)(a-b) c-a (a-b)(b-c) a-b (b-c)(c-a)' 

^ 2(c-a)(a-b) + (b-c) 2 + 2(a-b)(b-c) + (c-a) 2 + 2{b-c){c-a) + (a-b) 2 

(b-c){c~a)(a-b) 

_ {(b-c) + (c-a) + (a-b)} 2 
&c. 

2 



&C. (b-c)(c-a){a-b) ' 
it being of course supposed that the denominator does not vanish. 



150 DIRECT OPERATIONS WITH RATIONAL FRACTIONS CHAP. 

Example 5. 

« s b 3 -3 

■C =7 tt~, ; +7^ m r + 



(a-b){a-c) (b-c)(b~a) (c-a)(c-b) 

_ - a 3 {b - c)-b 3 {c- a) - <?{a-b) 
~ (b-c)(c- a) (a - b) 

Now we observe that when b = c the numerator of F becomes 0, hence b-c 
is a factor; by symmetry c-a and a-b must also be factors. Hence the 
numerator is divisible by (b -c){c- a) (a - b). Since the degree of the numer- 
ator is the 4th, the remaining factor, owing to the symmetry of the expression, 
must be Pa + PS + Pc. Comparing the coefficients of a s b in 

- a 3 (b - c) - b 3 (c - a) - <?{a - b) 
and F{a + b + c){b-c){c-a){a-b), 

we see that P= +1. 

Hence, finally, ¥ = a + b + c. 

Example 6. 

^_ a 2 +pa + q b 2 +pb + q c 2 +pc + q 

~ (a-b)(a-c)(x-a) {b-a) {b-c) (x-b) {c-a)(c-b){x-c)' 
■p_ (h-c) (a 2 +pa + q)(x- b) (x - c) + &c. + &c. 
(b - c) (c -a) (a- b) (x - a) (x - b) (x - c) ' 
_ (b - c) (a? + pa + q) {x 2 - (b + c)x + bc} +&c. +&c. 
&c. 

Now, collect the coefficients of x 2 , x, and the absolute term in the numerator, 
observing that the two &c.'s stand for the result of exchanging a and b and a 
and c respectively in the first term. We have in the coefficient of x 2 a part 
independent of p and q, namely, 

a 2 (b -c) + b 2 (c -a)+c*(a- b)= - (J - c) (c - a) (a - b) (1). 

The parts containing p and q respectively are 

{a(b -c) + b(c -a)+c{a-b))p = Q 
and {(b-c) + (c-a) + (a-b)}q = 0. 

The coefficient of a? therefore reduces to (1). 

Next, in the coefficient of x we have the three parts, 

- {a"-(b 2 - c 2 ) + b 2 (c 2 - a 2 ) + cV - b 2 )} = 0, 

- {a(b 2 -c 2 ) + b(c 2 - a 2 ) + c(a 2 - b 2 ))p 

= -(b-c)(c-a)(a-b)p (2), 

and - {(b 2 -c 2 ) + (c 2 -a 2 ) + {a 2 -b 2 )}q = 0. 

Finally, in the absolute term, 

abc {a(b -c) + b(c -a)+ c(a - b)} = 0. 
abc {{b -c) + (c -a) + (a- b)}p - 0, 
{bc(b - c) + ca(c -a)+ ab(a -b)}q 

= -(b-c)(c-u)(a-b)q (3). 






vin INVERSE METHODS 151 

Hence, removing the common factor (b-c) (c-a)(a- b), which now appears 
both in numerator and denominator, and changing the sign on both sides, we 
have 

,r- +px + q 



F = 



(x-a) (x- b) (x — e) 



The student should observe here the constant use of the identities on pp. 
81-S3, and the abbreviation of the work by two-thirds, effected by taking 
advantage of the principle of symmetry. In actual practice the greater part 
of the reasoning above written would of course be conducted mentally. 



INVERSE METHOD OF PARTIAL FRACTIONS. 

§ 6.] Since we have seen that a sum of rational fractions can 
always be exhibited as a single rational fraction, it is naturally 
suggested to inquire how far we can decompose a given rational 
fraction into others (usually called "partial fractions") having 
denominators of lower degrees. 

O 

Since we can always, by ordinary division, represent (and that 
in one way only) an improper fraction as the sum of an integral 
function and a proper fraction, we need only consider the latter 
kind of fraction. 

The fundamental theorem on which the operation of dissec- 
tion into " partial fractions " depends is the following : — 

If A/PQ be a rational proper fraction whose denominator contains 
two integral factors, P and Q, which are algebraically prime to each 
other, then we can always decompose A/PQ into the sum of two proper 
fractions, P'/P + Q'/Q. 

Proof. — Since P and Q are prime to each other, we can (see 
chap, vi., § 11) always find two integral functions, L and M, 
such that 

LP + MQ=1 (1). 

Multiply this identity by A/PQ, and we obtain 

A _AL AM 
PQ " Q + "P" ( ~ } ' 

In general, of course, the degrees of AL and AM will be higher 
than those of Q and P respectively. If this be so, transform 
AL/Q and AM/P by division into S + Q' Q and T + P'/P, so that 






152 PARTIAL FRACTIONS chap. 

S, T, Q', and P' are integral, and the degrees of P' and Q' less 
than those of P and Q respectively. We now have 

PQ P Q v ; ' 

where S + T is integral, and P'/P + Q Q a proper fraction. But 
the left-hand side of (3) is a proper fraction. Hence S + T must 
vanish identically, and the result of our operations will be simply 

A^ + g (4). 

PQ P Q w 

which is the transformation required. 

To give the student a better hold of the above reasoning, we 
work out a particular case. 



Consider the fraction 

x*+l 



F=: 



" (x 3 + &E 2 + 2x + 1 ) (X s + x + 1) ' 
Here A=a*»+1 > F = x 3 + Bx 2 + 2x + l, Q=x" + x + l. 

Carrying out the process for finding the G. C. M. of P and Q, we have 
1-1 + 1)1 + 3 + 2 + 1(1 + 2 
2 + 1-1 
-1-1)1 + 1 + l(-l + 
+ 1 
+ 1 

whence, denoting the remainders by Ri and Ro, 

P = (x + 2)Q + Ri, Q=- x&i + R* 

From these successively we get 

Ri = P-(a- + 2)Q, 
l = R, = Q + a ,R 1 , 

= Q + xP-x{x + 2)Q, 

= (-x n --2x+l)Q + .,;!> (1). 

In this case, therefore, 

M=-z*-2x+l, L=x. 

Multiplying now by A/PQ on both sides of (1), we obtain (putting in the 
actual values of P and Q in the present case) 

(af' + l)(-a 3 -2g + l) (z* + l)x . 
x 2 + Zx? + 2x+\ x*+x+l' 
_ -x e -2x 5 + x i -x 2 -2x+l j~ - x 

x 3 + 3x? + 2x+l + a? + x+l ' 

or, carrying out the two divisions, 






viii SPECIAL CASES 153 

or, seeing that the integral part vanishes, as it ought to do, 

which is the required decomposition of F into partial fractions. 

Cor. If P, Q, R, S, . . . be integral functions of x which are 

prime to each other, then any proper rational fraction A/PQRS . . . 
can be decomposed into a sum of proper fractions, P'/P + Q' Q + 

R'/R + sys + . . . 

This can be proved by repeated applications of the main 
theorem. 

§7.] Having shown a priori the possibility of decomposition 
into partial fractions, we have now to examine the special cases 
that occur, and to indicate briefer methods of obtaining results 
which we know must exist. 

"We have already stated that it may be shown that every 
integral function B may be resolved into prime factors with real 
coefficients, which belong to one or other of the types (x - a)'', 

(/ + 0b + 7 y. 

1st. Take the case where there is a single, not repeated, 
factor, X - a. Then the fraction F = A B may be written 

F * 
(•!• - a)Q 

say, where X - a and Q are prime to each other. Hence, by our 
general theorem, we may write 

f- F + Q ' m 

each member being a proper fraction. 

In this case the degree of P' must be zero, that is, P' is a 
constant. 

It may be determined by methods similar to those used in 
chap, v., § 21. See below, Example 1. 

P' determined, we go on to decompose the proper fraction 
Q Qi by considering the other factors in its denominator. 



154 SPECIAL CASES chap. 

2nd. Suppose there is a repeated factor (x - a) r ; say B = 
(.r - a) r Q, where Q does not contain the factor x - a. "We may, 
by the general principle, write 

r, P' + Q' 



(a; - a)'" Q* 

P' is now an integral function, whose degree is less than r\ 
hence, by chap, v., § 21, Ave may put it into the form 

P' = a + di(z - a) + . . . +a r _i(a;-a) r " 1 , 

and therefore write 

" + ...+^+g (2), 



(x-af (a; -a)'" 1 x-a Q 

where a , a lf . . ., a,._i are constants to be determined. See 
below, Example 2. 

3rd. Let there be a factor (x 2 + fix + y) g , so that 

b = (x 2 + fix + y yq, 

Q being prime to x 2 + fix + y. Now, we have 

P' Q' 



(x 2 + px + 7 y q- 

P' is in this case an integral function of degree 2s - 1 at most. 
We may therefore write, see chap, v., § 21, 

P' = (a + b x) + (</, + b x x) (x 2 + fix + y) 
t (a 2 + kjx) (x 2 + fix + y) 2 

+ (a g „ l + b 8 - 1 z)(x* + px + y y- 1 . 

We thus have 

_ a + h„r a, + b x x at^ + bg-tf Q' 

(x 2 + /3x+ y y> (x 2 + fix + y)*- 1 x 2 + (3x + y Q { ' ' 

where the 2s constants a m b , &c, have to be determined by any 
appropriate methods. See Examples 3 and 4. 

In the particular case where s = 1, we have, of course, merely 

_ a + &Q.T Q' 



Viil EXPANSION THEOREM 155 

By operating successively in the way indicated we can decompose 

every rational fraction into a sum of partial fractions, each of which 

belongs to one or other of the two types p r j{x - a) r , (a s + b s x)/(x 2 + ftx 

+ y) s , where a, ft, y, p r , a s , b 8 are all real constants, and r and s 

positive integers. 

It is important to remark that each such partial fraction 
has a separate and independent existence, and that if necessary 
or convenient the constant or constants belonging to it can be 
determined quite independently of the others. 

Cor. If P be an integral function of x of the nth degree, and 
a, a, . . ., a; ft, (3, . . . , ft ; y, y, . . . , y, . . . constants not less 
than n + 1 in number, r of which are equal to a, s equal to ft, t equal 
to y, . . . , then we can ahoays express P in the form 

P = 2{a + a,(x -«)+...+ ctr.^x - a) r " 1 }(a: - ft)%x - y) ( . . ., 

where a , a u . . . , a r _ u . . . are constants. In particular, if r = 1. 
s = 1 , t = 1, . . . , we have 

P = ?a (x-ft)(x-y) ... 

These theorems follow at once, if we consider the fraction 

p/(x - a y(x- fty(x - 7 y... 

There is obviously a corresponding theorem where x - a, 
x- ft, x - y are replaced by any integral functions which are 
prime to each other, and the sum of whose degrees is not less 
than n + 1. 

§ 8.] We now proceed to exemplify the practical carrying 
out of the above theoretical process ; and we recommend the 
student to study carefully the examples given, as they afford a 
capital illustration of the superior power of general principles as 
contrasted with "rule of thumb" in Akebra. 

o 

Example 1. It is required to determine the partial fraction, corresponding 
to x- 1, in the decomposition of 

(Iv 4 - 16a»+17a? - 8.r + 7)/(.r - 1) (a - 2) 2 (a- 2 + 1). 
We have 

r _ 4a^-16^ + 17x 2 -8a- + 7 _ p Q' 

(a;-l)(a;-2) 2 (a; 2 +l) a-l + (aj-2) 2 (£B 2 + l) ( >' 

and we have to find the constant p. 



156 EXAMPLES CHAP. 

From the identity (1), multiplying both sides by (x-1) (j;-2) 2 (a; 2 + l), we 
deduce the identity 

±x A - 16x 3 + 1 7 x s - 8.r + 7 =p{x - 2) 2 (« 2 + 1) + Q'(x - 1 ) (2). 

Now (2) being true for all values of x, must hold when o-=l ; in this case it 
becomes 

4 = 2^), that is,p=2. 

Hence the required partial fraction is 2/(x- 1). 

If it be required to determine also the integral function Q', this can be 
done at once by putting ^> = 2 in (2), and subtracting 2(.c-2) 2 (.£ 2 + l) from 
both sides. We thus obtain 

2x* - 8a? + 7,<" - 1 = Q'(x - 1 ) (3). 

This being an identity, the left-hand side must be divisible by x-1.* It is 
so in point of fact ; and, after carrying out the division, we get 

2,r 3 -6a: 2 + a:+l = Q' (4), 

which determines Q'. 

The student may verify for practice that we do actually have 
4^-16^ + 1 7.x- 2 - 8x + 7 _ _2_ 2 a 3 - 6x ° + x + l 
(x-l){x-2)*(x* + l) x - 1 + (x - 2) s (x 2 + 1 ) * 

Example 2. Taking the same fraction as in Example 1, to determine the 
group of partial fractions corresponding to (a; - 2) 2 . 
1°. "VVe have now 

4x * - 1 6^ + 1 7a; 2 - 8a: + 7 _ a n a x Q' .. 

(a;-l)(a;-2) 2 (a; 2 +l) ~ {x-2)- + (x^2) + (x-1) (x 2 +l) ^ " 
whence 

ix i -16x 3 + l7x' 2 -8x + 7 = a (x-l) (z 2 + 1) + <n(x- 2) (aj-1) (x 2 + 1) 

+ Q'(*-2) 2 (2). 

In the identity (2) put x = 2, and we get 

- 5 = 5«oi that is, « = - 1 . 
Putting now a = -1 in (2), subtracting ( - 1) (.«- 1) (.c 2 + l) from both sides 
and dividing both sides by x - 2, we have 

4r i -7v i + 2x-3 = a 1 (x-l)(x- + l) + Q'{x-2) (3). 

Put x = 2 in this last identity, and there results 

+ 5 = 5cti, that is, a,\ = + 1. 
The group of partial fractions required is therefore 

-l/(x-2T- + l/(x-2). 
If required, Q' may be determined as in Example 1 by means of (3). 
2°. Another good method for determining a and a\ depends on the use of 
"continued division." 

If we put x = y + 2 on both sides of (1), we have the identity 

4(y + 2) 4 -16(y + 2) 3 + 17(y + 2) 2 -8(?/ + 2) + 7 = a a x Q" 

(2/+l)2/ 2 l(y + 2) 2 +l} y 2 y + (y + l){{y + 2f + \\' 

* If it is not, then there lias been a mistake in the working. 



vin EXAMPLES 157 

that is, 

-5-4y + &c. _fl ffl _ Q" 

5y- + 9f + kc. y 2+ y (l+y)(3 + iy + y-) 
Now, by chap, v., § 20, the expansion of a rational fraction in descending 
powers of 1/y and ascending powers of y is unique. Hence, if we perform 
the operation of ascending continued division on the left, the first two terms 
must be identical with a /if + ai/y ; for Q"/(l +y) (3 + 4y + ?/ 2 ) will obviously 
furnish powers of y merely. 
AVe have 

-5-4+... 5 + 9+ . , . . 

+5+... -1+1+... 

therefore a = -1, ai= +1. 

The number of coefficients which we must calculate in the numerator and 
denominator on the left depends of course on the number of coefficients to be 
determined on the right. 

Example 3. Lastly, let us determine the partial fraction corresponding to 
7? + 1 in the above fraction. 
We must now write 

4.7;* -I6x? + l7x 2 -8x + 7 _ ax + b Q' . 

(x-l)(x-2)*(x 2 +lj~~x 2 + l + (x-l)(x-2f ( '• 

1°. Whence, multiplying by (x- 1) (x- 2) 2 , 
4ar 1 -16x 3 + 17x 2 -8^ + 7 _ jax + b) (x-1) {x- 2) 2 

x*+l ~ x-+l 

whence 



■ + Q' (2) 



4 ,,.2 _ 1 6i , + 13 + 8 4_J = {nx + b) ( x _ 5 + i£±l\ + Q' 
X'+l \ X 1 + 1 / 



, ,,, rN 7ax- + (7b + a)x+b _. 
= (ax + b) (x -5)4 \, + 1 - + < x >\ 

... r , „ (7b + a)x + (b-7a) _, . 
= (ax + b) .r-5 +7a + v ^—. \ ' + Q (3). 

ar + 1 

Now the proper fractions on the two sides of (3) must be equal — that u, we 
must have the identity 

( 7 b + a)x + (b - 7a) = Sx - 6, 
therefore 7b + a — 8, b-7a= - 6. 

Multiplying these two equations by 7 and by 1 and adding, we get 

506 = 50, that is, 6=1. 
Either of them then gives «=1, heuce the required partial fraction is 

(x+l)l(x*+l). 
2". Another method for obtaining this result is as follows. 
Remembering that x 2 + 1 = [x + i) (x - I) (see chap, vii.), we see that a; 2 +1 
vanishes when x—i. 
Now we have 

lx i -16x"+l7x 2 -8x + 7 = (ax+b){x-l)(x-2) 2 + Q'{x n -+l) 

= [ax + b) (a? - 5x 2 + 8x - 4) + Q'(.>" + 1 ) (4). 



158 EXAMPLES 



CHAP. 



Put in tins identity x=i, and observe that 

i i = Pxi 2 = (-l)x(-l)=+l, 
^—■Px i—( -l)xi= - i ; 
and we have 8i-G-(ai + b) (7i+l), 

= (7b + a)i+{b-7a) ; 
whence (7b + a - 8)i= - b + 7a- 6, 

an equality which is impossible * unless both sides are zero, hence 

7b + a-8 = 0, -b + 7a-6-0, 
from which a and b may be determined as before. 

3°. Another method of finding a and b might be used in the present case. 
We suppose that the partial fractions corresponding to all the factors 
except ic 2 + 1 have already been determined. We can then write 

„ 2 1 1 ax + b /c . 

From this we obtain the identity 
4.i 4 -16j 3 + 17a; 2 -8a; + 7 

= 2(x-2) 2 (x 2 + l)-(x-l)(x°+l) + (x-l)(x-2)(x 2 +l) 

+ (ax+b){x-rl){x- 2) 8 ; 

whence 

% A - Ao? + Sx* + ix - 4 = (ax + b) [x - 1) [x - 2) 2 ; 

and, dividing by (x - 1) (x - 2) 2 , 

x+l=ax + b. 
This being of course an identity, we must have 

a = l, b = l. 

Another process for finding the constants in all the partial fractions depends 
on the method of equating coefficients (see chap, v., § 16), and leads to their 
determination by the solution of an equal number of simultaneous equations 
of the 1st degree. 

The following simple case will sufficiently illustrate this method. 

Example 4. 

To decompose (3x-i)/(x- l)(„e-2) into partial fractions. 

We have 

Zx - 4 a b 

{x-\)[x~2)~x^l Jr x^~V 
therefore 3x- l = a(x-2) + b(x- 1), 

= (a + b)x - (2a + b). 
Hence, since this last equation is an identity, we have 

a + b = 3, 2a + b=i. 
Hence, solving these equations for a and b (see chap. xvi. ), we find «=i, 
b = 2. 

* For no real multiple (differing from zero) of the imaginary unit can be a 
real quantity. See above, chap, vii., § 6. The student should recur to this 
case again after reading the chapter on Complex Numbers. 



VIII 



EXERCISES XII 



159 



Example 5. We give another instructive example. To decompose 



F=,- 



x-+px + q 



we may write 



x- +px + q 



(x -a)(x- b) (x-c)' 
C 



A B 

+ - 



+ ■ 



(1), 



(x - a) (x - b) {x - c) x-a x-b x-c 
where A, B, C are constants. 
Now 

x 2 +px + q = A(x -b)(x-c) + B(x - e) {x- a) + C{x - a) (x - b) (2). 

Herein put x = a, and there results 

a 2 +pa + q= A(a- b)(a-c) ; 



whence 

By symmetry 

We have therefore 
x^+px + q 



A= a2+ P a + 9 
(a -b)(a-c)' 

b 2 + pb + q 



B= 



(x- a)(x- b)(x- c) 

a" +pa + q 



+ 



(b-a)(b-c)' 

c 2 +pc + q 

(c~a)(c- b)' 



b 2 +pb + q 



+ . 



c 2 +pc + q 



(a- b)(a-c)(x- a) (b-c){b-a)(x-b) (c-a)(c-b)(x-c) 



(3), 



an identity already established above, § 5, Example 6. It may strike the 
student as noteworthy that it is more easily established by the inverse than 
by the direct process. The method of partial fractions is in point of fact a 
fruitful source of complicated algebraical identities. 



Exercises XII. 
Express the following as rational fractions at their lowest terms. 



(1. 
(2. 

(3. 

(4. 
(5. 
(6. 
(7- 
(8. 

(9. 

(10. 



{x 3 + 2x 2 -x + 6)/(x* - x 2 + ix - 4). 
(9ar» + 53.T 2 - 9a- J 8)/(4x 2 + 44« + 120). 

z*+2x*-2x-l _ x*_+ ,< a - 3<" - 5x-2 
x* + a*-3x*-5x-2 a? 4 + 2x* - 2x - 1 " 

{3x*-x*-z- 1 )/(8a? + 5x 2 + Zx + 1 ) + (se* + 3a? + 5x + 3)/{x 3 + x 2 + x - 3). 
(a- 6 - 2a? + l)/(x 2 -2x + l) + {a? + 2a? + 1 )!(x 2 + 2x + 1 ). 
(6a? + IZax 2 - 9a 2 x - 10« 3 )/(9x 3 + \2ax 2 - Ua 2 x - 10a 3 ). 
(l-a-)l{(l+ax) 2 -(a + x) 2 }. 
{(w + x + z)(w + x)-y(y + z)}/{(w + x + z)(w + z)-y(x + y)}. 

(\-x)(\-x 2 fl { (T^p " (l-x)(l-x 2 ) + (l^tf) 2 ) ' 
{ (al + bm) 2 + (am - bl) 2 } / { (ap + bq) 2 + (aq - bpf } . 



160 



EXERCISES XII 



CHAP. 



(11. 

(12. 

(14, 
(15. 

(16. 

(17. 
(18. 
(19. 
(20. 

(21. 

(22. 
(23. 

(24. 
(25. 
(26. 

(27. 

(28. 

(29. 

(30. 
(31. 
(32. 

(33. 
(34. 
(35. 



{px 2 + (k-s)x + r} 2 - {px 2 +(k + s)x + r} 2 
\p3? + {k + t)x + r} 2 - {px 2 + (k-t)x + r} 2 ' 

x %-y 



2x -2y 2y-2x 



(13.) 



■ + ■ 



1 



a-b-(a-b)x a + b + (a + b)x 



l/(a -2b- lj(a -2b- l/(a - 2b) ) ). 
1/(6* + 6) - l/(2a3 - 2) + 4/(3 - 3a; 2 ). 
x^-y 3 



x*-y* 



x-y _ , / x + y 1_ ) 

x 2 -y 2 t \x 2 + y 2 x + y]' 
/ x l-x \ I / x \-x \ 
\\+x x )J \l+a: x )' 



6a; 
3a;-2 9a,- 2 + 4 



30a~+4a- Ax 




2x +1 
24(aY^l) ' 8(a;+l) ' 4(a- + l) 2 2(a;+iy 3 "3(a; 2 + a;+l)• 

_1 1_ _2_ _2_ 

(a;+l) 2 (a; + 2) 2 (x + 2f + x + l x+2' 

{a + b)j{x + a) + {a - b)/(x -a)- 2a{x + ty/ix 2 + a"). 

{(x-y)l{x+y)} + {{x-y)l{x + y)} 2 + {(x-y)/(x+y)}*. 

/ a- 3 -3a-+2 \ /a- 2 + 2a;+l \ 

\.r i + 2x 2 + 2x+l) X W-bx+i)' 

'a 2 + x 2 



/a 2 + x 2 \ 

(r^x- +1 ) x 



2ax 
x + y 



+ 



+ 1 
x-y 



ax- 



-5a; +4, 

4a(a + x, 

a 2 - ax + x 



2' 



-2- 



x' - %f 



aP-y 3 ar 5 + y 3 x i + x 2 y 2 + y 4 ' 
x 2 y 2 



1 1 

- + -. 
x- y< 



I 



Ht Hfi-' t ts)( t -.) 



x° y 



+ ; 



+ • 



2a(a-c)(x-a) 2a(a + c)(x + a) (c 2 - a 2 ) (x + c) 

( _20__180 420__280\ /i__20_ 2?2 _ i 2 ^ 280 \ 

(. + a-+l a- + 2 + a; + 3 a'+4j X t x-l + x-2~ x^S + x- 4j 

{(xy-l) 2 + (x + y-2)(x + y-2xy)}/{(xy + l) 2 -(x + y) 2 }. 

{l+y*+*-Syz)/(l+y+z). 

{a{a + 2b) + b(b + 2c) + c(c + 2a)} / [a 2 -b 2 -c 2 - 2bc) . 

(a + b) s +(b + c) s -(a + 2b + c) i 



(a + b)(b + c)(a + 2b + c) 

x 6 + a 6 



■+- 



orx* 



(I c + a 6 ) (a? - a 2 ) + a 2 x 2 (x* - a 4 ) X*- « G - « V V s - a 2 )' 

a 2 + (2ac - b 2 )x 2 + c 2 x A a 2 + (ac - b 2 )x 2 - bcx* 

a 2 + 2abx + {2ac+b- a ' •zhcjF+c 2 x A * a- + \ac ->> 2 + bex '■' 






VIII 



EXERCISES XII 



161 



X 2 + y 2 +x + y-xy + l | x 2 + y 2 + x - y + xy + 1 
x-y-1 x + y-1 



(37.) 



(38.) 



(■r 5 -10,cy + 5.CT/ 4 ) 2 + (5afy - lOieV + //•-/-' 

(b + c)- + 2(b 2 -c°-) + (b-cf 



(39.) 2(6 2 + c 2 -a 2 )/(a-&)(<w). 
(41.) 2(6 + c)/(c-o)(a-&). 
(43.) 2(& 2 + &c + c 2 )/(a-&)(«-c). 
(44.) {II(l-a?) + n(a;-y 2 )}/(l-aj^). 
(45.) {Z(J + c) s -3II(& + c)}/{2a 3 -3a&c}. 

(46.) ]~ x i g ~* y i y-1 1 ( 1 - x )( x -y)(y- 1 ) 



(40.) (Zx)CZx 2 )/xyz-Z(y + z)/x. 
(42.) 2,bc{a + h)/{a-b){a-c). 



(47. 



1+x x + y y + 1 (l+a:)(a: + ?/)(y + iy 
(y- 2 ) 2 +( z -a;) 2 + (.r-y) 2 | £ / 1 [ 1 1\ 

(y-2)(»-aj)(aj-y) Vy-z s-a: a;-y/ 



ras \ &-c , c-« , a-ft , (b-c)(c-a)(a-b) 

148.) 1 -\ 1-7 r-j re, ,. 

x - a x-b x-c (x-a)(x-b)(x-c) 

(49.) 2(a+p)(a + q)f{a-b)(a-c)[a + h). 

(50.) 2a 2 /(a-b){a-c)(h-a). (51.) 2a 2 /(a 2 -b°-)(o 2 -c 2 )(h 2 + a 2 ). 

(52. ) Z(y 2 + 0- x 2 )/yz(x - y) (x - z). 

a(b-c) 3 + b(c-a) 3 + c(a-b) s + {b 2 -c 2 )(b-c) + (c 2 -a 2 )(c-a) + (a 2 -b 2 )(a-b) 
a 2 (b-c) + b\c-a) + c 2 (a-b) 
{U) {{x + yf + {y + zf) {( z + x y> + (x + w) 2 } 

{(z + y){z + x) + (y + z)(x + w)} 2 +{(x+~y)(x + w)-{y + z){z + x)} 2 



(53.) 



Prove the following identities : — 
(55.) 2a 3 /(a-b){a-c) = Za. 
(56.) c(ic 2 -v) = au(l -uv), c{v 2 -u) = bv{l -uv), 
where tc = {ab-c 2 )j{bc-a 2 ), v=(ab-c 2 )/(ca-b 2 ). 

(57. ) 2 (a + a) (a + /3) (a + y)(a{a -b){a-c){a-d)= - aPyfabcd. 

abed 

. (b-c) 3 + (c-a) 3 + (a-bf K ' 



(59.) 



(60.) 



(ab - erf) (a 2 - 6 2 + c 2 - rf 2 ) + («c - bd) (a 2 + b 2 -c 2 - d 2 ) 
(a 2 - b- + c- - W) (a 2 + b 2 - c 2 - d 2 ) + 4.{ab - cd) (fflc - bd) 

(b + c)(a + d) 
~(b + c) 2 +(a + df 
a 5 (c -b) + b\a - c) + c 5 (6 - a) 



(c- b)(a-c)(b-a) 
(61.) {2{y-8)8}/{2(y-^}-4n(y- Z )a={2 a ?-2^}». 



Decompose the following into sums of partial fractions : — 
(62.) 0r 2 -l)/(x-2)( ; «-3). (63.) x 2 /(x- l)(x-2)(x- 3). 

(64.) 30a*/(a?-l)(a?-4). (65.) (.c 2 + 4)/(a; + l) 2 (x-2)(x + 3). 

VOL. I M 



1G2 



EXERCISES XII 



CHAP. VIII 



(66.) (&-2)l(a?-l). 

(68.) {2x-3)/(x-l)(x 2 + iy. 



(67.) (z 2 + * + l)/(a; + l)(z 2 + l). 

(69.) l/{x-a)(x-b)(x 2 -2px + q), p*<q. 



(70.) (l + aj + a?)/(l -»-»* + a 5 ). (71.) 18/(ar* + 4;e+8). 
(72.) (* + 3)/(^-l). (73.) lKafi + tf-rf-a*). 

(74.) Express (3a 2 + ic + l)/(a; 8 - 1) as the sum of two rational fractions 
whose denominators are x i -\ and x 4 + l. 

(75.) Expand 1/(3 -x) (2 + a?) in a series of ascending powers of x, using 
partial fractions and continued division. 

(76.) Expand in like manner 1/(1 -x) 2 (l+x 2 ). 

(77.) Show that 
2 (b + c + d)/(b-a)(c-ct)(d-a)(x - a) = (x-a-b-c-d)J(x - a)(x - b)(x - c)(x - d). 

abed 






CHAPTER IX. 
Further Application to the Theory of Numbers. 

ON THE VARIOUS WAYS OF REPRESENTING INTEGRAL AND 
FRACTIONAL NUMBERS. 

§ 1.] The following general theorem lies at the root of the 
theory of the representation of numbers by means of a systematic 
scale of notation : — 

Let r l} r.,, r 3 , . . ., r n , r n+1 , . . . denote an infinite series of 
integers* restricted in no way except that each is to be greater than 1, 
then any integer N may be expressed in the finite form — 

N =p +p lTl + p 2 r 1 r. 2 +iW 8 r a + • • • +iW a . . . r n , 

where p <r u p x <r s , p. 2 <r 3 , . . ., p n <r n+l . When r u r. 2 , r a , . . . 
are given, this can be done in one way only. 

For, divide N by r lt the quotient being N, and the remainder 
p ; divide N\ by r 2 , the quotient being N 2 and the remainder 
p l} and so on until the last quotient, say p n , is less than the next 
number in the series which falls to be taken as divisor. Then, of 
course, the process stops. We now have 

N =p + T$ 1 r l (j? <0 (1), 

N t = Pl + N 2 r 2 (p x <r 3 ) (2), 

N 3 =^ 2 + N 3 r 3 (p t <r a ) (3), 



N„ _ , = p n _ ! + p n r n { p n _ !<r„) (n). 

* In this chapter, unless the contrary is distinctly implied, every letter 
used denotes a positive integral number. 



164 FACTORIAL SERIES FOR AN INTEGER chap. 

From (1), using (2), we get 

N=Po + r 1 (p 1 + 'N,r 2 ), 
=Po + PiTi +rjrJS r 
Thence, using (3), 

N =p +P{>\ + PJV 2 + W.N,, 
and so on. 

Thus we obtain finally 
N =p +PS\ +iW g + pj\r 2 r 3 + . . . + p n r 1 r 2 . . . r n (A). 
Again, the resolution is possible in one way only. For suppose 
we also had 

N =p ' +p 1 'r 1 + p 2 'r 1 r 2 + p 3 'r 1 r 2 r 3 + . . . +p n 'r 1 r 2 . . . r n (B), 

then, equating (A) and (B), and dividing both sides by r lt we 
should have 

p 

- + (Pi +P*r a +p a r 2 r a + . . . +p n r 3 r a . . . r n ) 

' i 

= 7 + (ft'+ft''i+A'Vi + . • • +Pn'r a r a . . . r n ) (C). 
' i 

But the two brackets on the right and left of (C) contain integers, 
and p fr } and p '/r l are, by hypothesis, each a proper fraction. 
Hence we must have^,/^ -pdj^'i ', that is, 

Po*=Po, 

p x +p 2 r e + p 3 r 2 r 3 + . . .4 p n r 2 r 3 . . . r n 

= Pi +P*r* + p a 'r a r a + . . . + p n 'r 2 r 3 . . . r n (D). 

Proceeding now with (D) as we did before with (C), we shall 
prove Pi—pi'; and so on. In other words, the two expressions 
(A) and (B) are identical. 

Example. Let N = 719, and let the numbers r\, r%, r s , . . . be the natural 
series 2, 3, 4, 5, . . . Carrying out the divisions indicated above, we have 

2 )719 

3 )359 ... 1 

4 )119 ... 2 

5)29 ... 3 

5 ... 4. 



ix FACTORIAL SERIES FOR A FRACTION 165 

Hence Po = l, 2*1=2, #s=3, 2>s=4, Pi=5 ; 

and we have 719 = 1 + 2x2 + 3x2.3 + 4x2.3.4 + 5x2.3.4.5. 

§ 2.] There is a corresponding proposition for resolving a 
fraction, namely, r u r.,, . . ., r n , &c, being as before, 

Any proper fraction A/B can be expressed in the form 

± = Pi + r±. + J!±- + . . . + *>» +F 

B r, i\r 2 r,r, 2 r 3 i\r 2 . . . r n 

where p^r^ p. 2 <r 2 , . . ., p n <r n ; and F is either zero or can be made 
as small as we please by taking a sufficient number of the integers 
r u r 2 , . . ., r n . When o\, r 2 , . . ., r n , . . . are given, this resolution 
can be effected in one way only. 

The reader will have no difficulty in deducing this proposition 
from that of last paragraph. It may also be proved thus : — 

A _ Ar, _ Ar,/B 

b^bt; - "^ - ' 

Now we may put A^/B into the form p l + qJB, where g\<B. 
We then have 

A p x + g,/B 
B~ r> ' 

where pi<r Xi since, by hypothesis, A<B. 
Hence 

- = ^ + --^ (1) 

Treating the proper fraction ^,/B in the same way as we treated 
A/B, we have 



(.2). 



B r 2 + r 2 


°2 

B' 


where 


2>*<r ai &<B 


Similarly, 




I 2 = il + I 
B r 3 r a 


ft 

B' 


"'"• 


Pa<r a , 23<~B, &c 



(3). 



166 FACTORIAL SERIES FOR A FRACTION chap. 

And, finally, 

gn-i = Pn 2_ . ?» 

B 7- M r„ B' 
where Ihi^n, 8»<B (w). 

Now, using equations (1), (2), . . ., (n) in turn, we deduce 
successively 



B r, r x r 2 r 1 r a B f 



r, r,r 2 v/, W.B' 



^' + A + _J?i_ + ■ ^ 



+ q — p (A), 

r,r 8 . . . r n B 

where p^r,, p 2 <r. 2 , . . ., p?i<r n , q n <B. 

It appears therefore that F = q n \r{r % . . . r n B, which can 
clearly be made as small as we please by sufficiently increasing 
the number of factors in its denominator. This of course in- 
volves a corresponding increase in the number of the terms of 
the preceding series. 

It may happen, of course, that q n vanishes, and then F = 0. 
We leave it as an exercise for the student to prove that this case 
occurs when rfa . . . r n is a multiple of B, and that if A/B be 
at its lowest terms it cannot occur otherwise. He ought also to 
find little difficulty in proving that the resolution is unique when 
fii r„ • • ., r n , . . . are given. 

Example 1. Let A/B = 444/576, and let the numbers r\, r%, &c. , be 2, 4, 
6, 8, &c. 

We find 444 1 2 1_ 

57«~2 + 2.4 + 2.4.6' 
Example 2. A/B = 11/13, r 1; r«, &c., being 2, 3, 4, 5, 6, . . ., &c. 
11 1 2 1 3 3 



16 22,32. 8. 42. 3. 4. 52. 3. 4. 5. 62. 3. 4. 5. 6x13 

Since T\, r 2 , &c, are arbitrary, we may so choose them that the numer- 
ators p it ju, &c, shall each be unity. We thus have a process for decompos- 
ing any fraction into a sum of others with unit numerators. 



IX EXAMPLES 1G7 

Example 3. 11 5 1 

2 x 3x 13 x 

13)22(1 13)15(1 13~)l3(l 

13 13 13 

9 2 

2x 7 x 

13)18(1 13)14(1 

13 13 



Whence 



5 1 



n_i _i_ l _i i 

13~2 + 2.2 + 2.2.3 + 2.2.3.7 + 2.2.3.7.13" 



Here we have chosen at each step the least multiplier possible. When 
this is done, it may be shown that the successive remainders diminish down 
to zero, the successive multipliers increase, and the process may be brought 
to an end. If this restriction on the multiplier be not attended to, the reso- 
lution may be varied in most cases to a considerable extent. Since, however, 
we always divide by the sa me divisor B, there are only B possible remainders, 
namely, 0, 1, 2, . . ., B - 1 ; hence after B - 1 operations at most the remainder 
must recur if the operation has not terminated by the occurrence of a zero. 

Example 4. Thus we have 

2_1 1 

3~2 + 2.3 ' 

. Ill li 

also =o+?r- 7 + ^r-75+. • .+s— r„ + ; 



2 2.4 2. 4 2 2.4" 2. 4". 3' 



Example 5. 

U + JU 



29 5 5 . 5 5 . 5 . 29 ' 

als ° 4nV^ + ^ + CT + ^ + 5^ + 5T^ + &C - ; 

, 111 1 

also = - + t—z ■ + , „ „ + - 



66. 3 6. 3. 36. 3. 3. 29' 

and so on. 

§ 3.] The most important practical case of the proposition in 
§ 1 is that where r„ r 2 , . . . are all equal, say each =r. Then 
we have this result — 

Every integer N can be expressed, and that in one tvay only, in 
the form 

P*r n +Pn-S n ~ 1 + • • • +2hr+p , 
where _p , p iy . . ., p n are each < r. 

In other words, detaching the coefficients, and agreeing that 
their position shall indicate the power of r which they multiply, 
and that apposition shall indicate addition (and not multiplica- 
tion as usual), we see that, r being any integer whatever chosen 



168 SCALES OF NOTATION, INTEGERS CHAP. 

as the radix of a scale of notation, any integer whatever may be 
represented in the form p n p n -i • • • PiPol where each of the 
letters or digits p , p lf . . ., p n must have some one of the integral 
values 0, 1, 2, 3, . . ., r - 1. 

For example, if r = 1 0, any integer may be represented b.y 
PnJPn-i ■ • • P\Po where p ,Pi, • • -,p n have each some one of 
the values 0, 1, 2, 3, 4, 5, G, 7, 8, 9. 

The process of § 1 at once furnishes us with a rule for finding 
successively the digits p ,P\,p 2 , . . ., namely, Divide the given integer 
N by the chosen radix r, the remainder will be p ; divide the integral 
quotient of last division by r, the remainder will be p t , and so on. 

Usually, of course, the integer N will be given expressed in 
some particular scale, say the ordinary one whose radix is 10; 
and it will be required to express it in some other scale whose 
radix is given. In that case the operations will be carried on in 
the given scale. 

The student will of course perceive that all the rules of ordi- 
nary decimal arithmetic are applicable to arithmetic in any scale, 
the only difference being that, in the scale of 7 say, there are 
only 7 digits, 0, 1, 2, 3, 4, 5, 6, and that the "carriages" go by 
7's and not by 10's. 

If the radix of the scale exceeds 10, new symbols must of 

course be invented to represent the digits. In the scale of 12, 

for example, digits must be used for 10 and 11, say t for 10 

and e for 11. 

Example 1. To convert 136991 (radix 10) into the scale of 12. 

12)136991 







12)11415 


. . 


. e 






12)951 


. . 


. 3 






12)79 




. 3 






6 




. 7 


The result is 


6733e. 








Example 


2. To convert G733e (radix 


12) 


into the scale of r. 






r)6733e 










T)7el7 


. . 


. 1 






t-)9i;i 




. 9 






r)e4 


. . 


. 9 






r)ll 




. 6 






1 




. 3 


The result is 


136991. 









IX EXAMPLES OF ARITHMETICAL OPERATION 1G9 

Although this method is good practice, the student may very probably 
prefer the following : — 

6733e (radix 12) means 
6xl2 4 + 7xl2 3 + 3xl2 a + 3xl2 + ll. 

Using the process of chap, v., § 13, Example 1, we have 

6+ 7 + 3+ 3+ 11 

+ 72 + 948 + 11412 + 136980 
6 + 79 + 951 + 11415 + 136991. 

§ 4.] From one point of view the simplest scale of notation 
would be that which involves the fewest digits. In this respect 
the binary scale possesses great advantages, for in it every digit 
is either or 1. For example, 365 expressed in this scale is 
101101101. All arithmetical operations then reduce to the 
addition of units. The counterbalancing disadvantage is the 
enormous length of the notation when the numbers are at all 
large. 

"With any radix whatever we can dispense with the latter 
part of the digits allowable in that scale provided we allow the 
use of negative digits. For let the radix be r, then whenever, 
on dividing by r, the positive remainder p is greater than r/2, we 
can add unity to the quotient and take - (r - p) for a negative 
remainder, where of course r-p<r/2. For example, 3978362 
(radix 10) might be written 4022442, where 2 stands for -2; 
so that in fact 4022442 stands for 4-10° + O'lO 5 - 2"10 4 - 2-10 3 
+ 4-10 2 -4-10 + 2. 

Example 1. Work out the product of 1698 and 314 in the binary scale. 

1698 = 11010100010 
314= 100111010 



11010100010 
11010100010 
11010100010 
11010100010 
110101000100 

10000010001010110100 ( = 533172 radix 10). 

Example 2. Express 1698 and 314 in the scale of 5, using no digit greater 
than 3, and work out the product of the two transformed numbers. 



170 EXAMPLES OF ARITHMETICAL OPERATION chap. 

5)1698 5)314 



5)339 . . 
5)68 . . 
5)13 . 


. 3 
. 1 
. 3 




5)63 . . 

5)12 . . 

2 . 


. 1 
. 3 
. 2 


2 . 


. 3 


23313 

2231 

23313 
131111 
102111 

102111 







121121303 * 
The student may verify that 121121303 (radix 5) = 533172 (radix 10). 

Example 3. Show how to weigh a weight of 315 lbs. : first, with a series 
of weights of 1 lb. , 2 lbs., 2 2 lbs., 2 3 lbs., &c, there being one of each kind ; 
second, with a series of weights of 1 lb., 3 lbs., 3 2 lbs., 3 3 lbs., &c, there being 
one only of each kind. 

First. Express 315 in the binary scale. We have 
315 = 100111011, 
315 = l+2 + 2 3 + 2 4 + 2 5 + 2 8 . 
Hence we must put in one of the scales of the balance the weights 1 lb., 2 lbs., 
2 3 lbs., 2 1 lbs., 2 5 lbs., and 2 8 lbs. 

Second. Express 315 in the ternary scale, using no digit greater than 
unity. We have 

315 = 110100. 

Hence over against the given weight we must put the weights 3 4 lbs. and 3 5 
lbs. ; and on the same side as the given weight the weight 3 2 lbs. 

§ 5.] If we specialise the proposition of § 2 by making 

r, = r 2 - . . . = r n , each = r say, we have the following : — Every proper 

fraction A/B can be expressed, and that in one way only, in the form — 

A Pi , p* , p a , , p n 



+ " 



B r r 2 r 



4 . 3 v n > 



where p u p 2 , . . ., p n are each<r, and F either is zero, or can be 
made as small as we please by sufficiently increasing n. 

If r be the radix of any particular scale of notation, the fraction 



r r» r n 



* The arrangement of the multiplication in Examples 1 and 2 is purposely 
varied, because, although it is of no consequence here, sometimes the one order 
is more convenient, sometimes the other. A similar variety is introduced in 
§ 6, Examples 1 and 2. 



IX IN VARIOUS SCALES 171 

is usually called a radix fraction. We may detach the coeffi- 
cients and place them in apposition, just as in the case of 
integers, a point being placed first to indicate fractionality. * 
Thus we may write 

A 

where p x in the first place after the radix point stands for p x /r, 
p 2 in the second place stands for p a /r*, and so' on. 

Since the digits p x p. z p 3 . . . p n are the integral part of the 
quotient obtained by dividing Ar n by B, the radix fraction can- 
not terminate unless Ar n is a multiple of B for some value of n. 
Hence, if we suppose A/B reduced to its lowest terms, so that A 
is prime to B, we see that the radix fraction cannot terminate 
unless the prime factors of B (see chap, iii., § 10) be powers of 
prime factors which occur in r. For example, since r=10 = 2x5, 
no vulgar fraction can reduce to a terminating decimal fraction 
unless its denominator be of the form 2 1 "5'\ 

In all cases, however, where the radix fraction does not 
terminate, its digits must repeat in a cycle of not more than 
B - 1 figures ; for in the course of the division no more than B - 1 
different remainders can occur (if we exclude 0), and as soon as 
one of the remainders recurs the figures in the quotient begin 
to recur. 

Example 1. To express 2/3 as a radix fraction in the scale of 10 to within 
l/100000th— 

2_ 200000 _ 66666+j 
3~3xl0 5 ~ L0 5 ' 

_ 6xl0 4 + 6xl0 3 + 6xl0 2 + 6xl0 + 6 2/3 

10 5 + 100000' 



6 6 6 6 6^ 
10 10 2 10 J 10 4 10 3 ' 



where F = 



2/3 ^ 1 

1000U0 < 100000' 



* Napier of Merchiston was apparently the first who used the modern form 
of the notation for decimal fractions. The idea of the regular progression of 
decimals is older. Stevin fully explains its advantages in his ArithmMique 
(1585); and germs of the idea may be traced much farther back. According 
to those best qualified to judge, Napier was the first who fully appreciated the 



172 EXAMPLES OF RADIX FRACTIONS CHAP. 

In other words, we have to the required degree of accuracy 

~= -66666. 
o 

It is ohvious from the repetition of the figures that if we take n 6's after the 

point we shall have the value of 2/3 correct within l/10"th of its value. 

Example 2. Let the fraction be 5/64. Since 64 = 2 s this fraction ought 
to be expressible as a terminating decimal. "We have in fact 
5 5000000 78125 





64 


64 x 10 s 
= •078125 


10 6 


5 














Example 3 


. To 


express 


2/3 as 


a radix fraction 


in 


the 


scale 


of 


2 


to 


within l/2 3 th. 


2 
3 : 


2x2 6 
~3x2 B_ 


128/3 
2 6 


42 + 2/3 
2 6 * 














Neglecting 


2 / 3 i • i, ■ 

-p". which is < ^ and 


expressing 42 in 


the scale of 2, 


we 


have 




2 
3 : 


101010 
2 6 


= -101010 (radix 2). 















§ 6.] When a fraction is given expressed as a radix fraction 
in any scale, and it is required to express it as a radix fraction 
in some other scale, the following process is convenient. 

Let <f> be the fraction expressed in the old scale, r the new 
radix, and suppose 

. .Pi .iV-Pa , 

^ r r 2 r 

then r <b = p , + -+^ + . . . 

= [\ + <£i say. 
Now $, is a proper fraction, hence p x is the integral part of r<f>. 

A • Pa 

Again ?•</>! =p a + — + . . . 

=jp a + & say. 
So that p a is the integral part of r$ u and so on. 

It is obvious that a vulgar fraction in any scale of notation 
must transform into a vulgar fraction in any other ; and we shall 

operational use of the decimal point ; and in his Constructio (written long 
before his death, although not published till 1619) it is frequently used. See 
Glaisher, Art. " Napier, " Encyclopaedia Britannka, 9th ed.; also Eae's recent 
translation of the Constructio, p. 89. 



ix EXAMPLES 173 

show in a later chapter (see Geometrical Progression) that every 
repeating radix fraction can be represented by a vulgar fraction. 
Hence it is clear that every fraction which is a terminating or a 
repeating radix fraction in any scale can be represented in any 
other scale by a radix fraction which either terminates or else 
repeats. It is not, however, true that a terminating radix fraction 
always transforms into a terminating radix fraction or a repeater 
into a repeater. Non-terminating non-rejieating radix fractions 
transform, of course, into non-terminating non-repeating radix 
fractions, otherwise we should have the absurdity that a vulgar 
fraction can be transformed into a non-terminating non-repeating 
radix fraction. 

It is obvious that all the rules for operating with decimal 
fractions apply to radix fractions generally. 

Example 1. Reduce 3*168 and 11 "346 to the scale of 7, and multiply the 
latter by the former in that scale ; the work to be accurate to l/1000th 
throughout. 

The required degree of accuracy involves the 5th place after the radical 
point in the scale of 7. 

•346 
7 



•168 

_7 
1)-176 

_7 
1)232 

_7 
l)-624 

_7 
4)-368 

_7 
2V576 



2)-422 

_7 
2) -954 

_7 
6)-678 

_7 
4)746 

_7 
5) -222 

3-168 = 3-11142 (radix 7). 11-346 = 14-22645. 



14-22645 
3-11142 

46-01601 

1-42265 

14227 

1423 

632 

32 

50-64146 



174 REMAINDER ON DIVIDING BYf-1 chap. 

On account of the duodecimal division of the English foot into 12 inches, 
the duodecimal scale is sometimes convenient in mensuration. 

Example 2. Find the number of square feet and inches in a rectangular 
carpet, whose dimensions are 21' 3|" by 13' llf". Expressing these lengths 
in feet and duodecimals of a foot, we have 

21' 3|" = 19-36. 
13' ll|"=ll'e9. 

If, following Oughtred's arrangement, we reverse the multiplier, and put the 
unit figure under the last decimal place which is to be regarded, the 
calculation runs thus — 

19-36 
9ell 



19360 

1936 

1763 

13e 



20978 

209 (radix 12) = 288 + 9 = 297 (radix 10) feet. 
•78 (radix 12) = 7 x 12 + 8 = 92 square inches. 
Hence the area is 297 feet 92 inches. 

§ 7.] If a number N be expressed in the scale of r, and if we 
divide N and the sum of its digits by r - 1 , or by any factor of r— 1 , 
the remainder is the same in both cases. 

Let N = p + p t r + p 2 r* + . . . + p n r n . 
Hence N - (p +i> x + • • - + Pn) = Pi{r - I) + p. 2 (r 2 - 1) + . . . 

+ Pn(r n ~l) (1). 

Now, m being an integer, r m - 1 is divisible by r — 1 (see 
chap, v., § 17). Hence every term on the right is divisible by 
r - 1, and therefore by any factor of r - 1. Hence, p being r — 1, 
or any factor of it, and /x some integer, we have 

N-0'o +;>, + . • -+pn) = W (2). 

Suppose now that the remainder, when N is divided by p, is <r, 
so that N = vp + <t. Then (2) gives 

p + p, + . . . + p n = (v - fl)p + o- (3), 

which shows that when p +Pi + • • . +p n is divided by p the 
remainder is <r. 



IX CASTING OUT THE NINES 175 

Cor 1. In the ordinary scale, if we divide any integer by 9 or by 
3, the remainder is the same as the remainder we obtain by dividing 
the sum of its digits by 9 or by 3. 

For example, 31692-^9 gives for remainder 3, and so does 
(3 + 1 + G + 9 + 2) -+- 9. 

Cor. 2. It also follows that the sum of the digits of every midtiple 
of 9 or 3 must be a multiple of 9 or 3. For example, 

2x9-18 1 + 8 = 9 ■ 

13 x 9-117 1 + 1 + 7 = 9 

128x9 = 1152 1 + 1+5 + 2 = 9 
128x3 = 384 3 + 8 + 4 = 15 = 5x3. 

§ 8.] On Cor. 1 of § 7 is founded the well-known method 
of checking arithmetical calculations called " casting out the 
nines." 

Let L = MN ; then, if L = 19 + L', M = m9 + M', N = nd + N', 
so that L', M', N' are the remainders when L, M, N are divided 
by 9, we have — 

19 + L' = (m9 + M') (nd + N'), 

= mra81 + (M'w + N'm)9 + M'N', 
= (mn9 + Wn + N'm)9 + M'N' ; 

whence it appears that L' and M'N' must have the same re- 
mainder when divided by 9. L', M', N' are obtained in accord- 
ance with Cor. 1 of § 7 by dividing the sums of the digits in the 
respective numbers by 9. 

Example 1. Suppose we wish to test the multiplication 
47923x568 = 27220264. 
To get the remainder when 47923 is divided by 9, proceed thus: 4 + 7 = 11, 
cast out 9 and 2 is left ; 2 + 9 = 11, cast out 9; 2 + 2 + 3 = 7. The remainder 
is 7. Similarly from 563 the remainder is 1, and from 27220264, 7. Now 
7x1 + 9 gives of course the same remainder as 7+^9. There is therefore a 
strong presumption that the above multiplication is correct. It should be 
observed, however, that there are errors which this test would not detect ; if 
we replaced the product by 27319624, for instance, the test would still be 
satisfied, but the result would be wrong. 

In applying this test to division, say to the case L/M = N + P/M, since 
we have L = MN + P, and therefore L- P = MN, we have to cast out the nines 
from L, P, M, and N, and so obtain L', P', M', and N' say. Then the test is 
that L' - P' shall be the same as the result of casting out the nines from M'N'. 



176 LAMBERT'S THEOREM CHAF. 

Example 2. Let us test — 

27220662+-568 = 47923 + 398 + 568, 
or 27220662 = 47923x568 + 398. 

Here L' - P' = 0- 2= -2, 

M'.N'=7x 1=9-2. 
The test is therefore satisfied. 

§ 9.]* The following is another interesting method for ex- 
panding any proper fraction A/B in a series of fractions with 
unit numerators : — 

Let (?!, q,, q a , . . ., q n , and r x , r 2 , ?- 3 , . . ., r n , be the quotients and 
remainders respectively when B is divided by A, r lt r 2 , . . ., r n _, re- 
spectively, then 

- = + . . . + v i — + F (1), 

B ft q,q 2 q,q^ 3 q,q 2 ...q n 

where F = ( - 1 ) n i " n JMi • • • o n ~B, that is, F is numerically less than 
1/qfa . . . q n . 

For we have by hypothesis 

B = Aq x + r u therefore A/B = l/g 1 - rjqfi (2), 

B = i\q 2 + r s , therefore rjB = l/q 2 - r 2 /q 2 B (3), 

B = r 2 q 3 + r 3 , therefore r„/B = l/q 3 - r 3 Jq 3 ~B (4), 

and so on. 

From (2), (3), (4), we have successively 
A.l J_ J_AY\ 
B ~?i Mi 2&W' 

= ±-± + -l LfcV 

2i Mi MA Mzl^W' 
and so on. 

Since r n r 2 , . . ., r n go on diminishing, it is obvious that, if 

A and B be integers as above supposed, the process of successive 

division must come to a stop, the last remainder being 0. Hence 

* In Lis Essai d' Analyse Numirique sur la Transformation dcs Fractions 
(CEuvres, t. vii. p. 313), on which the present chapter is founded, Lagrange 
attributes the theorem of § 9 to Lambert (1728-1777). Heis, Sammlung von 
Bcispielen und Au/gaber. aits dcr allgemeinen Arithmetik und Algebra (1882), p. 
322, has applied series of this character to express incommensurable numbers 
such as logarithms, square roots, &c. In the same connection see also Syl- 
vester, American Jour. Math., 1880. Sec also Cyp. Stephanos, Bull. Soc. 
Math. Fr. 7 (1879), p. 81 ; G. Cantor, Zeitsch. f. Math. 14 (1869), p. 124 ; 
J. Liiroth, Math. Ann. 21 (1883), p. 411. 



IX 



EXERCISES XIII 177 



every vulgar fraction can be converted into a terminating series 
of the form 

1 JL JL 



Example. 



113 1 1 1 

: + 



244 2 2.13 2.13.24 2.13.24.61 



From this resolution we conclude that 1/2-1/2.13 represents 113/244 within 
l/26th, and that 1/2-1/2.13 + 1/2.13.24 represents 113/244 within l/624th. 



Exercises XIII. 

(1.) Express 16935 (scale of 10) in the scale of 7. 
(2.) Express 16-935 (scale of 10) in the scale of 7. 
(3.) Express 315 "34 (scale of 10) in the scale of 11. 
(4.) Express r7e9ee (scale of 12) in the scale of 10. 
(5.) Express Ir8e54 (scale of 12) in the scale of 9. 
(6.) Express 345"361 (scale of 7) in the scale of 3. 
(7.) Express 112/315 (scale of 10) as a radix fraction in the scale of 6. 
(8.) Express 3169 in the form ^ + g3 + r3.5+s3.5.7 + &c, where ]i<3, 
2<5, r<7, &c. 

(9.) Express 7/11 in the form pl2 + q/2.S + r/2.ZA + kc., where p<2, 
q<3, r<4, &c. 

(10.) Express 113/304 in the form ^3 + ?/3.5 + r/3 2 .5 + s/3 2 .5 2 -M/3 3 .5 2 f &r., 
where p< 3, q<5, r<3, &c. 

(11.) Multiply 31263 by 56341 in the scale of 7. 

(12.) Find correct to 4 places 31 -3432 x 150323, both numbers being in 
the scale of 6. 

(13.) Find to 5 places 31 -3432-7-2 67312, both numbers being in the scale 
of 12. 

(14.) Extract the square root of 365738 (scale of 9) to 3 places. 
(15.) Express 887/1103 in the form l/qi - l/qiq» + l/qiq«q3 - &c- 
(16.) Show how to make up a weight of 35 lbs. by taking single weights 
of the series 1 lb., 2 lbs., 2 2 lbs., &c. 

(17.) With a set of weights of 1 lb., 5 lbs., 5 2 lbs., &c, how can 7 cwt. be 
weighed ? First, by putting weights in one scale only and using any number 
of equal weights not exceeding four. Second, by putting weights in either 
scale but not using more than two equal weights. 

(18.) Find the area of a rectangle 35 ft. 3* in. by 23 ft. 6| in. 
(19.) Find the area of a square whose side is 17 ft. 4 in. 
(20.) Find the volume of a cube whose edge is 3 ft. 9} in. 
(21.) Find the side of a square whose area is 139 sq. ft. 130 sq. in. 
(22.) Expressed in a certain scale of notation, 79 (scale of 10) becomes 142 ; 
find the radix of that scale. 

VOL. I N 



178 EXERCISES XIII 



CHAP. 



(23.) In what scale of notation does 301 represent a square integer ? 

(24.) A number of 3 digits in the scale of 7 lias its digits reversed when 
expressed in the scale of 9 ; find the digits. 

(25.) If 1 be added to the product of four consecutive integers the result 
is always a square integer ; and in four cases out of five the last digit (in the 
common scale) is 1, and in the remaining case 5. 

(26.) Any integer of four digits in the scale of 10 is divisible by 7, pro- 
vided its first and last digits be equal, and the hundreds digit twice the. tens 
digit. 

(27. ) If any integer be expressed in the scale of r, the difference between 
the sums of the integers in the odd and even places respectively gives the 
same remainder when divided by r + 1 as does the integer itself when so 
divided. Deduce a test of multiplication by "casting out the elevens." 

(28.) The difference of any two integers which are expressed in the scale 
of 10 by the same digits differently arranged is always divisible by 9. 

(29.) If a number expressed in the ordinary scale consist of an even 
number of digits so arranged that those equidistant from the beginning and 
end are equal, it is divisible by 11. 

(30.) Two integers expressed in the ordinary scale are such that one has 
zeros in all the odd places, the other zeros in all the even places, the remaining 
digits being the same in both, but not necessarily arranged in the same order. 
Show that the sum of the two integers is divisible by 11. 

(31.) The rule for identifying leap year is that the number formed by the 
two last digits of the year must be divisible by 4. Show that this is a 
general criterion for divisibility by 4, and state the corresponding criterion 
for divisibility by 2". 

(32.) If the last three digits of an integer be^o^'o, show that the integer 
will be exactly divisible by 8, provided p$ + 2_£>i + 4p-2 be exactly divisible by 8. 

(33.) Show that the sum of all the numbers which can be formed with the 
digits 3, 4, 5 is divisible by the sum of these digits, and generalise the theorem. 

(34.) Itp/n and (n-j^/n, p<n, be converted into circulating decimals, find 
the relation between the figures in their periods. 

(35.) If, in converting the proper fraction ajb into a decimal, a remainder 
equal to b-a occurs, show that half the circulating period has been found, 
and that the rest of it will be found by subtracting in order from 9 the digits 
already found. Generalise this theorem. 

(36.) In the scale of 11 every integer which is a perfect 5th power ends in 
one or other of the three digits 0, 1, t. 

(37.) In the scale cf 10 the dilference between the square of every number 
of two digits and the square of the number formed by reversing the digits, is 
divisible by 99. 

(38.) A number of six digits whose 1st and 4th, 2nd and 5th, 3rd and 6th 
digits are respectively the same is divisible by 7, by 11, and by 13. 

(39.) Show that the units digit of every integral cube is either the same 
as that of the cube root or else is the complementary digit. (By the comple- 
mentary digit to 3 is meant 10 - 3, that is, 7.) 

(40.) If in the scale of 12 a square integer (not a multiple of 12) ends 



ix EXERCISES XIII 179 

with 0, the preceding digit is 3, and the cube of the square root ends with 
60. 

(41.) If a be such that a m + a — r, then any number is divisible by a m , 
provided the first m integers po, pi, ■ ■ ■ , p m -i of its expression in the scale of 
r are such that^ +2'i^+ • • • +Pm-i^ m ~ 1 is divisible by a" 1 . 

(42.) The digits of a are added, the digits of this sum added, and so on, 
till a single digit is arrived at. This last is denoted by <p(u). Show that 
<p(a + b) = <f> {</){a) + </>{b)} ; and that the values of <p(8n) for ft = 1, 2, . . ., a, 
successively consist of the nine digits continually repeated in descending 
order. 

(43.) A number of 3 digits is doubled by reversing its digits : show that 
the same holds for the number formed by the first and last digit, and that 
such a number can be found in only one scale out of three. 



CHAPTEK X. 
Irrational Functions. 

GENERALISATION OF THE CONCEPTION OF AN INDEX. 
INTERPRETATION OF z\ X^l, X~ m . 

§ 1.] The definition of an index given in chap, ii., § 1, be- 
comes meaningless if the index be other than a positive integer. 

In accordance with the generalising spirit of algebra we 
agree, however, that the use of indices shall not be restricted to 
this particular case. We agree, in fact, that no restriction is to 
be put upon the value of the index, and lay down merely that 
the use of the indices shall in every case be subject to the laws 
already derived for positive integral indices. Less than this we 
cannot do, since these laws were derived from the fundamental 
laws of algebra themselves, to which every algebraical symbol 
must be subject. 

The question now arises, What signification shall we attri- 
bute to x m in these new cases ? We are not at liberty to proceed 
arbitrarily, and give any meaning we please, for we have already 
by implication defined x m , inasmuch as it has been made subject 
to the general laws laid down for indices. 

§ 2 ] Case of x p! ? where p and q are any positive integers. Let 
z denote the value of x p to, whatever it may be ; then, since x^i is 
to be subject to the first law of indices, we must have — 
zV = zxzxzx . . . a factors, 

= xvli x xrti x xPlv x . . . q factors, 

- 3;P/?+P/?+.P/9+ • • • 1 terms, 
= XP. 



chap, x INTERPRET ATION OF X p,q 181 

Iii other words, z is such that its qt\\ power is x* } that is, z 
is what is called a qth root of xP, which is usually denoted 
by tfxP. 

Hence x.p'9 = *J&. 

In particular, if p - 1, 

We have now to consider how far an algebraical value of 
a 5th root of every algebraical quantity can be found. 

In the case of a real positive quantity k, since zi passes con- 
tinuously* through all positive values between and + x> as z 
passes through all positive values between and + 00 , it is clear 
that, for some value of z between and + 00 , we must have 
z? - k. In other words, there exists a real positive value of %Jk. 

Unless the contrary is stated we shall, when k is positive, 
take k 1! i as standing for this real positive value. 

The student should, however, remark that when q is even, 
= 2r say, there is at least one other real value of 0/k ; for, since 
( - z) 2r = z 2r , if we have found a positive value of z such that 
z 2r = k, that value with its sign changed will also satisfy the re- 
quirements of the problem. 

Next let k be a negative quantity. If q be odd, then, since 
z? passes through all values from - 00 to as z passes through 
all values from - 00 to 0, there must be some one real negative 
value of z, such that & = k. In other words, if q be odd, there 
is a real negative value of {/k. 

If q be even, then, since every even power of a real quantity 
(no matter Avhether + or - ) is positive, there is no real value 
of z. Hence, if k be negative and q even, %/k is imaginary. This 
case must be left for future discussion. 

It will be useful, however, for the student to know that 
ultimately it will be proved that */k has in every case q different 
values, expressions for which, in the form of complex numbers, 
can be found. Of these values one, or at most two, may be real, 
as indicated above (see chap, xii.) 

* For a fuller discussion of the point here involved see chaps, xv. and xxv. 



182 VERIFICATION OF THE LAWS FOR X rlq chap. 

Only iii the case where h is the pth power of a rational 
quantity can %/k be rational. 

Example. 

ltk=+h&>, 

2 £>/k has two real values, +h and -h. 

If k=+h*P+\ 

ip t}/k lias one real value, +h. 

HJc= -J&+ 1 , 

-v+Vk has one real value, - h. 

In all that follows in this chapter, we shall restrict the radicand, 
I; to be positive ; we shall regard only the real positive value of the qth 
root of k ; and this (ivhich is called the PRINCIPAL value of the 
root) is what ice understand to be the meaning of £ 1/ ? 

The theory of fractional indices could (as in the first edition of this volume) 
he extended so as to cover the case of a negative radicand, hut only so far as the 
order of the root is odd. The practical advantage gained by this extension is 
not worth the trouble which it causes by complicating the demonstrations. 
We think it better also, from a scientific and educational, as well as from a 
practical point of view, to consider the radication of negative radicands as a 
particular case of the radication of complex radicands (see chap, xii., § 19). 

§ 3.] We have now to show that the meaning just suggested 
for x p l q is consistent with all the Laws of Indices laid down in 
chap. ii. The simplest way of doing this is to re-prove these 
laws for the newly denned symbol x^i. 

We remark in the first place that it is necessary to prove 
only I. (a), II, and III. (a) ; because, as has been shown in chap. 
ii., I. (/8) can be deduced from I. (a), and III. (J3) from III. (a), 
without any appeal to the definition of x m . 

To prove I. (a), consider x pl ? and x r/s , where p, q, r, s are 
positive integers, and let 

z = xPb x r ' s . 

Then, since x? 1 * and x r ' s are, by hypothesis, each real and 
positive, z is also real and positive. Also 

z<i» = (xPlq x rl *)i s , 



X VERIFICATION OF THE LAWS FOR X p,q 183 

all by the laws for positive integral indices, regarding which 
there is no question. 

Now, by the meanings assigned to aP^ and x rls , we have 
(a^/«)9 = %* and (x rls ) a = x r . Hence 

= xP s xP, 
= K*"+« r ) 

hy the laws for positive integral indices. 

It now follows that is the qsth root of a^ JS +?'' ; and, since z 
is real and positive, it must be that qsth root which we denote 
by ofp*+9r)lv. Therefore 

z = z(2«+ <?'•)/</■', 

that is to say, 

z = ?pl<i+ r l s . 

The proof is easily extended to any number of factors. 

To prove Law II., consider (x^) r ' s , where p, q, r, s are posi- 
tive integers, 

and let z = (x^) r ' s . 

Then, since, by hypothesis, x*ti is real and positive, therefore 
(xPtey 1 *, that is z, is real and positive. Also 
z v = [(xPlqyisy^ 

= [{(ajP/s)*"/*} 8 ]?, 

by laws for positive integral indices ; 
= [(xPte) r ]i, 

by definition of a fractional index ; 
= (xvl*)9 r , 

by laws of positive integral indices ; 

by definition of a fractional index ; 
= xv, 

by laws of positive integral indices. 
Hence z is a qsth. root of »*"", and, since z is real and positive, we 
must have 

Z = o-Vrlqs, 

that is, z = a^/9)( r /»). 



184 PARADOXES chap. 

Lastly, to prove Law III. (a), let 

Then, since, by hypothesis, x^i and yvli are each real and 
positive, z is real and positive. Also 

z q = ( x plq yplqy^ 

by laws for positive integral indices ; 
= xPf, 

by definition for a fractional index ; 

by laws for positive integral indices. 

Hence z is a qi\\ root of (%y) p ; and, since z is real and positive, 

we must have 

a = (.vy)pli. 

The proof is obviously applicable where there is any number of 
factors, x, y, . . . 

§ 4.] Although it is not logically necessary to give separate 
proofs of Laws I. (/3) and III. (/?), the reader should as an 
exercise construct independent proofs of these laws for himself. 

It should be noticed that in last paragraph we have supposed 
both the indices pjq and r/s to be fractions. The case where 
either is an integer is met by supposing either q — 1 or s = 1 ; the 
only effect on the above demonstrations is to simplify some of 
the steps. 

§ 5.] Before passing on to another case it may be well to 
call attention to paradoxes that arise if the strict limitation as to 
sign of xPfo be departed from. 

By the interpretation of a fractional index 

x*l 2 = Z/x* = ± x\ 

But x^ = x\ 

which is right if we take x -4/2 to stand for the positive value of 
A 2 /V ; but leads to the paradox x* = - x~ if we admit the negative 
value. 

A similar difficulty would arise in the application of the law, 
(x m ) n = x mn = (.'■")"' ; 






x INTERPRETATION OF X° 185 

for example, (4*) 9 = (4*)' 

would lead to ( ± 2f = ± 4, 

that is, 4 = ± 4, 

if both values were admitted. Such difficulties are always apt 
to arise 'with x^i where the fraction pfq is not at its lowest terms. 
The true way out of all such difficulties is to define and 
discuss x 11 as a continuously varying function of n, which is called 
the exponential function. In the meantime fractional indices are 
introduced merely as a convenient notation in dealing with 
quantities which are (either in form or in essence) irrational ; 
and for such purposes the limited view we have given will be 
sufficient. 

§ 6.] Case of x°. This case arises naturally as the extreme 
case of Law I. (/3), when n = m; for, if we are to maintain that 
law intact, we must have, provided x 4= 0,* 

that is, X° = 1. 

This interpretation is clearly consistent with Law I. (a), for 

x m n° = x m +° 
simply means 

x m x 1 = x m , 

which is true, whatever the interpretation of x m may be. 

Again, a;'" = (x ,rt )°, 
that is x° = (x m )°, 

simply means 1 = 1 by our interpretation ; 

and x m0 = (x ) m , 
or x° = (x°) m , 

gives 1 = 1™ 

which is right, even if m be a positive fraction, provided we 
adopt the properly restricted interpretation of a fractional index 
given above. The interpretation is therefore consistent with II. 
The interpretation a; = 1 is also consistent with III. (a), for 

a?>f = (xyf 
simply means 1x1 = 1, 



* This provision is important since the form 0° is indeterminate (see chap, 
xxv.) 



186 NEGATIVE INDICES chap. 

§ 7.] Case of x~ m , where m is any real positive (or signless) 
number, and x 4= 0. 

Let z = x~ m , then, since x m ^0, Ave have 
z = x~ m x Z m -7-X m , 

if Law I. (u) is to hold for negative indices. Whence 

,0 /-•>»>' 



z = ar/x 



/a*"» 



by last paragraph. In other words, x~ m is the reciprocal of x m . 

As an example of the reconciliation of this with the other 
laws, let us prove I. (a), say that 



By 


our 


definition, 

X 


we 

~ m x 


have 

- n = (\fo: m )(ljx n ), 
= l/x m x n , 

s= l/x m + n , 

the last step by 


the laws 


already 










established for all positive 


indices ; 
























by definition of a 


negative index. 


Hei 


ice 


X' 


■m x 


-n _ r* - vi - n 







In like manner we could show that 

rwm v - n _ g.m - n 

The verification of the other laws may be left as an exercise. 

§ 8.] The student should render himself familiar with the 
expression of the results of the laws of indices in the equivalent 
forms with radicals ; and should also, as an exercise, work out 
demonstrations of these results without using fractional indices 
at all. 

For example, he should prove directly that 

Vx*/z= P !/xP+* (1); 

V{ V xP Y = V xPr = Vi V&Y ( 2 ) i 

yx&yyz=y(xyz) (3); 

y x mj y y m = "/(^jy)™ (4). 



EXAMPLES 187 



EXAMPLES OF OPERATION WITH IRRATIONAL FORMS. 

§ 9.] Beyond the interpretations x pl ^, x°, z~ m , the student has 
nothing new to learn, so far as mere manipulation is concerned, 
regarding fractional indices and irrational expressions in general. 
Still some practice will he found necessary to acquire the requisite 
facility. "We therefore work out a few examples of the more 
commonly occurring transformations. In some cases we quote 
at each step the laws of algebra which are appealed to ; in others 
we leave it as an exercise for the student to supply the omission 
of such references. 

Example 1. 

To express A v B in the form V P. 

A v / B = AB 1 /™ = (A m ) , /"'B 1/m , by law of indices II., 
= (A"'B) 1; »', by law of indices III. (a), 
= X /(A"'B). 
Example 2. 

'v/a=7a- 

for v^A = A 1/m = AP /m P, 

= m VAP. 
Example 3. 

sJxP m +<l = x(P m +tf'' m 
= x p+q!m i 

= xPx xn' m , by law of indices I. (a), 

m / 

Example 4. 

To express \/x»\ y/y as the root of a rational function of x and y. 

s/xp\ \/y r = xP ! v/y r !» = xP^v/yW'Q', 
= {xP'^lvftyvryiV, 

= {xP'/yiry/q^ 

= V{xP'ly r )- 
Example 5. 

V32=V(16x2), 
= Vl6xV2, 

= 4x V2. 



188 


EXAMPLES 


Example 6. 


2 x V2 x v"2 x \A 




= 2x2"x2 } x2', 

= 22, 
= 4. 



CHAP 



Example 7. 

= (« 2 )'"/ 2 »(l-a-2/a2)m/2» ) 

= a mln (l-x 2 /a 2 ) m ' 2n . 
Example 8. 

\/{yx + x") x ^(yz + ex) 

= V Hy + *)} x V {2(2/ + a-)} , 

= V-Z X V(y + *) X \/z X \/(2/ +^); 

= V(^)x{V(2/ + ^)} 2 , 

=(y+«)x V(a*)- 

Example 9. 

V240 + V40 

= V(16x3x5) + V(4x2x5), 
= V16V3V5+ VW2V5, 
= V5(4V3 + 2V2). 
Example 10. 

(V3 + 2V2 + 3V6)(V3-2 N /2 + 3V6) 

= (V3 + 3V6) 2 -(2V2) 2 , 
= (V3) 2 + 6V3V6 + (3V6) 2 -(2V2) 2 . 
= 3 + 6V(3x6) + 3 2 x6-2 2 x2, 
= 49 + 6\/18, 
= 49 + 18V2. 
Example 11. 

{V(l-a-) + V(l+.T)} 4 

= {(l-x) i + (l+x) i }*, 
= (l-xf + {l+ x ) n - 

+ i(l-x)*(l+x) i +4(l-x) i (l+x)i 
+ 6(l-x)(l+x), 

= 8 - 4x 2 + 4(1 -x) h (l + x)\l - x+l+x) 

= 8-4.t 2 +8V(l-tf 2 ). 
Example 12. 

V \x-y) V \x+y) 

_W(x + y)\ 2 +{s/(x-y)} 2 
\/{{x-y)(x + y)} 






RATIONALISING FACTORS 189 

_x+y+x~y 
2a; 



Example 13. 



V(* 2 ~2/ 2 )' 



(a^-a^+a: *- a; - *) x (;*:*+ 1+a: *) 
=x*-7? +a:*-aT* 

+ ar - a; 1 + x * - x * 

Tit "tC V X «C y 



5 1 _ 1 _ 5 

sx'+aT-a; -a; * 



Example 14. 
Show that 



We have 

g2 = a+s/W-b) + a-yW-5) + 2 /f {a + V(a 2 - 6)} (a - V(a 2 - 5)} ~| 

= a + V[« 2 -{V(« 2 -&)}'-'], 
= re + \/6. 

Hence, extracting the square root, we have 

S=V(a+V & )- 



RATIONALISING FACTORS. 

§ 10.] Given certain irrationals, say Jp, *Jq, y/r, we may 
consider rational, and it may be also integral, functions of these. 
For example, I *Jp + m \/q + n yV, and /( *Jp) s + m \/{pq) + n{ s/qf y 
are integral functions of Jp, *Jq, Jr, of the 1st and 2nd 
degrees respectively, provided /, m, n do not contain Jp, Jq, Jr. 
Again, (I Jp + m Jq)/(l \/g + m Jr) is a rational, but not integral, 
function of these irrationals. J (I \Jp + in \/q), on the other 
hand, is an irrational function of Jp and Jq. 

The same ideas may also be applied to higher irrationals, 
such us p llm , q lln , &c. 

§ 11.] Confining onrselves for the present to quadratic 
irrationals, we shall show that every rational function of a 
given set of quadratic irrationals, Jp, Jq, ^/r, &c, can be 



190 RATIONALISING FACTORS chap. 

reduced to a linear integral function of the square roots of p, q, r, 
and of their products, pq, pr, gr, pgr, &c. 

This reduction is effected mainly by means of rationalising 
factors, whose nature and use we proceed to explain. 

If P be any integral function of certain given irrationals, and Q 
another integral function of the same, such that the product QP is 
rational so far as the given irrationals are concerned, then Q is called 
a rationalising factor ofP with respect to the given irrationals. 

It is, of course, obvious that, if one rationalising factor, Q, 
has been obtained, we may obtain as many others as we please 
by multiplying Q by any rational factor. 

§ 12.] Case of Monomials. 

1°. Suppose we have only quadratic irrational forms to deal 
with, say two such, namely, pi and qK 

Then the most general monomial integral function of these 
is 

i = A(pi) 2m +\r) 2n+ \ 

where A is rational. There is no need to consider even indices, 
since (jpi) Zm =p m is rational. 

Now I reduces to 

I = (Ap'YOiM 
where the part within brackets is rational. 

Hence a rationalising factor is jj'-qK for we have 

Ipigi = (Aj> m q n )pq, 

which is rational. 

Example. A rationalising factor of 16 . 2 ? . 3* . 5* is 2*3-5', that is, (30) . 

2°. Suppose we have the irrationals p v *, q l!t , r^ u , say, and 

consider 

I = ApV s q m ^ ?'"/" * 

which is the most general monomial integral function of these. 
A rationalising factor clearly is 

„ .1 - l/s f ,\ - in It r l - n/u 

p q ' i 
or 4* - > (ft - "0. 't >•(« - »)/« 



* Where of course l<s, m<t, «<«, for if they were not they could be 
reduced by a preliminary process like that in case 1°. 



x RATIONALISING FACTORS 191 

Example. 

1=81. 8*. 6*. 7*, 
=31.3 1+ *.5*.7 i+i , 
=(31.3.7 4 ).3*.5 f .7*. 

5 2 1 

A rationalising factor is 3 . 5 . 7 . 

§ 13.] Case of Binomials. 

1°. The most general form when only quadratic irrationals 
are concerned is a \/p + b \/q, where a and b are rational ; for, if 
we suppose p a complete square, this reduces to the more special 
form A + B \Aj, where A and B are rational. 

A rationalising factor clearly is a s/p - b \/q. For, if 

I = a */p + b x /q, 

I(a \ > - b y/q) = (a ^p) 2 - (b s /q)\ 

= ap - b'q, 
which is rational 

The two forms a \/p + b \/q and a \/p - b *Jq are said to be con- 
jugate to each other with reference to y/q, and we see that any binomial 
integral function of quadratic irrationals is rationalised by multiplying 
it by its conjugate. 

2°. Let us consider the forms ap*l* ± bqW, to which binomial 
integral functions of given irrationals can always be reduced.* Let 

x = ap aly , y = bqffl, 
I = ap*b - hf r \ 
= « - y- 

Let m be the L.C.M. of the two integers, y, 8. Now, using 
the formula established in chap, iv., § 16, we have 

(z m - 1 + x m ~ 2 y+. . . + xy m ~ 2 + y m " 1V I = x m - y m . 

Here x m - y>" = (a'" p mx ^ - b"< tf&P), where ma/y and m/3/8 are 
integers, since m is divisible by both y and S, that is, x™ - y'" is 
rational. 

A rationalising factor is therefore x 1 " ~ 1 + x m ~ -y + . . . 
+ xy m ~ 2 + y m ~ l , in which x is to be replaced by ap , and y by b<f' 

* Tartaglia's problem. See Cossali Storia dell' Abjebra (1797), vol. ii. p. 266. 



192 RATIONALISING FACTORS chap. 

The form ap" ly + bq m may be treated in like manner by 
means of formulas (4) or (5) of chap, iv., § 16. 

Example. 

1 = 3.2*- 4.3*. 

Here m = 6, z=3.2*, 27 = 4.3*; 

and a rationalising factor is 

a 5 + xHj + x*if- + xhf + xy i + rf 

= 3 5 .2 S + 3 l .4.2 l .3* + 3 3 .4 2 .2.3* + 3 2 .4 3 .2 § .3* + 3.4 4 .2*.3 § + 4 s .3 i , 
= 3 5 .2.2 3 + 3 4 .8.2*.3* + 3 3 .32.3* + 3 2 .4 3 .2 S .3* + 3.4 4 .2*.3 § + 4 5 .3 f . 

§ 14.] Trinomials with Quadratic Irrationals. This case is 
somewhat more complicated. Let 

I = \/p + \/q+ Jr ; * 
and let us first attempt to get rid of the irrational Jr. This 
Ave can do by multiplying by the conjugate of \/p + sfq + s/r 
with respect to *Jr, namely, sip + \'q - sir. We then have 

( Jp + Vg - v/r)I = ( Jp + JqY - ( vV) 2 , 

= p + q-r+2j(pq) (1). 

To get rid of nj(pq) we must multiply by the conjugate of 
p + q - r + 2 s/(p<j) with respect to \/(pq). Thus finally 
{^ + 2-r-2 N \pq)}{ s/p + Jq ~ Jr)l = (p + q - r) 2 - {2 v /( pq)}\ 

=f + ? 2 + r 2 - 2j?2 - 2p- - 2jr. 

Hence a rationalising factor of I is 

{p + q-r- 2 \/(j>g)}( «/p + slq - Jr), 
or 
(Jp - Jq + */r)(vi> - s'q- Jr)(Jp + v'2 - v'r) (2). 

Ey considering attentively the factor (2) the student will see 
that its constituent factors arc obtained by taking every possible 
arrangement of the signs + and - in 

+ *Jp ± \fq± \ V, 

except the arrangement + + + , which occurs in the given trinomial. 

* This is really the most general form, for a\/]) + b\Jq + c\/r may be 
written V(^) + V(& 8 2) + vW- 



x REDUCTION TO LINEAR FORM 193 

Example 1. A rationalising factor of 

\/2 - V3 + V5 
is (V2 - \ft - V») (V2 + V3 + V5) ( v'2 + \/3- yf5). 

Example 2. A rationalising factor of 

1+2V3-3V2 
is (1 + 2V3+3V2)(1-2V3 + 3V2)(1-2V3-3V2). 

In actual practice it is often more convenient to work out 
the rationalisation by successive steps, instead of using at once 
the factor as given by the rule. But the rule is important, 
because it is general, and will furnish a rationalising factor for a 
sum of any number of quadratic irrationals. 

Example 3. A rationalising factor of 

1 + V2-V3 + V4 

is (1 + V2- V3- \/4)(l+V2+V3 + V4)(l+ V2 + V3- V*) 

x (1 - V2 - \/3 + \/4) (1 - V2 - V3 - V4) (1 - V2+ V3 + V4) 
x (1 - sJ2 + V3 - V4). 

Before giving a formal proof of the general truth of this 
rule, it will be convenient to enunciate one or two general pro- 
positions which are of considerable importance, both for future 
application and for making clear the general character of the 
operations which we are now discussing. 

§ 15.] Ever)/ integral function of a series of square roots, 

\/p, -Jq, \/r, &c, can be expressed as the sum of a rational term 

and rational multiples of \tp, \/q, \/r, &c, and of their products 

*J(P<l), J(pr)> v^Fi''). & i* 

First, let there be only one square root, say \/p, and consider 

any rational integral function of \jp, say <£( \''p). Every term 

of even degree in *Jp will be rational, and every term of odd 

degree, such as A( v//>) 2w+1 may be reduced to (Kp m ) \/p, that 

is, will be a rational multiple of s/p. Hence, collecting all the 

even terms together, and all the odd terms together, we have 

</>(^) = P + Qv> (l), 

where P and Q are rational. 

* Such a sum is called a " Linear Form." 
VOL. I 



194 RATIONALISATION OF ANY INTEGRAL FUNCTION chap. 

Next, suppose the function to contain two square roots, say 
4>( JPi *Jq)' First of all, proceeding as before, and attending 
to Jp alone, we get 

«/>( <Sp, \ /( L) = P + Qv / P, 

where P and Q are rational so far as p is concerned, but are 

irrational as regards q, being each rational integral functions of 

\/q. Reducing now each of these with reference to Jq we shall 

obtain, as in (1), 

P = F + QV2, Q = P" + Q"vA;, 
and, finally, 

<K Jp, Jq) = F + QVg + (P" + Q" Jq) y/p, 

= F + P" s/p + Q' Jq + Q" J(pq) (2), 
which proves the proposition for two irrationals. 

If there be three, we have now to treat P', P", Q', Q" by means 
of (1), and we shall evidently thereby arrive at the form 
A+B Jp + C Jq+T> vV+E^r) +F J(rp)+G J(pq) + li y/{pgr), 
and so on. 

Cor. It follows at once from the process by which we arrived 
at (1) that 

4>(- Jp) = P - Q yip. 

Hence if <f>( Jp) he any integral function of sip, <£( - Jp) is a 
rationalising factor of <£( \/p) ; and, more generally, if <f>( sjp, Jq, 
s/r, . . .) be an integral function of \/p, Jq, Jr, . . ., then, if we 
take any one of them, say Jq, and change its sign, the product 
</>( Jp> Jq> J r > • • •) x <K J Pi ~ J°> J r > • • •) ?s rational, so far 
as Jq is concerned. 

Example 1. If <p(.r)=:x 3 + .r 2 + x+l, find the values of 0(l + \/3) and 
0(1 - V3) and 0(1 + \J3) x 0(1 - \JZ). 

0(1 + V3) = (l + V3) 3 + (l + V3) 2 + (1 + V3)+ 1, 
= l + 3\/3 + 3.3 + 3\/3 

+1+2V3+3 

+ 1 + V3 
+ 1, 
= 16 + 9V3 

0(1 - \/3) is deduced by writing - \JZ in place of + \/3 everywhere in the 
above calculation. Hence 

0(1-V3) = 16-9V3; 

0(1 + V3) x 0(1 - V3) = (16) 2 - (9 V3) 2 , 
= 256-243, 
= 13. 



X LINEAR FUNCTION OF SQUARE ROOTS RATIONALISED 195 

Example 2. Find the value of a?+y i -\-7? — xyz i when x=sjq- \/r, 
y=sjr- \/P> z = s/p ~ \'<1- 

Since x + y + z=\/q- \Jr+\Jr- \/p+s/p-\/q 

= 0, 
we have (chap. iv. , § 25, IX.) 

Sb* - 3xyz = 2a(Za: 2 - 2xy), 
= 0. 
Therefore Zx 3 - xyz = 2xyz, 

= 2( V? - \Jr) ( V ~ Vi>) ( Vl> - VsOi 
= 2(2 - r)s/p + 2(r -^) >/? + %(p - ?)V r - 

Example 3. Evaluate (l+y+is)(l+2+a:)(l+as+y) when a:= V 2 > 2/ = V 3 > 
z=V5- 

(l+y+8)(l + «+a;)(l+a!+y) 

= 1 + 2(se + y + 2) + x 2 + (y + z)x + yz + &c. + &c. 
+ x{y- + z 2 ) + &c. + &c. + 2xyz, 
= 1 + x- + y 2 + z 2 + (2 + y 2 + z 2 )x + (2 + z 2 + x 2 )y 

+ (2 + a; 2 + y 2 )z + %yz + 3zx + Bxy + 2xyz, 
= 11 + 10 V2 + 9 V3 + 7 V5 + 3V!5 + 3\/10 + 3V6 + 2V30. 

§ 16.] We can now prove very easily the general proposition 
indicated above in § 14. 

If P be the sum of any number of square roots, say s/p, \/q, 
sfr, . . ., a rationalising factor Q is obtained for P by multiplying 
together all the different factors that can be obtained from P as 
follows : — Keep the sign of the first term unchanged, and tale every 
possible arrangement of sign for the following terms, except that which 
occurs in P itself. 

For the factors in the product Q x P contain every possible 

arrangement of the signs of all but the first term. Hence along 

with the + sign before any term, say that containing sjq, there 

will occur every possible variety of arrangement of all the other 

variable signs ; and the same is true for the - sign before \/q. 

Hence, if we denote the product of all the factors containing 

+ s!q by 4>( \/q), the product of all those factors that contain 

- s/q will differ from <£( \fa) only in having - \/q in place of 

+ s/q, that is, may be denoted by <£( - s/q). Hence we may 

write Q x P = </)( s/q) x <£( - s/q), which, by § 15, Cor. 1, is rational 

so far as v^ is concerned. The like may of course be proved for 

every one of the irrationals \/q, \ f r, . . . Also, for every factor 

in Q x P of the form sip + k there is evidently another of the form 



196 RATIONAL FUNCTION REDUCED TO LINEAR FORM chap. 

sip -k; so that Q x P is rational as regards Jp. Hence Q x P 
is entirely rational, as was to be shown. 

§ 17.] Every rational function, whether integral or not, of any 
number of square roots, s/p, sjq, s/r, . . ., can be expressed as the sum 
of a rational part and rational midtiples of \/p>, *Jq, Jr, &c, and of 
their products \/(pq), y/(pr), \/(qr), J(pqr), dr.* 

For every rational function is the quotient of two rational 
integral functions, say E/P. Let Q be a rationalising factor of 
P (which we have seen how to find), then 

R RQ 

P = PQ' 

But PQ is now rational, and RQ is a rational integral function of 
sip, sjq, \/r, . . ., and can therefore be expressed in the required 
form. Hence the proposition is established* 

Example 1. To express 1/(1 + \J2 + \J3) as a sum of rational multiples of 
square roots. Rationalising the denominator we obtain by successive steps, 

1 1 + V2j!_\/3 

1 + V2 + V3 (1 + \/2) 2 - (V3) 2 ' 

_ i+V2-ya 

2V2 

_ V2(l + y2-V3 ) 

2x2 
= 4 l (V2 + 2-V6), 

Example 2. Evaluate {x 2 -x+1 )/(« 2 + x + 1 ), where x = y3 + \/5. 
x--x + l _ 9 + 2V15- V3- V 5 
x 2 + x + 1 ~ 9 + 2 V15 + \J3 + v'5' 

_ (9 + 2Vl5) 2 - 2(9 + 2V15)(V3 + V5) + (V3 + V5) 2 
(9 + 2Vl5) 2 -(V3 + V5) a 
149 - 38 V3 - 30\/5 + 38 y/15 

133 + 34V15 
(149-38V3-30V5 + 88y i5)(133-34yi5) 
133 2 -34-xl5 ' 

_ + 437 + 46 y3 - 114y5 - 12yi5 
849 

* Besides its theoretical interest, the process of reducing a rational func- 
tion of quadratic irrationals to a linear function of such irrationals is important 
from an arithmetical point of view ; inasmuch as the linear function is in 
general the most convenient form for calculation. Thus, if it be required to 
calculate the value of l/(l + y2 + y3) to six places of decimals, it will be 
found more convenient to deal with the equivalent form ^ + jy2 - \\J6. 



THEORY FOR IRRATIONALS OF ANY ORDER 197 



GENERALISATION OF THE FOREGOING THEORY. 

§ 18.] It may be of use to the student who has already 
made some progress in algebra to sketch here a generalisation of 
the theory of §§ 13-17. It is contained in the following pro- 
positions : — 

I. Every integral function of p 1/n can he reduced to the form 
X + A 1 p l!n + A 2 p 2 l' l + . . . + A n . l p^-^ n , where A u , A,, . . ., 
A n _, are rational, so far as j)^ ,n is concerned. 

After what has been done this is obvious. 

II. Every integral function of p 111 , q 1,m , r lln , &c, can he ex- 
pressed as a linear function of p 1 ' 1 , p 2/l , . . ., jfl-W • q 1 *' 1 ", q- /m , . . ., 
£m-i)lm ; yl/w ,.2/n _ _ ^ ,<n-i)/w ; £ c ^ am i f t j ie products of these 

quantities, two, three, &c, at a time, namely, p 111 q 1/m , p 2 ' 1 q llm , &c, 
the coefficients of the linear function heing rational, so far as p 111 , q llm , 
r l!n , &c, are concerned. 

Proved (as in § 15 above) by successive applications of I. 

III. A rationalising factor of A + A,^ 1/rt + A. 2 p 2ln + . . . 
+ A n _ jj/'i-i)/" am always be found. 

We shall prove this for the case n = 3, but it will be seen 
that the process is general. 

Let P = A + A^ + A 2 p% ( 1 ), 

then p$P =pA 2 + A p^ + A^ (2), 

and p%P =joAi + pA. 2 p* + A p ?s (3). 

Let us now put x for jp«, and y for p%, on the right-hand sides of 
(1), (2), and (3) ; we may then write them 

(A o -P) + A,*; + A 2 y = (1'), 

(M 2 -/P) + A^ + A 1 ^0 (2'), 

(M. -i»*P) +M^ + A y = (3'), 

whence, eliminating x and y, we must have (see chap, xvi., § 8) 

(A - P) (A 2 -pA.A.,) + (pA. 2 -pi?) (pAf - A A,) 

+ ( P A l -piF)(A°-A u A 2 ) = (4). 



198 



THEORY FOR IRRATIONALS OF ANY ORDER 



C1IAV. 



Whence 

{(A 2 -^AA) + (pV - A A t )/ + (A x 2 - AA^JP 

= A (A 2 -j?AA) +pA 2 (pA 2 2 - A A,) + Mi(A, 2 - A A 2 ) 
= A 3 + Mi 3 + /A/ - 3MAA,, (5). 

Hence a. rationalising factor of P is 

(A 2 -MA) + (M/ - AA>* + (A, 2 - A A. 2 )/ (6), 

and the rationalised result is 

A 3 + M. 3 + p 2 A. 2 3 - 3;>Ao A, A 2 (7). 

The reader -who is familiar with the elements of the theory 
of determinants will see from the way we have obtained them 
that (6) and (7) are the expansions of 



and 



1 


A, 


A 2 


1 


A 


A, 


2 


M« 


A 


A 


A, 


A 2 


pK 


A 


A, 


Mi 


pA, 


A 



(6'), 



(n 



and will have no difficulty in writing down the rationalising 
factor and the result of rationalisation in the general case. 

IV. A rationalising factor can be found for any rational integral 
function of p 1/l , q llm , r lln , . . . , &c, by first rationalising with 
respect to p 111 , then rationalising the result with respect to q l,m , 
and so on. 

V. Every rational function of p 1/l , q Vm t r^*, whether integral or 
not, can be expressed as a linear function of p 1!l , pr 11 , • • • , p^ 1 ' 1 ^ 1 ; 
q Vm ,q 2lm , . . ., (fi™-**!™, &C ; and of the products p*' 1 qt /n r* lm . . ., 
the coefficients of the function being rational, so far as p 1!l , q 1!m , r 1/n 
are concerned. 

For every such function has the form P/Q where P and Q 
are rational and integral functions of the given irrationals ; and, 
if R be a rationalising factor of Q, PR QR will be of the form 
required. 



X EXERCISES XIV 199 

Example 1. Show that a rationalising factor of a; + y + z is 

(a; 3 + y 3 + z 8 - y V - s 3 a; 3 - a: y 3 ) 

/I 3 i ,, 4 J 1 i 4 4> 
x (a- + y + z + 2y 3 3 3 - z 3 x 3 - x 6 y ) 

, i 3 9 4 4^44 i is 

x (ar + 1/ 3 + ; 3 - y V + 2z 3 ar - ary 3 ) 

,3 9 3 44 4Jo4Sn 
x (a; 3 + y 3 + 2 3 - y V - z 3 a; 5 + 2x 6 if) ; 

and that the result of the rationalisation is (x + y + z) 3 - 27 xyz. 

Exercises XIV. 
Express as roots of rational numbers — ■ 

(1.) 2 ! x3-*x4 i x9*. (2.) {!/(**)}*+ {S/tf)}* 

(3.) 3V(Z/7)-2W?/7). (4.) {1/{W*)} x #(1S»)}. 

Simplify the following — 
(5.) {(a- a /6- 2 )VW 6 )} x ^ / (« 1; <"- 1, i 1/( "- 1) )- 

(6.) (/fl/IHWM) x (a;(c+«)'(<»-»))l/(»-c) X (x(»+*)/(»-e))V(«-a). 

(7.) Showthat n/("%/ ,!+ n/(''%= ^v' + ^x"^ 

Simplify — 

(S.) [x^-y'^ix-t-y-h (9.) (ar»- 2 + *-»)/(*- 2 4-ar 1 ). 

(10.) {x + x'^^-x'^-f. 

(11.) (a^-a^+l) (ar*+af*+ l){x-x h + l). 

(12.) (x i - x ~ V+ 3 *V ) (■** - 2T )• 

(13.) (x l + 4*/ 2 )/(x 3 + 2^ 3 + 2y). 

(14. ) 2 3 - 1 + 2 3 /(2 3 - 1) + 1/(2* + 1/(2 3 + 1)). 

(15. ) {x? " - 2./ 1 " //' " + y 2/ ") (x 1 " + x l '-'" y 1 "-" + y 1/n ) (aj 1 '" - y 1 ")• 

(16.) Show that x/{x i -l)-x i /{x i + l)-l/(x^-l) + l/(x h +l)=x i + 2. 
(17.) If 0(a-) = (a x -«"*)/(«* + «"*)> F(a-) = 2/(a* + «T*), 
then tf>(a; +y)= {<fi(x) + <p(y) } / {1 + £(a-)0(y) } ; 

F(* + y)= F(*)F(y)/ {1 + tfaMy) } . 
(18.) UxP/y=l, then xP- m /y9- n =x n J''i- m =y"- m '''''. 
(19.) If m-a x , n = ay, m«n x = a- il , then xy; = l. 

Transform the following into sums of simple irrational terms : — 

(20. ) V«/( \Ja + \/b) + V&/( V« ~ V 6 )- 
(21.) (2V5-3V'^ + W6) 2 . 

(22. ) {x + 1 - V2 + \/3) (a: + 1 + y/2 - V 3 ) (a: + 1 - V2 - V 3 ), arranging ac- 
cording to powers of x. 



200 EXERCISES XIV chap. 

(23. ) (1/ V»+ 1/ sja) (x i - a h )l {( V« + \/xf - ( V« " V*) 2 } ■ 
r24 v V(« 2 + {«(! - »0/3V»t} 8 ) + o ( to- 1)/2V ot 

K '' \/(a*+{a(\-m)lZsJm,y)-a{m-\)l2\/vi 

(C} r v \/(p/a + x) + * s /(p/a-z) 2pq 

(■to- ) „ t— — r — -,> —, {, where a,x'=: =- — „. 

s/(p/a + x) - \/{p/a -x)' 1 + cf 

ton \ P-a p-b p + a p + b 

{to-) 1 r--— — r, where x-\/(nb). 

x-a x-b x + a x + b vv ' 

(27.) 1/ { X /( P -q) + y/ p + V<?} + 1/ ! Mp -q)-\/p- \/?} + 1/(VJP - V?)- 

(28) /( l-*-^(2x + x*) ) / / 1 -x+ V(2.r + ^) ) . 

v '' V U-a+V(2»+a*)/" r V ll-aj-V^+a: 2 )/ 

(29. ) ( V(« + & + c) + V(« - 6 + '0)7( V(« + * + c ) - V(« - 6 + c)) 2 . 

(30.) {U(p-q) + */p-<s/q)Ksf(p-q)-y/p-\/q)}. 

(31.) (2.r 5 -6.r + 5)/(4 / 2x'+ vA + l). 

(32. ) (a? + 3 ^23 + 1 )/(.r + ^2 - 1). 

(33. ) { l/{a + b)- l/(a - b) } { [ y/{a + b)f + [ l/{u - i)] 2 - $/{<# - b-) \ . 

Show that 

<"•> (r^r# Gt^)-^^- ve-*»- 

(35.) (V(y- + i) + V(p 2 --i))- 3 + (V(^ 2 + i)-V(y-i))- 3 =(y--i)V(r+i)- 

(36.) V[\/{« 2 + \V& 2 )} + VF+ ^V& 4 )}] = (« § + *<¥- 

Express in linear form — 

(37.) {Z/x-Z/y)/(l/x+i/y). 

(38. ) (1 + V3 + V5 + V7)/(l - V3 - \ 7 5 + \/7)- 

(39.) 2(v&+ Vc)/(V & + Vc - V«) - 42«(Vft + Vc)/n(\/ft + V<--- V«)- 

Rationalise the following : — 

(40.) 3.5*- 4^. (41.) 2V(Hc-«). 

(42.) V5 + V3 + V4 - V6- (43.) 3.2* + 4.2 4 -l. 

(44.) a* + & J + c*. (45.) 2*+2* + l. 

(46.) If« = ; rV(l+i/ 2 ) + 2/\/(l+* 2 ).tlienV(l+«-) = -''2/ + \/((l+» 2 )( 1 +2/ 2 ))- 

(47.) Show that V(y .. ) + >/0 ,l, ) + V(a> . y j 

= ( »/ - z) % + (z - aQ j + (a - y) 1 + (y - z)\z - x)\x - y Y 
■'" + y 2 + z 2 — yz-zx- xy 
(48. ) If jr = l/( Vi + \/c - \fa), y = l/(\/c + s/a - \/£), z=lf(s/a + V* 

- /y/c), U = l/( V« + Vft + V c )> 

then Tl(-x+y + z + m)/(Ssb - ?t) 3 = n ( ^ + c - a)/8abc. 

xyz 

Historical Note. — The use of exponents began in the works of the German 
"Cossists," Rudolf! (1525) and Stifel (1544), who wrote over the contractions 



x HISTORICAL NOTE 201 

for the names of tbe 1st, 2nd, 3rd, . . . powers of the variable, which had been 
used in the syncopated algebra, the numbers 1, 2, 3, . . . Stifel even states 
expressly the laws for multiplying and dividing powers by adding and subtracting 
the exponents, and indicates the use of negative exponents tor the reciprocals of 
positive integral powers. Bombelli (1579) writes _., ^, „£,, 3> • • •> where 
we should write .v°,.r, .-/;-, .*.•■'', . . . Stevin (1585) uses in a similar way @, ©, 
©i ©>•••> an d suggests, although he does not practically use, fractional 
powers such as ©, ©, which are equivalent to the x , x , of the present day. 

Viete (1591) and Oughtred (1631), who were in lull possession of a literal 
calculus, used contractions for the names of the powers, thus, Aq, Ac, Aq</, to 
signify A' 2 , A 3 , A 4 . Harriot (1631) simply repeated the letter, thus, aa, aaa, 
acuta, for ft 2 , « :i , a 4 . Herigone (1634) used numbers written after the letter, 
thus, A, A2, A3, . . . Descartes introduced the modern forms A, A 2 , A 3 , . . . 
The final development of the general idea of an index unrestricted in magnitude, 
that is to say, of an exponential function a x , is due to Newton, and came in 
company with his discovery of the general form of the binomial coefficients as 
functions of the index. He says, in the letter to Oldenburg of 13th June 1676, 
"Since algebraists write a 2 , ft 3 , «', &c, for aa, aaa, aaaa, &c. , so I write 

a , a", a 3 , for \Ja, \/a s , \/c.a s : and I write a- 1 , or-, ar", &c, for -, — , 
1 « aa 

—,kr." 

aaa 

The sign \/ was first used by Rudolff.; both he and Scheubel (1551) used 
/fj to denote 4th root, and wJ to denote cube root. Stifel used both •sj'fr. 
and \J to denote square root, \/%%. to denote 4th root, and so on. Girard 

(1633) uses the notation of the present day, \J, \/, &c. Other authors of the 
17th century wrote y/2 :, \/3 : , &c. So late as 1722, in the second edition of 
Newton's Arithmetica Universalis, the usage fluctuates, the three forms \/3:, 

/3 3/ 

V : , v' all occurring. 

In an incomplete mathematical treatise, entitled 1)e Arte Logistica, &c, 
which was found among the papers of Napier of Merchiston (1550-1617 ; pub- 
lished by Mark Napier, Edinburgh, 1839), and shows in every line the firm grasp 
of the great inventor of logarithms, a remarkable system of notation for irrationals 

is described. Napier takes the figure " [_, and divides it thus 4J |_5j [6. 

' •' t-| m it 

He then uses J, \_J, I , &c. , which are in effect a new set of symbols for the 

nine digits 1, 2, 3, &c, as radical signs. Thus UlO stands for \A0, I- 10 for 
^10, _i°10 for ^/lO, J! 10 for ^10, _] ^j or =] for "Ao ; and so on. 

Many of the rules for operating with irrationals at present in use have come, 
in form at least, from the German mathematicians of the 16th centuiy, more 
particularly from Scheubel, in whose Algebra Compendiosa Fo.cilisqiie Descriptio 
(1551) is given the rule of chap, xi , § 9, for extracting the square root of a 
binomial surd. Iu substance many of these rules are doubtless much older (as 
old as Book X. of Euclid's Elements, at least) ; they were at all events more or 
less familiar to the contemporary mathematicians of the Italian school, who did 
so much for the solution of equations by means of radicals, although in symbol- 
ism they were far behind their transalpine rivals. See Hutton's Mathematical 
Dictionary, Art. "Algebra." 

The process explained at the end of next chapter for extracting the square or 



202 HISTORICAL NOTE chap, x 

cube root by successive steps is found in the works of the earliest European 
writers on algebra, for example, Leonardo Fibonacci (c 1200) and Luca Pacioli (c. 
1500). The first indication of a general method appears in Stifel's Arithmetica 
Integra, where the necessary table of binomial coefficients (see p. 81) is given. 
It is not quite clear from Stifel's work that he fully understood the nature of the 
process and clearly saw its connection with the binomial theorem. The general 
method of root extraction, together with the triangle of binomial coefficients, is 
given in Napier's De Arte Logistica. He indicates along the two sides of his 
triangle the powers of the two variables (prsecedens and succedens) with which 
each coefficient is associated, and thus gives the binomial theorem in diagram- 
matic form. His statement for the cube is — " Supplementum triplicationis tribus 
constat numeris : primus est, duplicati prsecedentis triplum multiplicatum per 
succedens ; secundus est, prsecedentis triplum multiplicatum per duplicatum suc- 
cedentis ; tertius est, ipsum triplication succedentis." In modern notation, 



CHAPTER XL . 
Arithmetical Theory of Surds. 

ALGEBRAICAL AND ARITHMETICAL IRRATIONALITY. 

§ 1.] In last chapter we discussed the properties of irra- 
tional functions in so far as they depend merely on outward 
form ; in other words, we considered them merely from the 
algebraical point of view. We have now to consider certain 
peculiarities of a purely arithmetical nature. Let p denote any 
commensurable number ; that is, either an integer, or a proper or 
improper vulgar fraction with a finite number of digits in its 
numerator and denominator ; or, what comes to the same thing, 
letp denote a number which is either a terminating or repeating 
decimal. Then, if a be any positive integer, H/p will not be 
commensurable unless p be the nth power of a commensurable 
number ;* for if %/p = k, where k is commensurable, then, by the 
definition of %/p, p = k n , that is, p is the nth power of a commen- 
surable number. 

If therefore p be not a perfect nth power, %/p is incommensur- 
able. For distinction's sake %/p is then called a surd number. 
In other words, we define a surd number as the incommensurable 
root of a commensurable number. 

Surds are classified according to the index, n, of the root to 

be extracted, as quadratic, cubic, biquadratic or quartic, quintic, 

. , . w-tic surds. 

The student should attend to the last phrase of the definition of a surd ; 
because incommensurable roots might be conceived which do not come under 

;; This is briefly put by saying that p is a perfect «th power. 



204 CLASSIFICATION OF SURDS chap. 

the above definition ; and to them the demonstrations of at least some of the 
propositions in this chapter would not apply. For example, the number e (see 
the chapter on the Exponential Theorem in Part II. of this work) is incom- 
mensurable, and \/c is incommensurable ; hence \/e is not a surd in the exact 
sense of the definition. Neither is V( V 2 + 1), for \/2 + 1 is incommensurable. 
On the other hand, \/( V 2 )> which can be expressed in the form tyl, does come 
under that definition, although not as a quadratic but as a biquadratic surd. 

He should also observe that an algebraically irrational function, say \/x, 
may or may not be arithmetically irrational, that is, surd, strictly so called, 
according to the value of the variable x. Thus \J1 is not a surd, but V 2 is - 



CLASSIFICATION OF SURDS. 

§ 2.] A single surd number, or, what comes to the same, a 
rational multiple of a single surd, is spoken of as a simple mono- 
mial surd number ; the sum of two snch surds, or of a rational 
number and a simple monomial surd number, as a simple binomial 
surd number, and so on. 

The propositions stated in last chapter amount to a proof of 
the statement that every rational function of surd numbers can 
be expressed as a simple surd number, monomial, binomial, 
trinomial, &c, as the case may be. 

§ 3 ] Two surds are said to be similar when they can be expressed 
as rational multiples of one and the same surd ; dissimilar when this 
is not the case. 

For example, >/18 and N /8 can be expressed respectively in 
the forms 3 N /2 and 2 J 2, and are therefore similar ; but J 6 
and s/2 are dissimilar. 

Again, ^/54 and £/16, being expressible in the forms 3^2 
and 2 ^/2, are each similar to 1/2. 

All the surds that arise from the extraction of the same nth root 
are said to be equiradical. 

Thus p*, p*, p*, P*° are all equiradical with p*. 

There are n - 1 distinct surds equiradical with 2> v '\ namely, p l/n , 
P 2!n , . . ., £>("■- D/'^ and no more. 

For, if we consider p mln where m>n, then Ave have p m l n = 
^*+»/n -where /x and v are integers, and v < n. Hence p m,n = 
pft pin _ a rational multiple of one of the above series. 

All the surds equiradical with p l l* are rational functions (namely, 



xi CLASSIFICATION OF SURDS 205 

positive integral powers) of p lln ; and every rational function of p l,n 
or of surds equiradieal with p l,n may be expressed as a linear function 
of the n - 1 distinct surds which are equiradieal with p l,n , that is, in 
the form A + A^ 11 ' 1 + A.,p 2ln + . . . + A M _ 1 ^ n_1 ) /n , where A , A l5 
. . . , A n _ i are rational so far as p lln is concerned. 

This is merely a restatement of § 18 of chap. x. 

§ 4.] The p'oducl or quotient of two similar quadratic surds is 
rational, and if the product or quotient of the two quadratic surds is 
rational they are similar. 

For, if the surds are similar, they are expressible in the 
forms A s/p and B s/p, where A and B are rational ; therefore 
A \/p x B s/p - ABp ; and A \/p/B s/p = A/B, which proves the 
proposition, since ABp and A/B are rational. 

Again, if s/p x \/q = A, or \/p/ \/q = B, Avhere A and B are 
rational, then in the one case s/p - (AJq) s ^q, in the other s/p 
= B \'q. But A/q and B are rational. Hence s/p and \/q are 
similar in both cases. 

The same is not true for surds of higher index than 2, but 
the product of two similar or of two equiradieal surds is either rational 
or an equiradieal surd. 

INDEPENDENCE OF SURD NUMBERS. 

§ 5.] If p, q, A, B be all commensurable, and none of them zero, 
and s/p and sjq incommensurable, then we cannot have 

s/p = A + B sjq. 
For, squaring, we should have as a consequence, 

p = A 2 + B 2 q + 2AB s/q ; 
whence 

s/q = (p - A 3 - BV)/2AB, 

which asserts, contrary to our hypothesis, that Jq is commen- 
surable. 

Since every rational function of s/q may (chap, x., § 15) be 
expressed in the form A + B Jq, the above theorem is equivalent 
to the following : — 

One quadratic surd cannot be expressed as a rational function of 
another which is dissimilar to it. 



206 INDEPENDENCE OF SURD NUMBERS chap. 

Since every rational equation between <Jp and Jq which is 
not a mere equation between commensurables (for example, 
( \ / 3) 2 + ( \/2) 2 = 5) is reducible to the form 

A sj(pq) + B x/jp + C s/q + D = 0, 

where A, B, C, D are rational ; and, since this equation may im- 
mediately be reduced to another of the form 

Jp = L + M */q, 

where L and M are rational, it follows that 

No rational relation,, which is not a mere equation between rational 
numbers, can subsist between two dissimilar quadratic surds. 

§6.] 7/" the quadratic surds >Jp, \/q, \/r, \/{qr) be dissimilar to 
each other, then >Jp cannot be a rational function of Jq and ijr. 

For, if this were so, then we should have 

s/p = A + B *Jq + C Jr + D \/(gr), 

where A, B, C, D are all rational. 

Now we cannot, by our hypotheses, have three of the four 
A, B, C, D equal to zero. 

In any other case, we should get on squaring 

p = {A + Bjq + C Jr + D J(qr)}°-, 

which would either be a rational equation connecting two dis- 
similar quadratic surds, which is impossible, as we have just seen ; 
or else an equation asserting the rationality of one of the surds, 
which is equally impossible. 

An important particular case of the above is the following : — 

A quadratic surd cannot be the sum of two dissimilar quadratic 
surds. 

It will be a good exercise for the student to prove this 
directly. 

§ 7.] The theory which we have established so far fur 
quadratic surds may be generalised, and also extended to surds 
whose index exceeds 2. This is not the place to pursue the 
matter farther, but the reader who has followed so far will find 
the ideas gained useful in paving the way to an understanding 
of the delicate researches of Lagrange, Abel, and Galois regarding 






xi INDEPENDENCE OF SURD NUMBERS 207 

the algebraical solution of equations whose degree exceeds the 
4 th. 

§ 8.] It follows as a necessary consequence of §§ 5 and 6 
that, if we are led to any equation such as 

A + Bs'p + C v/g + D ^(pq) = 0, 
where Jp and *Jq are dissimilar surds, then we must have 

A = 0, B = 0, C = 0, D = 0. 
One case of this is so important that we enunciate and prove it 
separately. 

If x, y, z, u be all commensurable, and Jy and sju incommen- 
surable, and if x + Jy = z + Ju, then must x-z and y = u. 
For if x =# z, but = a + z say, where a 4= 0, then by hypothesis 

a + z + \/y = z + \''u, 
whence a + \/y — *Ju, 

a 3 + y + 2a s/y - u, 

s/y = (« - a 2 - y)/2a, 
which asserts that Jy is commensurable. But this is not so. 
Hence Ave must have x = z ; and, that being so, we must also 
have Jy — \ f u, that is, y = u. 

SQUARE ROOTS OF SIMPLE SURD NUMBERS. 

§ 9.] Since the square of every simple binomial surd number 
takes the form p + Jq, it is natural to inquire whether J(j> + *Jq) 
can always be expressed as a simple binomial surd number, that 
is, in the form *Jx + \'y, where x and y are rational numbers. 
Let us suppose that such an expression exists ; then 

*J(jp+ s 'q)= \/x+ >Jy, 
whence p + \'q = x + y + 2 *J(xy). 

If this equation be true, we must have, by § 8, 

x + y=p (1), 

2^)= s'q (2); 

and, from (1) and (2), squaring and subtracting, we get 
(x + y) 3 -4xy=p* - q, 

that is, (z-y) 2 =p 3 ~q ( 3 > 



208 LINEAR EXPRESSION FOR </(« + \^) chap. 

Xow (3) gives either 

x-y = + s!{/-q) (4), 

or x-y= - *S(p'-q) (4*). 

Taking, meantime, (4) and combining it with (1), we have 

(x + y) + (x-y)=p + s/{p*-q) (5), 

(x + y) - (x -y)=p- s '{f -q) (6) ; 

Avhence 2x =p + \,'(p* - q), 

2y=p- s/(p*-q); 
that is, x = h{p+ s'df-q)} (7), 

V = HP - n ; (/ " ?)} (8). 

If we take (4*) instead of (4), we simply interchange the values 
of x and y, which leads to nothing new in the end. 

Using the values of (7) and (8) we obtain the following 
result : — 

Since, by (2), 2 s/x x v /y = + ^> we must take either the two 
upper signs together or the two lower. 

If we had started with \/(p - */q), it would have been 
necessary to choose \/x and \fy with opposite signs. 

Finally therefore we have 

(9), 

(9*). 

The identities (9) and (9*) are certainly true ; we have in fact 
already verified one of them (see chap, x., § 9, Example 14). They 
will not, however, furnish a solution of our problem, unless the 
values of x and y are rational. For this it is necessary and 
sufficient that p* - q be a positive perfect square, and that p be 






XI 



EXAMPLES 209 



positive. Hence the square root of p + *Jq am he expressed as a 
simple binomial surd number, provided p be positive and p* - a be a 
positive perfect square. 

Example 1. Simplify >/(19-4V21). 

Let V( 19 - 4 V 21 )=V / a ; + Vy- 

Then aj+y=19, 

(cc-y) 2 = 361-336 

= 25, 

x-y= +5 say, 

x + y-19 ; 
whence # = 12, y=7, 

*Jx - ± VI 2, \ // = T \''7, 
so that V(l 9 -W21)=±(Vl2-\/7)- 

Example 2. To find the condition that \Z(VP+ V?) ma } r De expressible 

4 / 

in the form (\/x + \ f y) *Jp we have 

•s/NP + V?) = fa x V U + V(?/l>)} • 

Now V{1 + V((?/i°)} "ill t> e expressible in the form \/x+\Jy, provided 
1 - q/p be a positive perfect square ; this, therefore, is the required condition. 
For example, 

V(5V7 + 2V42)= \/7x V(5 + 2v'6) 

= ± 4/7K/3 + V2). 

Example 3. It is obvious that in certain cases V(.P + V?+ V r + N/' s ) roust 
be expressible in the form \'x + \/y+ \ ~, where a 1 , y, z are rational. To find 
the condition that this may be so, and to determine the values of x, y, z, let 

V(P + \fl + V'' + V s ) = \ /x + \'y + V s C 1 ). 

then }i+\/q+\/r+\Js = x + y + z + 2\/(yz) + 2s/(zx) + 2\/(xy) (2). 

Now let us suppose that 

2V(yz)=V? (3), 

2sJ{zx) = s /r (4), 

2V(«y)=V* (5). 

From (4) and (5) we have by multiplication 

4x\/(yz)=*/(rs); 

whence, by using (3), x = hy/{>'s/q) (6). 

Proceeding in like manner with y and r, we obtain 

y=W(q*M (7), 

~~=W(<7>7*) («)• 

It is further necessary, in order that (2) may hold, that the values (6), (7), 
(8) shall satisfy the equation 

x + y + z=p (9), 

VOL. I P 



210 ARITHMETICAL SQUARE ROOT chap. 

that is, we must have 

V(W?) + VWO + s/iqr/s) = 2p (10), 

where the signs throughout must be positive, since x, y, z must all be positive. 
Also, since x, y, z must all be rational, we must have 

rs o 9 s as W 
q r s 

where a, /3, y are positive rational numbers, such that 

a + p + y = 2p, 
whence, in turn, we obtain 

q = §y, r = ya, s = ap. 



ARITHMETICAL METHODS FOR FINDING APPROXIMATE RATIONAL 
VALUES FOR SURD NUMBERS. 

§ 10.] It has already been stated that a rational approxima- 
tion, as close as we please, can always be found for every surd 
number. It will be well to give here one method at least by 
which such approximations can be obtained. We begin with 
the approximation to a quadratic surd ; and we shall afterwards 
show that all other cases might be made to depend on this. 

§ 11.] First of all, we may point out that in every case we 
may reduce our problem to the finding of the integral part of 
the square root of an integer. Suppose, for example, we wish 
to find the square root of 3*689 correct to five places of decimals. 
Then, since ^3-689 = ^36890000000/10*, we have merely to 
find the square root of the integer 36890000000 correct to the 
last integral place, and then count off five decimal places. 

§ 12.] The following propositions are all that are required 
for the present purpose : — 

I. The result of subtracting (A + B) 2 from N is the same as the 
result of first subtracting A 2 , then 2AB, and finally B 2 . 

This is obvious, since (A + B) 2 = A 2 + 2AB + B 2 . 

II. If the first p out of the n digits of the square root of the 
integer N have been found, so that P10 n_ ^ is a first approximation 
to VN, then the next p - 1 digits ivill be the first p - 1 digits of the 
integral part of the quotient {N - (Pl0 n -^) 2 }/2Pl0' 1 -^ with a possible 
error in excess of 1 in the last digit, 



xi ARITHMETICAL SQUARE ROOT 211 

Let the whole of the rest of the square root be Q. Then 

x /N = P10»-" + Q, 
where 1 0* - x < P < 1 0*, Q < 1 n -* ; 

whence N = (P10 w -*>) 2 + 2PQ10*-* + Q 2 , 

N-(P10"-^) 2 _ Q 2 (1). 
— \°l + 



2P10""^ ^ 2P10 n "^ 



Now 



Q72PlO n -^<10 2 ( ,l -^/2 x 10J , - 1 10 n -*'<10 n -* +1 /2. 

Hence Q 2 /2P10 n --P will at most affect the (n - 2p + l)th place, 
and the error in that place will be at the utmost 5 in excess. 
Therefore, since Q contains n-p digits, the first p - 1 of these 
will be given by the first p - 1 digits of {N - (P10 n -^) 2 }/2P10 n "-P 
with a possible error in excess of 1 in the last digit.* 

§ 13.] In the actual calculation of the square root the first 
few figures may be found singly by successive trials, Proposi- 
tion I. being used to find the residues, which must, of course, 
always be positive. Then Proposition II. may be used to find 
the succeeding digits in larger and larger groups. The approxi- 
mation can thus be carried out with great rapidity, as will be 
seen by the following example : — 

Let it be required to find the square root of N = 680100000000000000, 
which, for shortness, we write 6801(14). 

Obviously 8(8)<\/N<9(8) ; in other words, \/N contains 9 digits, and the 
first is 8. 

Now N- (8(8)) 2 = 401(14), which is the first residue. We have now to 
find the greatest digit x which can stand in the second place, and still leave 
the square of the part found less than N, that is (by Proposition I.), leave the 
residue 401(14) -2 x 8(8) xx(7) - {x(7)} 2 positive. It is found by inspection 
that» - = 2. Carrying out the subtractions indicated, that is, subtracting 
|16(8) + 2(7)} x2(7) = 162x2(14) from 401(14), we have now as residue 
7700(12). 

* The effect of such an error would be to give a negative residue in 
the process of § 13 ; so that in practice it would be immediately discovered 
and rectified. As an example of a case where the error actually occurs, the 
reader might take the square root of 5558(12), namely, 74551995, and attempt 
to deduce from 745 the two following digits. He will find by the above rule 
52 instead of 51. If it be a question of the best approximation, the rule gives 
here, as always, the best result ; but this is not always what is wanted. 



212 



EXAMPLES 



CHAP. 



The double of the whole of the part of \J~N now found is 164(7) ; and 

we have next to find y as large as may be, so that 7700(12)- {164(7) + 2/(6)} 

x ?/(6) shall remain positive. This value of y is seen to be 4. It might, of 

course, be found (by Proposition II.) by dividing 7700(12) by 164(7), and 

taking the first figure of the quotient. 

The residue is now 112400(10). The process of finding the first four 
digits in this way may be arranged thus : — 

8(8) 6801(14) 8(8) 

16(8) 6400(14) 

162(7) 401(14) +2(7) 

164(7) 324(14) 

1644(6) 7700(12) +4(6) 

1648(6) 6576(12) 

16486(5) 112400(10) +6(5) 

16492(5) 98916(10) 

134840(9) 

We might, of course, continue in the same way, figure by figure, as long as 
we please ; and we might omit the records in brackets of the zeros in each line. 

Havino-, however, already found four figures, we can find three more by 
dividing the residue 134840(9) by 16492(5), which is the double of 8246(5), 
the part of V N already found. 



16492(5) 134840(9) 
131936 



817(2) 



29040 
16492 

125480 
115444 



10036000(4) 
The next three digits are therefore 817. 10036000(4) is not the residue ; 
for we have only subtracted from V N as yet {8246(5)} 2 and 2x8246(5) 
x 817(2). Subtracting also {817(2)} 2 we get the true residue, namely, 
93685110000. We may now divide this by 2x8246817(2), that is, by 
1649363400, and thus get the last two figures. We have then 

10036000(4) 
667489(4) 



1649363400 I 



93685110000 
8246817000 
11216940000 
9896180400 

1320759600 



56 



We have now found the whole of the integral part of V 68 01(14), namely, 
824681756. 



XI 



RATIONAL APPROXIMATION TO ANY SURD 



213 



If it were desired to carry the approximation farther, 8 places after the 
decimal point could at once be found by dividing the true residue 
(1320759600 - 56 2 ) by 2 x 824681756. If we require no more places than those 
8 places, then the residue is of no importance, and we may save labour by 
adopting the abbreviated method of long division (see Brook Smith's Arith- 
metic, chap. vi. , § 153). Thus 



nftiW$$V$ 



1320759600 
3136 

1320756464 
1319490810 

1265654 
1154554 



80076736 



111100 

98962 

12138 
11545 



593 
495 

"98 
98 





We thus find V6801(14) = 824681756 '80076736. 
will find that in point of fact 



On verifying, the reader 



(824681756 -80076736) 2 = 680100000000000001 -82 . . . 

It will be a good exercise for him to find out how many decimal places of 
the square root of a given integer must be found before the square of the 
approximation ceases to be incorrect in the last integral place. 

§ 14.] By continually extracting the square root (that is to 
say, hy extracting the square root, then extracting the square 
root of the square root, and so on), we may bring any number 
greater than unity as near unity as we please. In other words, 
by making n sufficiently great, W 1 * may be made to differ from 1 
by less than any assignable quantity. 

For let it be required to make N I/2 less than 1 + a, where a 
is any positive quantity. This will be done if 2 n be made such 
that (1 + af l > N. Now (chap, iv., § 1 1) (1 + a) 2 " = 1 + 2' l a + a 
series of terms, which are all positive. Hence it will be sufficient 
if we make 1 + 2"u> N, that is, if we make 2 n a>N - 1, thut is, 



214 EXAMPLES chap. 

if we make 2 n >(N- l)/a, which can always be done, since by 

making n sufficiently great 2 n may be made to exceed any 

quantity, however great. 

Example. How many times must we extract the square root beginning 
with 51 in order that the final result may differ from 1 by less than - 001 ? 

We must have 

2» > (51-1)/ -001, 
2" > 50000. 
Now 2 15 = 32768, 2 16 = 65536, 

hence we must make ?i = 16. 

In other words, if we extract the square root sixteen times, beginning 
with 51, the result will be less than 1 "001. 

§ 15.] It follows from § 14 that we can approximate to any 
surd whatever, say p lln , by the process of extracting the square 
root. For (see chap, ix., § 2) let 1/n be expressed in the binary 
scale, then we shall have 

1/n = a/2 + /3/2 2 + y/2' + . . ,+fjL, 

where each of the numerators a, ft, y, . . . is either or 1, and 
ix is either absolutely or < l/2 r , where r is as great as we 
choose. 

Hence 

= pt/2 x ^/S/22 x ^/23 x x ^» ^y 

Now, excepting the last, each of these factors is either ] , or of 
the form p* 1 "' , which can be approximated to as closely as we 
please by continued extraction of the square root. If /i = 0, the 
last factor is 1 ; and if ll< \/2 r , since r may be as great as we 
choose, we can make it differ from 1 by as small a fraction as 
we choose. It follows therefore that the product on the right 
hand of (1) may be found in rational terms as accurately as Ave 
please. 

Example. To find an approximate value of 5V 9 . 
We have 

i i_ i i i L L L L L L 

3 2' 2 2* 2 6 2 8 "*"2 10 "*~2 12 "^ 2 U "^2 16 + 2 18 "^2-""^' U ' 
where fi < 1/2 20 . 



XI ALGEBRAICAL SQUARE ROOT 215 

Now we have, correct to the fourth decimal place, the following values :— 
5V*= 2-67234, 51^"= 1-00096, 

51 1/24 = 1 -27857, 51 1 "-' 4 = 1 -00023, 

51^=1-06336, 51' 2,6 = 1-00006, 

51 1 - 8 = 1 -01548, 51 1 "-' 8 = 1 -00002, 

5H/2"> - 1 -00385, 51 1 / 220 < 1 -00001. 

Hence, multiplying the first nine numbers together, we get 

5V 3 = 3 70841 
The correct value is 3-708429 . . . 

§ 16.] The method just explained, although interesting in 
theory, would be very troublesome in practice. 

The method given in § 13 for extracting the square root may 
be easily generalised into a method for extracting an ?it\\ root 
directly, figure by figure, and group by group of figures. The 
student will be able to establish for himself two propositions, 
counterparts of I. and II., § 12, and to arrange a process for 
the economical calculation of the residues. A method of this 
kind is given in most arithmetical text-books for extracting the 
cube root, but it is needless to reproduce it here, as the extrac- 
tion of cube and higher roots, and even of square roots, is now 
accomplished in practice by means of logarithmic or other tables 
(see chap, xxi.) Moreover, the extraction of the nth root of a 
given number is merely a particular case of the numerical solu- 
tion of an equation of the nth degree, a process for which, called 
Homer's Method, will be given in a later chapter. 

Our reason for dwelling on the more elementary methods of 
this chapter is a desire to cultivate in the mind of the learner 
exact ideas regarding the nature of approximate calculation — 
a process which lies at the root of many useful applications of 
mathematics. 

SQUARE BOOT OF AN INTEGRAL FUNCTION OF X. 

§ 17.] When an integral function of x is a complete square 
as regards .r, its square root can be found by a method analogous 



216 ALGEBRAICAL SQUARE ROOT chap. 

to that explained in § 12, for finding the square root of a number. 
Although the method is of little interest, either theoretically or 
practically, we give a brief sketch of it here, because it illustrates 
at once the analogy and the fundamental difference between 
arithmetical and algebraical operations.* 

I. We may restate Proposition I. of § 1 2, understanding now 
A and B to mean integral functions of x. 

II. IfF =p x 2n + p 1 x 2n ~ 1 + . . . +p, n , and if JF = {q„x n + q x x n ~ x 
+ . . . + q n - p + x x n - p+1 ) + {q n -p* n ~ p + • • • + q ) = p + Q,say; and if 

we suppose the first p terms, namely, P = q x n + q x x n ~ l + . . . + q n - P +i 

x n ~P +1 , of \/F to be biown, then the next p terms will be the first p 

terms in the integral part of (F - P 2 )/2P. 

for we have 

F = F + 2PQ + Q 2 ; 

, F-F Q 2 

hence —^ = Q + ^ 

Now the degree of the integral part of Q 2 /2P is 2(n-p)-n 
= n- 2p. Hence Q 2 /2P will at most affect the term in x n ~ 2p . 
Hence (F - P 2 )/2P will be identical with Q down to the term in 
x n-2p+i inclusive. In other words, the first n-p - (» - 2p) =p 
terms obtained by dividing F - P 2 by 2P will be the p terms of 
the square root which follow P. 

We may use this rule to obtain the whole of the terms one 
at a time, the highest being of course found by inspection as the 
square root of the highest term of the radicand ; or we may ob- 
tain a certain number in this way, and then obtain the rest by 
division. t 

The process will be understood from the following example, 

* The method was probably obtained by analogy from the arithmetical 
process. It was first given by Recorde in The Whetstone of Witte (black 
letter, 1557), the earliest English work on algebra. 

t Just as in division, we may, if we please, arrange the radicand according 
to ascending powers of x. The final result will be the same whichever arrange- 
ment be adopted, provided the radicand is a complete square. If this is not 
the case the operation may be prolonged indefinitely just as in continued 
division. We leave the learner to discover the meaning of the result obtained 
in such cases. The full discussion of the matter would require some refer- 
ence to the theory of infinite series. 



XI 



ALGEBRAICAL SQUARE ROOT 



217 



in which we first find three of the terms of the root singly, and 
then deduce the remaining two by division : — 

Exam] ile. 

To find the square root of 

a;io + 6^9 + 13Z 8 + 4a; 7 - 18.x 6 - 12a: 5 + 14ar* - 12k 3 + 9a- 2 - 2x + 1. 



1 

2 + 3 
2 + 6 + 2 
2+6+4-4 

2+6+4-8 



1 + 6 + 13+ 4-18-12 + 14-12 + 9-2 + 1 

1 



6 + 13+ 4-18-12 + 14-12 + 9-2 + 1 

6+ 9 

4+ 4-18-12 + 14-12 + 9-2 + 1 
4 + 12+ 4 



- 8-22-12 + 14-12 + 9-2 + 1 

- 8-24-16 + 16 



2+ 4- 2-12+9-2+1 
2+ 6+ 4- 8 



+ 1 
+ 3 
+ 2 



+ 1-1 



- 2- 6- 4 + 9-^2 + 1 

- 2- 6- 4 + 8 

1-2 + 1 
Hence the square root is x 5 + 3a; 4 + 2a* 3 - 4a.* 2 + x - 1 ; and, since the residue 
x*-2x+\ is the square of the two last terms, namely, a;— 1, we see that the 
radicand is an exact square. Of course, we obtain another value of the square 
root by changing the sign of every coefficient in the above result. 

A similar process can be arranged for the extraction of the 
cube root ; but it is needless to pursue the matter further. 

§ 18.] The student should observe that in the simpler cases 
the root can be obtained by inspection ; and that in all cases the 
method of indeterminate coefficients renders any special process 
for the extraction of roots superfluous. This will be understood 
from the following example. 



Example. 

To extract the square root of 



,.io 



+ 6a; 9 + 13a: 8 + 4a; 7 - 18a: 6 - 12a; 5 + Ux 4 - 12a; 3 + 9a: 2 - 2x+ 1 



(1). 



If the radicand be a complete square, its square root must be of the form 

ar 5 +px i + qx* + rx" +sx+ 1 (2 ). 

The square of (2) is 

x it) + '2px 9 +(p 2 + 2q)x s + (22)q + 2r)x 7 + (2pr + q ii + 2s)x e + . . . (3). 

Now this must be identical with (1) ; hence we must have 

2p = G, p 2 + 2-7=13, 2pq+2r=4, 2pr+g a +2s= -18. 



218 EXERCISES XV 



CHAP. 



The first of these equations gives p = S ; p being thus known, the second 
gives q=2 ; p and q being known, the third gives r— -4 ; andp, q, r being 
known, the last gives 8=1. We could now find I in like manner ; but it is 
obvious from the coefficient of £ that t= — 1. 

Hence one value of the square root is 

x 5 + 3X 4 + 2x* - Ax 2 + x - 1 . 

N.B. — The equating of the coefficients of the remaining terms of (1) and 
(3) will simply give equations that are satisfied by the values of p, <7, r, s 
already found, always supposing that the given radicand is an exact square. 

A process exactly similar to the above will furnish the root of an exact 
cube, an exact 4th power, and so on. 



Exercises XV. 

Express the following as linear functions of the irrationals involved. 

(1.) l/(VH + V3 + \/14). (2.) x /12/(l + V2)(\ / 6-V3). 

(3.) (1 - V2 + V3)/(l + V2+ V3) + (1 - V2 - V3)/(l + \/2 - V3). 

(4.) (3-V5)/(V3 + V5) 2 + (3 + \/5)/(V3-\/5) 2 . 

(5.) V5/(V3 + V5-2V2)-V2/(\/3 + V2-\/5). 

(6.) (7 - 2 V5)(5 + V7)(31 + 13 v '5)/(6 - 2V7)(3 + V5)(H + 4 V")- 

(7.) V(25 + 10V6). (8.) V(3/2 + v'2). 

(9.) V(123-22V2). (10.) V(44V2 + 12V26). 

(11.) V{(8 + 4VlO)/(8-4VlO)}. 
(12.) V(7 + 4\/3)+V(5-2V6). 
(13.) V(15-4V14) + 1/V(15+4V14). 
(14. ) 1/V(16 + 2 V63) + l/\/(16 - 2 V63). 
(15.) l/V(16V3 + 6V21) + V(16v'3-6V21). 

(16.) Calculate to five places of decimals the value of {\/(5 + 2\/6) 

-V(5-2V6)}/{V(5 + 2\/6)+V(5-2V<5)}- 

(17.) Calculate to seven places of decimals the value of \/(\/15 + \/13) 
+ VW15-V13). 

Simplify— 

(IS.) V{3 + V(9- i ; 2 )} + V{3-V(9-^ 2 )}. 
(19.) ^{a + b-c+2^(b(a-c))}. 
(20.) % /{« 2 -2 + «V(« 2 -4)}. 

<-U/{(>-^Xr^-)}- 

(22.) Show that V{2 + V(2 - V2)} = ./ { 2 + V( ^ +n/2) } 

(23. ) Express in a linear form \/( 5 + V 6 + V^O + V 15 )- 
(24.) „ ,, V(25-4V3-12 x /2 + 6V6). 



XI 



EXERCISES XV 219 



(25.) If a-d = bc, then \/{a + \Jb + \Jc + \Jd) can always be expressed in 
the form {\/x + sfy){\JX + sJY). Show that this will be advantageous if 
a 2 - b and cr-c are perfect squares. 

(26.) If f/(a + \Jb) = z+\/y, where a, b, x, y are rational, and \Jb and 
s/y irrational, then f/{a - \Jb) = x - \/y. Hence show that, if a 2 -b = z 3 i 
where z is rational, and if x be such that 4x z -Zxz = a, then ^/(a + sjb) 
=x+»/(x*-z). 

(27.) Express in linear form <i/{99 - 35\/8). 

(28.) ,, „ 4/(395 + 93 V18). 

(29.) „ „ 4/(117V2 + 74V5). ' 

(30.) Show that 4/(90 + 34V7)- ^(90-Mv7) = 2>/7. 

(31.) If x= J/{p + q)+ </{p-q), and f-q-^r 3 , show that ar 3 -3ra;-2^ 
= 0. 

(32.) If py* + qi/ + r = 0, wherep, q, r, y are all rational, and y irrational, 
then p=0, q = Q, r=0. Hence show that, if x, y, z be all rational, and 

x , y , z all irrational, then neither of the equations x+y=z, x+y=z is 
possible. 

(33.) Find, by the full use of the ordinary rule, the value of \/10 to 5 
places of decimals ; and find as many more figures as you can by division 
alone. Use the value of \/10 thus found to obtain \/ - 004. 

Extract the square root of the following : — 

(34.) (yz + zx + xy) 2 -4xyz(z + x). 

(35. ) 25a; 2 + 9r/ 2 + z 2 + 6yz - lOzx - SOxy. 

(36.) 9ar l + 24ar 3 +10a: 2 -8a;+l. (37.) af 4 - 4ar 3 + 2a: 2 +4a; + l. 

(38.) 4x i -l2x i y + 25xY-2ixy 3 + l6y i . 

(39.) ar s -6ar 4 + 4ar 3 + 9a; 2 -12a; + 4. 

(40. ) 4a; 6 - 12a* + 5ar* + 22x? - 23a; 2 - 8a; +16. 

(41. ) 27(;a + qf(p" + q 2 ) 2 - 2( 2 r + ipq + q 2 ?. 

(42.) a;" 3 -2a:Va; + 3a;-2v'a;+l- 

(43.) Extract the cube root of 

8a; 9 - 1 2a; 8 + 6a; 7 - 37a; 6 + 36;^ - 9a,- 1 + 54a- 3 - 27a; 2 - 27. 

(44.) Extract the cube root of 

18(p 3 +p 2 q +pq 2 + y 5 ) ±2 V3(5/ + 3p 2 q - Zpq 2 - 5? 3 ). 

(45.) Show that X can be determined so that a 4 + 6a? + 7a; 2 - 6a; + X shall be 
an exact square. 

(46.) Find a, b, c, so that x R - 8z s + ax 4 + bx* + ex 2 - Ux + i shall be an 
exact square. 

(47.) If ax 4 + bx 3 + ca- 2 be subtracted from (x 2 + 2a- + 4) 3 the remainder is an 
exact square ; find a, b, c. 

(48. ) If a^ + ax 5 + tar 4 + car 3 + dx 2 + ex +/ be an exact square, show mat 
d = ^\« 4 - %a 2 b + \ b 2 + \ac, 
e = - ^a 5 + \a?b - \a 2 c - \ab 2 + \be, 



-< 



220 EXERCISES XV 



CHAP. XI 



And that the square root is 

ar* + iaa? + ( - |«. 2 + \b)x + (&a? -\db+ |c). 

(49.) 4a5 6 +12.r i + 5,r 4 -2.*: :, are the first four terms of an exact square ; find 
the remaining three terms. 

(50.) If x^ + Sdx^ + ex^fx^ + gx^ + hx + P be a perfect cube, find its cube 
root ; and determine the coefficients e,f, g, h, in terms of d and k. 

(51.) Show that 

b%a -b)(c -b){(a- 6) 2 + (c-6) 2 } - ffJM^ + O + **(«•- & + c) 
is an exact cube. 

(52.) Express \/{l+x + a?+a?+. . . ad »} in the form a + bx + cx- + 
... as far as the 4th power of x. To how many terms does the square of 
your result agree with 1 + x + x 2 + x 3 +. . .? 

(53.) Express, by means of the ordinary rule for extracting the square 
root, xAl ~ ■*') "s an ascending series of integral powers of x, as far as the 
4 th power. 

(54.) Express \J(x+l) as a descending series of powers of x, calculating 
six terms of the series. 

(55.) Show that Lambert's theorem (chap, ix., § 9) can be used to find 
rational approximations to surd numbers. Apply it to show that \/'2 = \ + 1/2 
- 1/2.5 + 1/2.5.7 - 1/2.5.7.197 approximately ; and estimate the error. 



CHAPTEE XII. 
Complex Numbers. 

ON THE FUNDAMENTAL NATURE OF COMPLEX NUMBERS. 

§ 1.] The attempt to make certain formulae for factorisation 
as general as possible has already shown us the necessity of in- 
troducing into algebra an imaginary unit i, having the property 
i" = — 1. It is obvious from its definition that i cannot be equal to 
any real quantity, for the squares of all real quantities are positive. 
The properties of i as a subject of operation are therefore to be 
deduced entirely from its definition, and from the general laws 
of algebra to which, like every other algebraical quantity, it 
must be subject. 

Since i must, when taken along with other algebraical quan- 
tities, obey all the laws of algebra, we may consider any real 
multiples of i, say yi and y'i, where y and y' are positive or 
negative, and we must have yi = iy, yi + y'i - (y + y')i = i(y + y'), 
and so on ; exactly as if i were a real quantity. 

By taking all real multiples of i from - go i to + go i, we have 
a continuous series of purely imaginary quantity, 

- oo i . . . - i . . . Oi . . . + i . . . + cc i I., 

whose unit is i, and which corresponds to the series of real 
quantity, 

-co .. .-1. . .0. . . + 1.. . + oo II., 

whose unit is 1. 

No quantity of the series I. (except Oi) can be equal to any 
quantity of the series II., for the square of any real multiple of ?', 
say yi, is y 2 i 2 = y 2 ( - 1) = -if, that is, is a negative quantity. 



222 FUNDAMENTAL CHAKACTER OF COMPLEX NUMBERS chap. 



Hence no purely imaginary quantity except Oi can be equal to a real 
quantity. Since Oi = ( + a - a)i = + (ai) - (ai) = 0, if the same 
laws are to apply to imaginary as to real quantity, Ave infer that 
Oi = 0. Hence is the middle value of the series of purely 
imaginary, just as it is of the series of real quantity ; it is, in 
fact, the only quantity common to the two series. 

Conversely, if yi = 0, we infer that y = 0. For, since yi = 0, 
yixyi = 0, that is, - y* = ; hence y = 0. 

§ 2.1 If we combine, by addition, any real quantity x with a 
purely imaginary quantity yi, there arises a mixed quantity x + yi, 
to which the name complex number is applied. 

We may consider the infinite series of complex numbers 
formed by giving all possible real values to x, and all possible 
real values to y. We thus have a doubly infinite series of com- 
plex quantity. The student should note at the outset this double 
character of complex quantity, on account of the contrast which 
thus arises between purely real or purely imaginary quantity 
on the one hand, and complex quantity on the other. Thus there 
is only one way of varying z continuously (without repetition of 
intermediate values) from - 1 to + 2, say, if z is to be always 
real ; and only one way of varying z in like manner from - i to 
+ 2i, if z is to be always purely imaginary. But there are an 
infinite number of ways of varying z continuously from -1+4 
to 2 + 3i, say, if there be no restriction upon the nature of z, 
except that it is to be a complex number. 

This will be best un- 
derstood if we adopt 
the diagrammatic method 
of representing complex 
numbers introduced by 
Argand. 

Let XOX', YOY' be 
two rectangular axes. We 
shall call XOX' the axis of 
Fig. i. real quantity, or z-axis ; 

and YOY' the axis of purely imaginary quantity, or y-axis. To 







xil argand's diagram 223 

represent any complex number x + yi we measure from (called 
the origin) a distance OM, containing x units of length, to the 
right or left according as x is positive or negative ; and we draw 
MP, containing y units of length, upwards or downwards accord- 
ing as y is positive or negative. The point P, or, as is more 
convenient from some points of view, the " radius vector " OP, is 
then said to represent the complex number x + yi. It is obvious 
that to every conceivable complex number there corresponds one 
and only one point in the plane of XX' and YY' ; and, conversely, 
that to every one of the doubly infinite series of points in that 
plane there corresponds one and only one complex number. P 
is often called the ajfixe of x + yi, or simply the " Point x + yi." 

If P lie on the axis XX', then y = 0, and the number x + yi is 
wholly real. If P lie on the axis YY', then x = 0, and x + yi is 
wholly imaginary. Now there is only one way of passing from 
any point on XX' to any other point, if we are not to leave the 
axis, namely, we must pass 
along the rr-axis ; and the 
same is true for the axis YY'. 
If, however, we are not re- 
stricted as to our path, there 
are an infinity of ways of 
passing from one point in the X' O 

plane of XX' and YY' to any 
other point in the same plane. Y 

If we draw any continuous Fi o- - 

curve whatever from P to Q, and imagine a point to travel along 
it from P to Q, the value of x corresponding to the moving point 
will vary continuously from the value OM to the value ON, and 
the value of y in like manner from MP to NQ. Hence there are 
as many ways of varying x + yi from OM + MPi to ON + NQi as 
there are ways of drawing a continuous curve from P to Q. 

Similar remarks apply when P and Q happen, as they may 
in particular cases, to be both on the x-axis, or both on the 
y-axis, provided that there is no restriction that the varying 
quantity shall be always real or always imaginary. There are 



Q 



PM QN 



224 RATIONAL OPERATIONS WITH COMPLEX NUMBERS CHAi'. 

many other properties of complex numbers, which are best under- 
stood by studying Argand's diagram, and we shall return to it 
again in this chapter. In the meantime, however, to prevent 
confusion in the mind of the reader, we shall confine ourselves 
for a little to purely analytical considerations. 

§ 3.] If x + yi = 0, then x = 0, y = 0.* For it follows from 
x + yi = that x = - yi. Hence, if y did not vanish, we should 
have a real quantity x equal to a purely imaginary quantity - yi, 
which is impossible. We must therefore have y = ; and conse- 
quently x— - Oi = 0. 

Cor. Hence if x + yi = x + y'i, then must x = x and y = y . 

For x + yi = x' + y'i gives, if we subtract x + y'i from both sides, 
(x — x) + (y — y')i = 0. 
Hence x — x' — 0, y — jf = 0, 

that is, x = x', y = y'. 

RATIONAL FUNCTIONS OF COMPLEX NUMBERS. 

§ 4.] We have seen that so long as we operate upon real 
quantities, provided we confine ourselves to the rational opera- 
tions — addition, subtraction, multiplication, and division, we 
reproduce real quantities and real quantities only. On the 
other hand, if we use the irrational operation of root extraction, 
it becomes necessary, if we are to keep up the generality of 
algebraical operations, to introduce the imaginary unit i. We are 
thus led to the consideration of complex numbers. The ques- 
tion now naturally presents itself, " If we operate, rationally or 
irrationally, in accordance with the general laws of algebra on 
quantities real or complex as now defined, shall we always re- 
produce quantities real or complex as now defined ; or may it 
happen that at some stage it will be necessary in the interest of 
algebraic generality to introduce some new kind of imaginary 
quantity not as yet imagined 1 " The answer to this question 
is that, so far at least as the algebraical operations of addition, 

* Here and hereafter in this chapter, when we write the form x + yi, it is 
understood that this denotes a complex number in its simplest form, so that 
x and y are real. 



XII RATIONAL FUNCTIONS OF COMPLEX NUMBERS 225 

subtraction, multiplication, division, and root extraction are 
concerned, no further extension of the conception of algebraic 
quantity is needed. It is, in fact, one of the main objects of the 
present chapter to prove that algebraic operations on complex 
numbers reproduce only complex numbers. 

§ 5.] The sum or product of any number of complex numbers, and 
the quotient of two complex numbers, may be expressed as a complex 
number. 

Suppose we have, say, three complex numbers, a;, + yj,, x 2 + y 2 i, 
x 3 + y 3 i, then 

(«i + ffii) + (x 2 + y 2 i) - (x 3 + y 3 i) = (.r, + x 2 - x 3 ) + (y, + y 2 - y 3 )i, 
by the laws of algebra as already established. 

But x l + x 2 - x 3 and y x + y 2 - y 3 are real, since x x , x 2 , x 3 , y x , y 2 , y 3 
are so. Hence (#, + x, 2 - x 3 ) + (y x +y 2 - y 3 )i is in the standard form 
of a complex number. The conclusion obviously holds, however 
many terms there may be in the algebraic sum. 

Again, consider the product of two complex numbers, x x + yj, 
and x 2 + yj. We have, by the law of distribution, 

(«i + yd) («» + yd) = x&i + y<ij£ + w + <w- 

Hence, bearing in mind the definition of i, we have 

(a + yd) (- r 2 + yj) = to - ysj») + to + «Wi)i, 

which proves that the product of two complex numbers can be 
expressed as a complex number. 

To prove the proposition for a product of three complex 
numbers, say for 

P = (.r, + yj) (x, + yj) (x 3 + y 3 i), 

Me have merely to apply the law of association, and write 

P = {(»i + yj) («s + yd)} («3 + yd). 

We have already shown that the function within the crooked 
brackets reduces to a complex number ; hence P is the product 
of two complex numbers. Hence, again, by what we proved 
above, P reduces to a complex number. In this way we can 
extend the theorem to a product of any number of complex 
numbers. 

VOL. I Q 



226 RATIONAL FUNCTIONS OF COMPLEX NUMBERS chap. 

Lastty, consider the quotient of two complex numbers. We 
have 

Si + ffii _ («i + yj) («» - yJ) * 
x 3 + y 2 i (x 2 f - (y 2 if ' 

= (^gg + vm) - fay» - %2!ti)i 
a* + y* 

x x x 2 + y^/nX f^iV-i ~ 



2 

Xo t y^ / \ *^2 "■" 



— S~ ]' 

y 2 / 



This proves that the quotient of two complex numbers can 
always be reduced to a complex number. 

Cor. 1 . Since every rational function involves only the opera- 
tions of addition, subtraction, multiplication, and division, it 
follows from what has just been shown that every rational 
function of one or more complex numbers can be reduced to a com- 
plex number. 

Cor. 2. If <f>(x + yi) be any rational function of x + yi, having all 
its coefficients real, and if 

<j)(z + yi) = X + Yi, 
then 

<f>(x - yi) = X - Yi, 

X and Y being of course real. 

Cor. 3. Still more generally, if <£(», + yj, x 2 + y 2 i, . . . , x n + y n i) 
be any rational function of n complex numbers, having all its coefficients 
real, and if 

4>{x, + y t i, x 2 + y 2 i, . . . , x n + y n i) = X + Yi, 

then 

^{x.-y.i, x 2 -y 2 i, . . . , x n -y, l i) = X-Yi. 

Cor. 4. If all the coefficients of the integral function <j>(z) be 
reed, and if cj>(z) vanish when z = x + yi, then <f>(z) vanishes wlien 
z = x-yi. 

' Here we perform an operation which we might describe as "realising" 
the denominator ; it is analogous to the process of rationalising described in 
chap. x. 






xir EXAMPLES 227 

For, by Cor. 1, <f>(x + yi) = X + Yi where X and Y are real. 
Hence, if cf>(x + yi) = 0, we have X + Yi = 0. Hence, by § 3, X = 
and Y = 0. Therefore <j>(x - yi) = X - Yi = - 0/ = 0. 

Cor. 5. If all the coefficients of the integral function <£(2„ 
z s , . . . , z n ) be real, and if the function vanish when z u z.,, . . . , z n 
are equal to x x + yj,, x 2 + y 2 i, . . . , x n + y n i respectively, then the 
function will also vanish when s u z. 2 , . . . , z n are equal to x x - y x i, 
x s - yi 1 , ' • ■ j x n - Vni respectively. 

Example 1. 

3(3 + 20 - 2(2 - 3i) + (6 + 80 = 9 + 6i- 4 + 6J+6+ Si, 

til + 2Qi. 
Example 2. 

(2 + 30 (2 - 50 (3 + 20 = (2 - 50 (6 - 6 + 9i + 40, 
= (2 -5013;', 
= 26i + 65, 
= 65 + 26i. 
Example 3. 

(b + c - ai) [c + a - bi) (a + b- ci) 

= {U(b + c)-2bc(b + c)\ + {abc-Za{a + b){a + c)}i, 

- 2abc + {abc - 2a 3 - 2a 2 (6 + c) - Zabc} i, 

— 2abc - {a 3 + b 3 + c 3 + (b + c) [c + a) {a + b)} i. 
Example 4. 

To show that the values of the powers of i recur in a cycle of 4. 

We have i=i, i 2 =-l, i 3 = i 2 xi= -i, i i =(i~)-=+l ! 

i s =i 4 x i = i, i 6 = i 4 x i-= - 1, i 7 = i 4 x i 3 — - i, i 8 = { 4 x i 4 = + 1 ; 

and, in general, 

i4n+l = i } i*H-2=-l, $*"+•= -t, l' 4 ("+ 1 )=+l. 

Example 5. 

3 + 5i _ (3 + 5Q(2 + 3Q _ 6-15 + 19i_ 9 19. 

2-3i~ 4 + 9 13 13 + 13*' 

Example 6. 

(x + yi)" = x" + „Ci x n ~Hyi) + „C 2 8»- 2 (yi) a + • • • . 

=(.?;«-, a z"-v 2 +»c.iz"-y-. • ■) 

+ (nC 1 x»- 1 y- n C 3 x n - 3 y 3 + „C 5 x n -hj 5 -. . .)*■ 

In particular 

(jb + yt)*={a? - Sx-y° + 2/ 4 ) + (4afy - to*/ 8 )*. 
Example 7. 

If <p(z) = Z " ' 



then «/>(2 + 30 : 



'z- + z + V 
(2 + 3Q 2 -(2 + 3Q + l 

: "(2 + 30 2 + (2 + 30 + l' 
-5 + 12t-2-3i + l 
-5 + 12i + 2 + 3i+l' 



228 CONJUGATE COMPLEX NUMBERS CHAP. 

_j-6 + 9i _ 3(-2 + 3 Q(-2-15Q 
-2 + 15i~ 229 

= 2§9 {4 + 45_6 ' +30 ^' 
_U7 _72 . 
229 229 *' 
From this we infer that 

,., ... 147 72 . 
^ (2 - 3l) = 229-229 i; 
a conclusion which the student should verify by direct calculation. 

CONJUGATE COMPLEX NUMBERS, NORMS, AND MODULI. 

§ 6.] Two complex numbers which differ only in the sign 
of their imaginary part are said to be conjugate. Thus — 3 — 2i 
and - 3 + 2t are conjugate, so are - ii and + ii ; and, generally, 
x + yi and x - yi. 

Using this nomenclature we may enunciate Cor. 3 of § 5 
as follows : — 

If the coefficients of the rational function <£ be real, then 
the values of 

4fa + yj, x 2 + y.j, . . ., x n + y n i) 

and </>(' r i-*/A «»-yA ■ • •, - r n ~ ?/n0> 

where the values of the variables are conjugate, are conjugate 

complex numbers. 

The reader will readily establish the following : — 

The sum and the product of two conjugate complex numbers are real. 

Conversely, if both the sum and the product of two complex numbers 
be real, then either both are real or they are conjugate. 

§ 7.] By the modulus of the complex number x + yi is meant 
+ J(x 2 + y 2 ). This is usually denoted by | x + yi \* 

It is obvious that a complex number and its conjugate have the 
same modulus; and thai this modulus is the positive value of the 
square root of their product. 

Examples. 

|-3 + 4i|=+ N /{(-3) 2 M 2 }=5. 
|-3-4;|= + v /{(-3) 2 + (-4) 2 }=5. 

|l+i|= + v /(l 2 +l 2 )=v/2. 

* Formerly by mod {x + yi). 



XII 



MODULI 229 



It should be noticed that if y = 0, that is, if the complex 
number be wholly real, then the modulus reduces to + Jx 2 , 
which is simply the value of x taken with the positive sign, or, 
say, the numerical value of x. For example, | - 3 | = + \/( - 3) 2 
= + 3. For this reason Continental writers frequently use | x | 
where x is a real quantity, as an abbreviation for " the numerical 
value of SB." We shall occasionally make use of this convenient 
contraction. 

For reasons that will be understood by referring once more to § 2, the 
ordinary algebraical ideas of greater and less which apply to real quantities 
cannot be attached to complex numbers. The reader will, however, find that 
for many purposes the measure of the " magnitude " of a complex number is 
its modulus. We cannot at the present stage explain precisely how "magni- 
tude" is here to be understood, but we may remark that, in Argand's diagram, 
the representative points of all complex numbers whose moduli are less than 
p lie within a circle whose centre is at the origin and whose radius is p. 

§ 8.] If a complex number vanish its modulus vanishes ; and, 
conversely, if the modulus vanish the complex number vanishes. 

For if x + yi = 0, then by § 3, x = and y = 0. Hence 
*J(x* + f) = 0. ' 

Again, if J(x" + y 2 ) = 0, then x 2 + y 2 - ; but, since both x 
and y are real, both x 2 and y 2 are positive, hence their sum cannot 
be zero unless each be zero. Therefore x = and y = 0. 

If two complex numbers are equal their moduli are equal ; but the 
converse is not true. 

For, if x + yi = x' + y'i, then, by § 3, x = x\ y = y. Hence 
J(x 2 + f) = V(x' 3 + y' 2 ). 

On the other hand, it does not follow from J(x 2 + y 2 ) 
= J(x' 2 + y' 2 ) that x = x', y=- y. Hence the converse is not true. 
§ 9.] Provided all the coefficients in <£(x' + yi) be real, we have 
seen (§ 5, Cor. 2) that if 

<p(x + yi) = X + Yi, 
where X and Y are real, then 

c/,(.c - yi) = X - Yi. 
Now \cf>(x + yi)\= v/(X 2 + Y 2 )= J{(X + Yi)(X-Yi)}, 

= J{<f>(x + yi) <}>(x-yi)}, 



230 MODULI 



CHAP. 



In like manner it follows from § 5, Cor. 3, that 

I <t>fa + yj, %a + thh • ■ ■, x n + yd) | 

= I <M«i - Vih x, - yj, . . ., x n - y n i) \ 
= + s /[<i>{x x + y x i, x 2 + y 2 i, . . ., x n + yj) 

x <f>( Xl - y,i, x 2 - yj, . . ., x n - y n i)] (2). 

The theorems expressed by (1) and (2) are very useful in 
practice, as will be seen in the examples worked below. 

It should be observed that (1) contains certain remarkable 
particular cases. For example, 

I («! + yj) fa + yd) • • • fa + yj) I 

= + ^[(jc, + yj) (x 2 + yj) . . . fa + y n i) 

* ( g i - yj) fa - y a i) • • • fa - y j)\ 

= + Vfa + y?) fa + y 2 ) . . . (x n 2 + y n 2 ), 

= I fc + yj) I x I fa + yd) I x • • • x I fa + yj) I (3). 

In other words, the modulus of the 'product of n complex numbers 
is equal to the product of their moduli. 
Also 



g, + yj> 

x 2 + y 2 i 



= \/fa* + y. 2 ) _ ( g. + yj 1 

" \/(x 2 +y 2 2 ) \x 2 + y 2 i\ (4). 



In other words, the modulus of the quotient of two complex numbers 
is the quotient of their moduli. 

§ 10.] The reader should establish the results (3) and (4) of 
last paragraph directly. 

It may be noted that we are led to the following identities : — 

fa + y?) fa + y.f) = fac, - yfo)' + {x,y 2 + ay/,) 2 . 

If we give to x„ y u x 2 , y 2 positive integral values, this gives us 
the proposition that the product of hoo integers, each of which is the 
sum of two square integers, is itself the sum of two square integers ; 
and the formula indicates how one pair of values of the two 
integers last mentioned can be found. 

Also 

fa 2 + y?) fa 2 + y 2 ) fa + y.*) = zflp, - x,yjj 3 - xjj,y, - x^y,)* 

+ (yiWe + y&&i + y<FiX e - 2W 3 ) 2 - 



XII 



MODULI 231 



This shows that the product of three sums of two integral squares 
is the sum of two integral squares, and shows one way at least of 
finding the two last-mentioned integers. 

Similar results may of course be obtained for a product of 
any number of factors. 

Example 1. 

Find the modulus of (2 + 3i) (3 - 2i) (6 - 4i). 
|(2 + 8i)(3-2i)(6-4t)| 

=|(2+3*)[x|(3-2*)[x|(6-4i)| > 

= s f\lZ)x ^(13) x ^(52), 

=26^/(13). 
Example 2. 
Find the modulus of ( ^2 + ^3) (-s/3 + » V 5 )/(V2 + W&)- 

( v /2 + rV3)(V3 + tV5) | 

/r /( v / 2 + i v /3)(V3 + ^ v /5) \ r (N/2-W3)( x /3-i\/5) \-| 
= VLl" >/2 + iV5 J l V2-W5 <U' 

- V[ (?± l^]= 7(f)- 

Example 3. 

Find the modulus of {(/3 + 7) + (/3 - 7)1} {(7 + a) + (7 - a) i} {(a + /3) + (a - /3)i} . 
The modulus is J({((H-yT + (P-y)-} {(y + *? + {y-a.f\ {(a + (3f + (a- p)*}) 
= v/{8(^ + 7 2 )(7 2 +a 2 )(a 2 + ^)}. 
Example 4. 

To represent 26 x 20 x 34 as the sum of two integral squares. 
Using the formula of § 10 we have 
26 x 20 x 34 = (l 2 +5 2 ) (2 2 + 4 2 ) (3 2 + 5 2 ), 

= (1.2.3 -1.4.5- 2.5.5 -3.5.4) 2 + (5.2.3 + 4.3.1 + 5.1.2 -5.4.5) 2 , 
= 124 2 + 48 2 . 

§11.] The modulus of the sum of n complex numbers is never 
greater than the sum of their moduli, and is in general less. 

This may be established directly ; but an intuitive proof will 
be obtained immediately from Argand's diagram. 

§ 12.] We have seen already that, when PQ = 0, then either 
P = or Q = 0, provided P and Q be real quantities. It is 
natural now to inquire whether the same will hold if P and Q 
be complex numbers. 

If P and Q be complex numbers then PQ is a complex 
number. Also, since PQ = 0, by § 8, | PQ | = 0. But | PQ | 
= | p i*IQI» b y § 10 - Hence |P| x IQI = 0. Now |P| and 



232 argand's DIAGRAM chap. 

I Q | are both real, hence either | P | = or | Q | = 0. Hence, by 
§ 8, either P = or Q = 0. 

We conclude, therefore, that if PQ = 0, then either P = or 
Q = 0, whether P and Q be real quantities or complex numbers. 

discussion of complex numbers by means of 
argand's diagram. 

§ 13.] Returning now to Argand's diagram, let us consider 
the complex number x + yi, which is represented by the radius 
vector OP (Fig. 1). Let OP, which is regarded as a signless 
magnitude, or, what comes to the same thing, as always having 
the positive sign, be denoted by r, and let the angle XOP, 
measured counter-clock-wise, be denoted by 6. 

We have seen that if OP represent x + yi, then x and y are 
the projections of r on X'OX and Y'OY respectively. Hence 
we have, by the geometrical definitions of cos 6 and sin 9, 

r= + V(r + /) (1), 

x/r-cos$, y/r = sin d, (2). 

From (1) it appears that r, that is OP, is the modulus of the 
complex number. The equations (2) uniquely determine the 
angle 6, provided we restrict it to be less than 2ir, and agree 
that it is always to be measured counter-clock- wise from OX.* 
We call 6 the amplitude of the complex number. It follows 
from (2) that every complex number can be expressed in terms 
of its modulus and amplitude ; for we have 

x + yi = r(cos 6 + i sin 6) (3). 

This new form, which we may call the normal form, possesses 
many important advantages. 

* Sometimes it is more convenient to allow 8 to increase from - ir to + w ; 
that is, to suppose the radius OP to revolve counter-clock-wise from OX' to 
OX' again. In either way, the amplitude is uniquely determined when the 
coefficients x and y of the complex number are given, except in the case of a 
real negative number, where the amplitude apart from external considera- 
tions is obviously ambiguous. 



XII 



COMPOSITION OF VECTORS 



233 



Since two conjugate complex numbers differ only in the sign 
of the coefficient of i, it follows 
that the radii vectores which re- 
present them are the images of 
each other in the axis of x (Fig. 3). 
Hence two such have the same 
modulus, as we have already shown 
analytically ; and, if the amplitude 
of the one be 6, the amplitude of 
the other will be 2tt - 6. In other 
words, the amplitudes of two con- 
jugate complex numbers are con- 
jugate angles. 




Example. 



- 1 - i= V 2 ("^2-^') = V2 ( C0S T + l ' sin T)- 



§ 14.] If OP, OQ' (Fig. 4) represent the complex numbers 

Y 

Q -,Q 




x + yi and x' + y'i, and if 
— > 
PQ be drawn parallel and equal 



to OQ', then OQ will represent 
the sum of x + yi and x' + y'i. 

For the projection of OQ on the 
z-axis is the algebraic sum of the 



Yi 

I'lii. 4. 



projections of OP and PQ on the 
— > 
same axis, that is to say, the projection of OQ on the a-axis is 

— > 
x + x. Also the projection of OQ on the y-axis is, by the same 
reasoning, y + y'. Hence OQ represents the complex number 
(x + x) + (y + y')i, which is equal to (x + yi) + (x' + y'i). 

By similar reasoning we may show that if OP„ OP 2 , OP 3 , OP 4 , 

OP 5 , say, represent five complex numbers, and if P,Q, be parallel 



234 



| 2i + Z 3 + 



+ Zn\< 



+ 



+ \Zm 



CHAP. 



and equal to 0P 2 , Q 2 Q 3 , parallel and equal to 0P 3 , and so on, then 
0Q 5 represents the complex number which is the sum of the 
complex numbers represented by OP,, OP„, OP 3 , OP 4 , OP 5 . 

This is precisely what is known as the polygon law for com- 
pounding vectors. Since OQ 5 is never greater than the peri- 
meter OPjQ.jQaQ.jQs, and is in general less, Fig. 5 gives us an 
intuitive geometrical proof that the modulus of a sum of complex 
numbers is in general less than the sum of their moduli. It is 
equally obvious from Fig. 4 that the modulus of the sum of two com- 







Fig. 5. 



plex numbers is in general greater than the difference of the moduli. 
" Sum of complex numbers " in these theorems means, of course, 
algebraic sum. 

§ 15.] If we employ the normal form for a complex 
number, and work out the product of two complex numbers, 
r,(cos 0, + i sin : ) and r t (cos 9 2 + i sin 2 ), we have 

r^cos 0, + i sin 0])r»(cos 9 2 + i sin B 2 ) 

= r,r a {(cos 0, cos 6. 2 - sin 0, sin $,) + (sin 0, cos 6. 2 + cos 9^ sin 9 2 )i), 
= r,r 2 {cos (0, + 2 ) + i sin (0, + 9. 2 )} (1). 

We thus prove that the product of two complex numbers is a 
complex number, whose modulus r{r 2 is the product of the moduli 



xii demoivre's theorem 235 

of the two numbers, a result already established ; and we have 
the new theorem that the amplitude of the product is, to a multiple 
of 2tt, the sum of the amplitudes of the factors. For we can always 
find an angle <£ lying between and 2rr, such that cos <f> = 
cos (0, + 2 ) and sin (f> - sin {d x + 2 ), and we then have 6 Y + 6 3 
= 2mr + cf>. 

This last result is clearly general ; for, if we multiply both 
sides of (1) by an additional factor, r 3 (cos 3 +• i sin 3 ), we have 

r^cos Q x + i sin 1 )r 2 (cos 0„ + i sin 2 )r 3 (cos 3 + i sin 3 ) 

= r^cos (0, + 6 2 ) + i sin (0 t + 6 2 )} r 3 (cos 6 3 + i sin $ 3 ), 
= r t r 2 r 3 {cos ($ x + d 2 + d 3 ) + i sin (0 1 + 6 2 + 3 )}, 

by the case already proved, 

= ?W 3 {cos (0, + B 2 + 9 3 ) + i sin (0 : + 2 + 3 )}. 
Proceeding in this way we ultimately prove that 

r,(cos $! + i sin tf^r^cos 2 + i sin 2 ) . . . r n (cos 9 n + i sin 6 n ) 

= r{r 2 . . . ?- n {cos(#! + 2 + . . . + 9 n ) + i sin(#, + 6 2 + . . . + 6 n )} (2). 

This result may be expressed in words thus — 

The pvduct of n complex numbers is a complex number whose 
modulus is the product of the moduli, and whose amplitude is, to a 
multiple of 2ir, the sum of the amplitudes of the n complex numbers. 

If we put r l = r 2 = . . . = r n , each = 1 say, we have 

(cos #! + i sin 0j) (cos # 3 + i sin 6 2 ) . . . (cos 6 n + i sin d n ) 

= cos (6, + 6 2 + . . . + tl ) + i sin (0, + 2 + . . . + 6 n ) (3). 

This is the most general form of what is known as Demoivre's 
Theorem. 

If we put 0, = 6 2 - . . . - 6 n , each = 0, then (3) becomes 
(cos + i sin 0) n = cos nd + i sin nd (4), 

which is the usual form of Demoivre's Theorem.* It is an analy- 
tical result of the highest importance, as we shall see presently. 

* This theorem was first given in Demoivre's Miscellanea Analytica 
(Lond. 1730), p. 1, in the form 

^ = i\ / {^ + \/(^-l)}+i/\/{^+v'(^-l)}, where ar = cos0, f = cos«0. 



236 QUOTIENT OF COMPLEX NUMBERS CHAP. 

Since cos - i sin - cos (2r - 0) + i sin (2ir - 0), 

we have, by (3) and (4), 

lT(cos 0, - i sin 0,) = cos (20,) - i sin (20,) (3') ; 

and (cos - i sin 9) n = cos n0 - i sin «0 (4'). 

The theorem for a quotient corresponding to (1) may be 
obtained thus — 

r(cos + i sin 0) _ r(cos + i sin ) (cos 0' - i sin 0') 
/(cos 0' + i sin 0') = r'(cos 2 0' + sin a 0') ' 

= - {(cos cos 0' + sin sin 0') 

+ (sin cos 0' - cos sin 0')i}, 

= -, {cos (0 - 0) + * sin (0 - $')} (5). 

Hence $e quotient of two complex numbers is a complex number 
ichose modulus is the quotient of the moduli, and whose amplitude is 
to a multiple of 2ir the difference of the amplitude of the two complex 
numbers. 

IRRATIONAL OPERATIONS WITH COMPLEX NUMBERS. 

§ 16.] Since every irrational algebraical function involves 
only root extraction in addition to the four rational operations, 
and since we have shown that rational operations with complex 
numbers reproduce complex numbers and such only, if we could 
prove that the nth. root of a complex number has for its value, 
or values, a complex number, or complex numbers and such only, 
then we should have established that all algebraical operations 
with complex numbers reproduce complex numbers and such 
only. 

The chief means of arriving at this result is Demoivre's 
Theorem ; but, before resorting to this powerful analytical engine, 
we shall show how to treat the particular case of the square root 
without its aid. 

Let us suppose that 

J(x + yi) = X + Yi (1). 



xii SQUARE ROOTS OP A COMPLEX NUMBER 237 

Then x + yi = X 2 -Y 2 +2XYi. 

Hence, since X and Y are real, we must have, by § 3, 

X 2 -Y 2 = .x (2), 

2XY = y (3). 

Squaring both sides of (2) and (3), and adding, we deduce 

(X 2 + Y 2 ) 2 = x 2 + f • 
whence, since X 2 + Y 2 is necessarily positive, we deduce 

X 2 + Y 2 = + V'(/ + /) (4). 

From (2) and (4), by addition, we derive 

2X 2 = + >J(x 2 + y 2 ) + x, 

u. v ■ Y* + \ (x 2 + y 2 ) + x 
that is, A = - -• 

We therefore have X= ± y j 2 ) ^' 

In like manner we derive from (2) and (4), by subtraction, &c, 

T .;y{±i^b'} (6 ). 

Since x 2 + y 2 is numerically greater than x 1 , + ^(x 2 + y 2 ) is 
numerically greater than x. Hence the quantities under the sign 
of the square root in (5) and (6) are both real and positive. The 
values of X and Y assigned by these equations are therefore real. 

Since 2XY = y, like signs must be taken in (5) and (6), or 
unlike signs, according as y is positive or negative. 

We thus have finally 

(7). 

(8), 
if y be negative. 

Example 1. 

Express \/(8 + 6i) as a complex number. 

Let ^{8 + 6i) = x + yi. 

Then a~-i/ 2 = 8, 2xy=6. 

Hence (* 2 + ?/ 2 ) 2 = 64 + 36 = 100 

Hence x ! + if=\0; 

and x~-y 2 = 8; 



\ 



if y be positive ; 

= ± 




238 EXAMPLES chap. 

therefore 2x- 2 = 18, 2y° = 2. 

Hence x=±3, y==hl. 

Since 2a*y = 6, we must have either x — +3 and y=+l, or ar=-3 and 
y=-l. 

Finally, therefore, we have 

V(8 + 6i)=±(3 + i); 
the correctness of which can be immediately verified by squaring. 
Example 2. 

v<3-7o=±{y(»>^/(»)}. 

Example 3. 

Express \/( + i) and \/( ~ *) as complex numbers. 
Let \/{+i)=3:+#*; 

then i = ar - 2/ 2 + 2a;?/i. 

Hence a.- 2 -2/ 2 = (a), 2asy=l (0). 

From (a) we have (x + y)(x-y) = ; that is, either ?/= -a or y = x. The 
former alternative is inconsistent with (/3) ; hence the latter must be accepted. 
We then have, from (/3), 2a?=l, whence £ 2 = l/2 and a;=±l/\/2. Since 
!/=£, we have, finally, 

V+*=±^" (7)- 

Similarly we show that 

Example 4. 

To express the 4th roots of + 1 and - 1 as complex numbers. 

^/ + 1 = V(V + 1) = V±l = V + l or «s/-l = ±l or dbi. 
Hence we obtain four 4th roots of + 1, namely, +1, - 1, + i, - i. 

Again v - 1 = V(V - l)= V^- 

Hence, by Example 3, 

§ 17.] We now proceed to the general case of the rcth root 
of any complex number, r(cos 6 + i sin 6). 

Since r is a positive number, Vr has (see chap, x., § 2) 
one real positive value, which we may denote by r 1/n . 

Consider the n complex numbers — 

r Vn [ cos - + i sin - ) (1), 

\ n ii/ x 

2-rr+e . . 2tt + 0N 

A In I 



( 2tt + 9 . . 2tt + 9\ /t . x 

cos h i sin ) (2), 

\ n n J 



xii MTH BOOTS OF ANY COMPLEX NUMBER 239 

.. / Att + . . 4- + 0\ ... 

i ' " cos + t sin ) (o), 

\ n n / 



2stt + 6 . . 2sir + 8\ , . . 

r 1 '" cos - + * sin - - ) (s + 1 ), 

n n J 



r Vn ( 



2(>i-l)- + e . . 2(n- l)ir+0\ . , 

cos — + % sin — hi). 

n n / 



i r Vnf 



No two of these are equal, since the amplitudes of any two 
differ by less than 2ir. The ?ith power of any one of them is 
r(cos 6 + i sin 6) ; for take the (s + l)th, for example, and we have 

2$ir + 6 . . 2stt + 6 
cos + ^ sm 

n a 

( ^.\ n ( 2STT+0 . . 2S7T+8\ n 

= I r xjn ) [ cos + i sin 

\ / \ n n 

( 2sw + 6 . . 2S7T + 8 

= r [ cos n + i sin n 

\ n n 

by Demoivre's Theorem, 

= r(cos (25tt + 6) + i sin (2stt + 6) ), 

= r(cos + i sin 6). 

Hence the complex numbers (1), (2), . . ., (n) are n different 

nth roots of r(cos 6 + i sin 6). 

We cannot, by giving values to 5 exceeding n— 1, obtain 
any new values of the ?tth root, for the values of the series 
(1), (2), . . ., (h) repeat, owing to the periodicity of the 
trigonometrical functions involved. We have, for example, 
r ,/H (cos.(2?i7r + 6)1 n + i sin.(2?i- + 6)jn) = r ln (cos.6!n + i sin. 6jn) ; 
and so on. 

We can, in fact, prove that there cannot be more than n 
values of the ?ith root. Let us denote the complex number 
r(cos + i sin 6) by a, for shortness ; and let z stand for any 
mth root of a. Then must z n = a, and therefore z n - a = 0. 
Hence every »th root of a, when substituted for z in z n - a, 
causes this integral function of z to vanish. Hence, if «„ z s , . . ., 



240 JITH ROOTS OF ± 1 CHAP. 

~ s be s nth roots of a, z-z u z-z 2 > • • ■, % — %* will all be 
factors of z n - a. Now z 11 - a is of the ?(th degree in z, and 
cannot have more than n factors (see chap, v., § 10). Hence s 
cannot exceed n ; that is to say, there cannot be more than n 
nth roots of a. 

We conclude therefore that every complex number has n nth 
roots and no more ; and each of these nth roots can be expressed as a 
complex number. 

Cor. 1. Since every real number is merely a complex num- 
ber whose imaginary part vanishes, it follows that every real 
number, whether positive or negative, has n nth roots and no more, 
each of which is expressible as a complex number. 

Cor. 2. The imaginary nth roots of any real number can be 
arranged in conjugate pairs. For we have seen that, if x + yi be 
any nth root of a, then (x + yi) n - a = 0. Hence, if a be real (but 
not otherwise), it follows, by § 5, Cor. 4, that (x - yi) n - a = ; 
that is, x - yi is also an nth root of a. 

N.B. — This does not hold for the roots of a complex number 
generally. 

§ IS.] Every real positive quantity can be written in the 
form 

r(cos + i sin 0) (A) ; 

and every real negative quantity in the form 

r(cos 7r + i sin 7r) (B) ; 

where r is a real positive quantity. Hence, if we know the n 
nth roots of cos + ism 0, that is, of +1, and the n nth roots of 
cos 7r + isin7r, that is, of - 1, the problem of finding the n nt\i 
roots of any real quantity, whether positive or negative, is 
reduced to finding the real positive value of the nth root of a 
real positive quantity r (see chap, xi., § 15). 

By means of the nth roots of + 1 we can, therefore, com- 
pletely fill the lacuna left in chap, x., § 2. In addition to their 
use in this respect, the nth. roots of ± 1 play an exceedingly im- 



XII WTH ROOTS OF ±1 241 

portant part in the theory of equations, and in higher algebra 
generally. We therefore give their fundamental properties here, 
leaving the student to extend his knowledge of this part of 
algebra as he finds need for it. 

Putting r-1 and = in 1, . . ., n of § 17, and remem- 
bering that V ln = 1, we obtain for the n nth roots of + 1, 

cos + i sin 0, cos — + % sin — , , 

n n 

2(/i-1)tt . . 2(?i-1)tt 

cos — J - — v i sin — — • 

n n 

Putting r=l, 6 = tt, we obtain for the nth roots of - l, 

TV . . 7T StT . . StT 

cos - + i sin -, cos — + i sin — , ...,.., 
n n n n 



(2w- 1W . . (271-1W 

cos — + i sin • 

n n 

Cor. 1. Since cos . 2(?i - l)ir/w = cos . 27r/«, sin . 2(n - l)irjn = 

- sin . iTrjn; cos . 2(w - 2)7r , ?i = cos . iir/n, sin . 2(u - 2)7rjn = 

- sin . 47r/», and so on, we can arrange the roots of +1 as 
follows: — 

nth roots of + 1, n even, =2m say, 

2tt . . 2tt 4ir . . 4-7T 

+ 1, cos — ± x sin — , cos — ± % sin — , . . ., 
n n n n 

^ a 2(^-iy 2(>u-lV . 

cos ±isin- , -1 (L) ; 

7ith roots of + 1, n odd, = 2m + 1 say, 

, 2ir . . 2tt 4:tt . . 4tt 

+ 1, cos — ±zsin — , cos — ±tsin — , . . ., 
n n n n 

2nnr . . 2nnr tir .. 

cos ± i sin (D). 

n n v ' 

vol. I R 



242 WTH ROOTS OF ± 1 chap. 

Similarly we can arrange the roots of - 1 as follows: — 
reth roots of - 1, « even, = 2m say, 

cos - ± i sm -, cos — ± i sin — , . . . , 
n n n n 

(2m- 1W . . (2m- l>r 

cos '— ± i sin ^- (E) : 

n n v ' 

?i th roots of - 1, n odd, = 2m + 1 say, 

Ti . . 7T 37T . . 3~ 

cos - ± i sin -, cos — ± i sin — , .... 
n n n n 

(2m- 1W L . . (2m- 1W 

cos v '— ± t sin v — — . - 1 (F). 

11 n v ' 

From (C), (D), (E), (F) we see, in accordance with chap, x., 
§ 2, that the ??th root of + 1 has one real value if n be odd, and 
two real values if n be even ; and that the »th root of - 1 has 
one real value if n be odd, and no real value if n be even. 

We have also a verification of the theorem of § 17, Cor. 2, 
that the imaginary roots of a real quantity consist of a set of 
pairs of conjugate complex numbers. 

Cor. 2. The first of the imaginary roots of + 1 in the series 
(1), . . .,(»), namely, cos. 2wjn + i sin. 2ir/n, is called a primitive* 

rtth root of + 1. Let us denote this root by w. 
Then since, by Demoivre's Theorem, 

•2tt . . 2ttY -2stt . . 2S7T 

cos — + i sin — - cos — + % sm — , 
n n / it n 

and, in particular, 



1% n 



it 



2tt . . 2ir\ 
(o n = ( cos — + i sin — I == cos 2-n- + i sin 2tt, 



1, 



* By a primitive imaginary ?tth root of + 1 in general is meant an ?;th root 
which is not also a root of lower order. For example, cos.27r/3 + isin.27r/3 
is a 6th root of +1, but it is also a cube root of +1, therefore cos.27r/3-f 
»sin.2r/3 is not a primitive 6th root of +1. It is obvious that cos.27r/« + 
■i sir\.2ir/n is a primitive «th root ; but there are in general others, and it may 
be shown that any one of these has the property of Cor. 2. 



XII FACTORISATION OF x n ± A 243 

we see that, if <o be a primitive imaginary nth root of + 1, then the 
n nth roots of + 1 are 

to , u) , u) , . . . , w n (Gr). 

Similarly, i/" w' - cos.ir/n + i sin.7r/?t, w/tic/i we may call a primitive 
imaginary nth root of - I, then the n nth roots of - 1 are 



,./2n-l /fj\ 



OJ 



§ 19.] The results of last paragraph, taken in conjunction 
with the remainder theorem (see chap, v., § 15), show that 

Every binomial integral function, x n ± A, can be resolved into n 
factors of the 1st degree, whose coefficients may or may not be wholly 
real, or into at most two real factors of the 1st degree, and a number 
of real factors of the 2nd degree* 

Take, for example, x 2m - a 2m . This function vanishes whenever we sub- 
stitute for x any 2»ith root of a 2m ; that is, it vanishes whenever x has any 
of the values aw, aw 2 , . . . , aw 2 '" where w stands for a primitive 2?)ith root 
of +1. 

Hence the resolution into linear factors is given by 

To obtain the resolution into real factors, we observe that, corresponding 
to the roots +a and —a, we have the factors x-a, x + a; and that, corre- 
sponding to the roots a(cos.STr/m±i sin .sir/m), we have the factors 



(St . . sir\ / 
x-a cos ai sin — II 
m m J \ 



sir . . sw\ 

x-a cos \-ai sin — I, 

vi m J 



( SW\" „ . S7T 

= I x - a cos — ) + a- sin" — , 
\ to / TO 

o Sir „ 

=x- - 2ax cos h a~. 

m 

Hence the resolution into real factors is given by 

x 2m - a 2m =(x -a)(x + a) (x 2 - lax cos — + a 2 ) lx 2 - lax cos ha 2 )... 

m m 

We may treat x 2m + a im , x 2m + l - a~ m+1 , and x 2m + 1 + a 2m+l in a similar way. 

Example 1. 

To find the cube roots of + 1 and - 1. We have +1 =1 {cos O + i sin 0}. 
Hence the cube roots of+1 are 

cos + i sin 0, cos . 27r/3 ± i sin . 27r/3, 
that is to say , + 1 , - 1/2± i*j3/2. 

* The solution of this problem was first found in a geometrical form by 
Cotes ; it was published without demonstration in the Harmonia Mensurarum 
(1722), p. 113. Demoivre (Misc. Anal., p. 17) gave a demonstration, and also 
found the real quadratic factors of the trinomial 1 + 2 cos Ox" + x' 2 ". 



244 EXAMPLES chap. 

Again -1 = 1 {cos ir + i sin jt} . Hence the cube roots of - 1 are 
cos . ir/3 ± i sin . ir/B, cos it + i sin w, 
that is to say, l/2±i\j3f2, - 1. 

Example 2. 

To find the cube roots of 1+i. We have l+t=V 2 (Vv^+ ll /V 2 ) 
= \/2(cos 45° + i sin 45°). Hence the cube roots of 1 + i are 

2 J (cos 15° + i sin 15°), 2*(cos 135° + i sin 135°), 2*(cos 255° + i sin 255°), 
that is, 

2*(cos 15° + i sin 1 5°), 2 h { - cos 45° + i sin 45°), 2*( - cos 75° - i sin 75°), 
that is, 

i/V3 + l V3-l\ 1/ 1 . 1 \ 2 1/ V3-1 -\/3 + l\ 
2 r2V2 _+ l "2V2~> H"V2 + V2> U~~2^ l 2V2> 

«,„*;. (V3 + l) + (V3-l)f -1+t (V8-l) + CV3 + l)t 

mat is, j , j — , -2 . 

93 93 9? 

Here it will be observed that the roots are not arranged in conjugate pairs, as 
they would necessarily have been had the radicand been real. 

Example 3. 

To find approximately one of the imaginary 7th roots of + 1. One of the 
imaginary roots is 

cos 51°25'43" + i sin 51°25'43". 

By the table of natural sines and cosines, this gives 

•6234893+ -7818318?; 
as one approximate value for the 7th root of + 1. 

Example 4. 

If a be one of the imaginary cube roots of + 1, to show that 1 + w + co 2 = 
and that {ax + a 2 y){a 2 x + ay) is real. 

AVe have 1 +w + w 2 = (l - w 3 )/(l - w) = 0, since co 3 =l and l-w#0. 
Again, 

{ax + a 2 y) {a 2 x + ay) = a*x 2 + {a 4 + a 2 )xy + uPy' 2 . 

Now u 3 =l ; and w 4 + w 2 =w 3 u> + w a =« + a 2 — -1, since l+« + w 2 = 0. 

Hence 

( ax + a 2 y) {a 2 x + ay) = x 2 — xy + y 2 . 



FUNDAMENTAL PROPOSITION IN THE THEORY OF EQUATIONS. 

§ 20.] If /(2)sA + A,« + A/+ • • • +A„; n be an integral 
function of z of the nth degree, whose coefficients A , A,, . . . , A w 
are given complex numbers, or, in particular, real numbers, where, of 
course, A a + 0, then f(z) can always be expressed as the product of n 
factors, each of the 1st degree in z, say z - z l} z-z.,, z - z 3 , . . ., 
z - z n , ;.',, z 2 , . . ., z n being in general complex numbers. 



xii FACTORISATION OF ANY INTEGRAL FUNCTION 245 

It is obvious that this proposition can be deduced from the 
following subsidiary theorem : — 

One value of z, in general a complex number, can always he found 
which causes f{z) to vanish. 

For, let us suppose that /(s,) = 0, then, by the remainder 
theorem, f(z) =f(z) (z - «,), where f(z) is an integral function of z 
of the (n - l)th degree. Now, by our theorem, one value of z at 
least, say z,, can be found for which f(z) vanishes. We have, 
therefore, /,(&) = ; and therefore /,(*) =f a (z) (z - z 2 ), where f 2 (z) 
is now of the (» - 2)th degree ; and so on. Hence we prove 

finally that 

f(z) s k(z - *,) (z - z 2 ) ...(z- z tl ), 

where A is a constant. 

§21.] We shall now prove that there is always at least one 
finite value of z, say z = a, such that by taking z sufficiently near 
to a, that is by making \z - a\ small enough, we can make 
\f(z) | as small as we please. So that in this sense every 
integral equation/^) = has at least one finite root. 

Let | z I = R. Then, since 

\f(z) | = | A. | R" 1 1 + A n _ JA n z + . . . + A jA n z n | , 
we have, by § 14, 

\f(z) | > | A n | R»{1 - | A n _,/A n z + . . . + A /A,,*" | }, 
provided | z |, (i.e. R), be large enough ; therefore 

|/(*)|>|A*|R»{1-C(1/R+ . . . + 1/R»)}, 

where C is the greatest of |A n _,/A n |, . . ., [A /A w |. There- 
fore, taking provisionally R>1, we have 

\f(z) | > | A. | R»{1 - CXI - 1/R»)/R(1 - 1/R)}, 

>|A n |R»{l-C/(R-l)} (1), 

provided R > C + 1 . 

Hence, by taking \z\ sufficiently large, we can make \f(z)\ as large 
as we please ; and we also see that there can be no root of f(z) - 
whose modulus exceeds C + 1 . 

Let now w be the value of z at any finite point in the 
Argand Plane, so that \f(ic) | is finite. It follows from what has 



246 EXISTENCE OF A ROOT CHAP. 

just been proved that we can describe about the origin a circle 
S of finite radius, such that, at all points on and outside S, 
|/(~) I > \f( w ) | • Then, since \f(w) ] is real and positive, if we 
consider all points within S, we see that there must be a finite 
lower limit L to the value of \f(w) \ ; that is to say, a quantity 
L which is not greater than any of the values of \f(w) | within S, 
and such that by properly choosing u we can make \f(w) | = L + e, 
where e is a real positive quantity as small as we please. 

We shall show that L must be zero. For, suppose L > 0, 
and choose w so that \f(w) | = L + e. Let h be a complex number, 
say r(cos + i sin 6). Then 

f(w + h) - A + A (w + h) + . . . + A n (w + h) n , 

=f(w) + BJi + BJf+ . . .+A n h n (2), 

where A n is independent both of w and h, and by hypothesis 
cannot vanish, but B„ . . ., B n _! are functions of w, one or more 
of which may vanish. Suppose that B m is the first of the B's 
that does not vanish, and let b m (cos a m + i sin <x m ), etc., be the 
normal forms of the complex numbers B m /f(w), etc. Then, 
since \f{w) | is not zero, b mi etc., are all finite. Also we have, 
by Demoivre's Theorem, 

f(w + h)ff(w) = 1 + b m r™e m + b m+1 r™+iQ m+i + . . . + 6 n r»6 n (3), 
where m = cos {md + a m ) + i sin (rnd + a m ), etc. 

We have h, and therefore both r and 6 at our disposal. Let 
us first determine so that cos {m6 + a m ) = - 1, sin (mO + a m ) = ; 
that is, give 6 any one of the m values {it - a m )/m, (Stt — a m )/w, . . ., 
(2m - 1. 7r - a m )/m, say the first. Then we have Q m = -1; 
and Q m+l , etc., assume definite values, say, G' m+1> etc. We now 
have 
f(w + h)/f(w) = 1 - b m r™ + b m+1 f m + 1 0' m+1 + . . . + b n r"Q' n (A). 

Considering the right hand side of (4) as the sum of 1 - b m r m 
and b m+l r m+i Q' m+l + . . . + b n r n Q' n , we see, by § 14, that the 
modulus of f(w + h)jf(w) lies between the difference and the sum 
of the moduli of these two. Also 



XII 



EXISTENCE OF A ROOT 247 



\b m+l r™+ l Q , m+l + . . . + b n O' n r n \<b m+l r m+1 + • • • +&»'* 

<b(r m+1 + . . . +r n ), 

where b is the greatest of b m+l , . . ., b n , 

< (« - m)br m +\ 

provided we take r< 1, so that r m+i >r m+2 > _ # § >r n 

Therefore we have 

I - b m r m -(n- m)b?- m+l < \f(w + h)/f(w) | 

< 1 - 6 TO r m +{n - m)br m+l (5) ; 

provided r be so chosen that 1 - b m r m and 1 - b m r m - (n - m)br m+1 
are both positive. Let us further choose r so that (n - m)h m+1 
<b m r m . All these conditions will obviously be satisfied if we 
give a finite value to r less than the least of the three, 

1, l/(26 m ) lM . b m /(n-m)b. (6). 

When r is thus chosen, \f(w + h)/f(w) | will lie between two 
positive proper fractions, so that \f(w + h)jf(w) | = 1 — /x, where /x 
is a positive proper fraction ; and we have 
|/(w + A)| = (l-/t)|/HI = (l-/*)(L + € ) = L + 6 - /t (L + e)(7). 

which, since e may be as small as we please, is less than L by a 
finite amount. L is therefore not a finite lower limit as sup- 
posed; in other words, L must be zero. Our fundamental 
theorem is thus established. 

By reasoning as above we can easily show that, if 

f(z) = A + A il z*+ . . . +A n z n , 
then 
\Ao\{l-(n-s+l)d\z\»}<\f(z)\ 

<\A \{l+(n-s+l)d\z\°} (8), 

where d is the greatest of | A«/A 1 , . . ., | A n /A 1 , provided [ z \ 
is less than the lesser of the two quantities 1, l/{(n - s + l)d} l > s . 

Combining this result with one obtained incidentally above, 
we have the following useful theorem on the delimitation of the 
roots (real or imaginary) of an equation. 

Cor. 1, The equation A + AgZ 8 + . . . + A n z n = can have no root 
whose modulus exceeds the greatest of the quantities 1 + | A /A w | , 



248 GRADIENT AND EQUIMODULAR CURVES CHAP. 

1 + | A s jA n | , . . ., 1 + | A n _JA n | , or whose modulus is less than the 
least of 1, l/{(n - s + 1) | A s /A 1 }V°, . . ., l/{(« - s + 1) | A n /A 1 }**. 

Cor. 2. JFe ca» always assign a positive quantity ?;, such that, if 
| A|<?7, |/(2 + h) -f(z) |<e, w/tere e is a positive quantity as small as 
we please. 

This is expressed by saying that the integral function f(z) is 
continuous for all complex values of its argument which have a 
finite modulus. The proof is obvious after what has already 
been done. 

The above demonstration is merely a version of the proof given by 
Argand in his famous Essai, * amplified to meet some criticisms on the briefer 
statement in earlier editions of this work. The criticisms in question touch 
the formulation, but not the essential principle of Argand's proof, which is 
both ingenious and profound. As some of the critics appear to me to have 
missed the real point involved, perhaps the following remarks, which the 
student will appreciate more fully after reading chap. xv. §§ 17-19, and 
chap, xxix., may be useful. 

Taking any value of z=x + yi, let f(z) — u + vi, where u and v are real 
functions of the real variables x and y. Plot u- v- and x- y- Argand 
diagrams. Then to each point (x, y) there corresponds one point (u, v), 
although it may happen that to one (u, v) point there correspond more than 
one (x, y) point. If we start with any given point (x, y), and consider 
\/( z ) I = \f( u2 + v2 )> it is obvious that the direction in which \f(z) \ varies most 
rapidly is obtained by causing (x, y) to move in the x- y- plane, so that 
(u, v) moves along the radius vector towards the origin in the u- v- plane ; 
also that the rate of variation in the perpendicular direction is zero. If, 
therefore, we trace one of the curves v\u = constant in the x- y- plane, the 
tangent to this curve at every point z on it is the direction of most rapid 
variation. We may call these curves the Gradient Curves of/(s).t 

* See in particular his amplified demonstration given in a note in Ger- 
gonne's Ann. de Math. t. v. pp. 197-209 (1814-15). 

t These curves, together with the curves u^ + v 2 — constant, which we 
may call the Equimodular Curves of /(c), possess a number of interesting 
properties. Since the Equimodulars and Gradients are orthomorphosed (see 
chap. xxix. § 36) from a series of concentic circles and their pencil of 
common radii, they form two mutually orthogonal systems. Through 
every given point (x' t y'), which is not the affixe of a root of /(s) = 0, there 
passes one equimodular u 2 + v 2 = u" 1 + v' 2 and one gradient u'v-v'u — 0. 
Every gradient u'v-v'u — passes through all the intersections of u — 0, 
v — 0, i.e. through the affixes of all the roots of /(z) = 0. Near the root 
points the equimodulars take the form of small ovals enclosing these points. 



XII ARGAND'S PROGRESSION TO A ROOT 249 

It may readily be shown that the process by which we pass from v to 
w + h (—w v say) in the above demonstration simply amounts to passing a 
certain distance along a tangent to the gradient curve through the point w. 
We may repeat this process, starting from w x and passing along the tangent 
to the gradient through w 1} and so on. We shall thus ha.vef(to)>f(w 1 )> 
f{w ) > . . . This process we may call Argand's Progression towards the root 
of an equation. 

Since b n <b, b m r m + (n- m)br m+l <\, and (n-m)br m+1 <b m r'», it follows 
that 

r < { \A W ) I / 2 (' 1 - m ) I A » I !• ' /(m+1) and r < ! B ™ I A* _ m ) I A » I • 

The first of these conditions shows that the longest admissible steps of 
Argand's Progression become smaller and smaller as we approach a root. 
This is expressed by saying that the Progression becomes asymptotic as we 
approach a root. 

But the second condition shows that Progression also becomes asymptotic 
as we approach a point at which Bj = ; or Bi = 0, B. 2 =:0 ; or B 1 = 0, B 2 =0, 
B 3 = 0, etc. Such points are stationary points for the variation of \J\z) | on 
any path which passes through them. They are also multiple points on the 
gradient curves which pass through them ; so that at them we have 2, 3, 4, 
etc., directions of most rapid variation of \f{z) \ . 

If the original progression leads towards one of these points (for which of 
course \f{iv) |4=0), we must infer its existence from the asymptotic approach, 
and start afresh from that point along one of the tangents to the gradient 
that passes through it ; or we may avoid the point altogether by starting 
afresh on a path which docs not lead to it. 

An interesting example is to take z 2 - z + 1 = 0, and start from a point on 
the real axis in the x- y- diagram. Argand's Progression will lead first to 
the minimum point (1/2, 0) on the x- axis ; then along a line parallel to the 
y- axis to the points, (1/2, ± *j3/2), which are the affixes of the two imaginary 
roots. The diagram of chap. xv. § 19 will also furnish a curious illustration 
by taking initial points on one or other of the two dotted lines. 

It should be noted that the question as to whether |/(a) | actually reaches 
its lower limit is not essential in Argand's proof, if we merely propose to 
show that a value of z can be found such that \f(z) | is less than any assigned 
positive quantity, however small. Nor do we raise the question whether 
the root is rational or irrational, which would involve the subtle question of 
the ultimate logical definition of an irrational number (see vol. ii. (ed. 1900) 
chap. xxv. §§ 28-41). 

§ 22.] We have now shown that in all cases 

f(z) = A(z - 2,) (z - gg) . . . (z - z n ), 

where A is a constant. 

z lf z,, . . ., Zn may be real, or they may be complex numbers 
of the general form x + yi. They may be all different, or two 



250 EQUATION OF WTA DEGREE HAS n ROOTS CHAP. 

or more of them may be identical, as may be easily seen by 
considering the above demonstration. 

The general proposition thus established is equivalent to the 
following : — 

If f(z) be an integral function of z of the nth degree, there are 
n values of z for which f(z) vanishes. These values may be real or 
complex numbers, and may or may not be all unequal. 

We have already seen in chap, v., § 16, that there cannot be 
more than n values of z for which f(z) vanishes, otherwise all its 
coefficients would vanish, that is, the function would vanish for 
all values of z. We have also seen that the constant A is equal 
to the coefficient A n . We have therefore the unique resolution 

f(z) = An(z - z t ) (z - z 2 ) . . . (z - z n ). 

§ 23.] If the coefficients of f(z) be all real, then we have 
seen that if f(x + yi) vanish f(x - yi) will also vanish. In this 
case the imaginary values among z u z 2 , . . ., z n will occur in 
conjugate pairs. 

If a + fii, a - (3i be such a conjugate pair, then, correspond- 
ing to them, we have the factor 

(Z - a - (3i) (z-a + (3i) = (z- af + ft, 

that is to say, a real factor of the 2nd degree. 

It may of course happen that the conjugate pair a ± fti is 
repeated, say s times, among the values z„ z,, . . ., z n . In that 
case we should have the factor (z - a) 2 + (3 s repeated s times ; so 
that there would be a factor {(z - a) 2 + (3 2 } s in the function f(z). 

Hence, every integral function of z, wlwse coefficients are all real, 
can be resolved into a product of real factors, each of which is either 
a positive integral power of a real integral f miction of the 1st degree, 
or a positive integral power of a real integral function of the 2nd 
degree. 

This is the general proposition of which the theorem of § 19 
is a particular case. 






XII 



EXERCISES XVI 951 



Exercises XVI. 



Express as complex numbers — 

(1.) (a + bi) 8 + (a-bif. 

(2) 1+i l 1 ~ i 

2 + 36i f 7-26i 
{4m) 6 + 8i + 3-4i' 

u) (p±si\\(p^iY 

\p-qij \p + v i J 

69-7V(15) + (V3-6V5)i 

{ ' 3-(V3-3V5)i 

(6.) Show that 

if n be any integer which is not a multiple of 3. 

(7.) Expand and arrange according to the powers of a; 

(as - 1 - i\J2) (x-\ + i\/2){x-2 + i^/3) (a; - 2 - i\/3). 

(8. ) Show that 

\(2a-b-c)+i{b-c)^B} 3 ={{2b-c-a) + i(c-aWS} 3 . 

(9.) Show that 

{(V3 + 1) + (V3-- W=16(l + i). 

(10.) If £ + J?t be a value of x for which ax 1 + bx + c = 0, a, 6, c being all 
real, then 2a£rj + br) = 0, a?? 2 = a£ 2 + &£ + c. 

(11.) If #(a?+2/*)=X.+Y*, show that 4(X 2 -Y 2 ) = x/X + y/Y. 

(12.) If n be a multiple of 4, show that 

l+2i + 3i 2 + . . . + (n + l)i n =%(n + 2-ni). 

(13.) Showthat|o + o 1 5 + . . . +a n z n \^\a n \ \z\ n (l-nc/\z\), provided \z\ 
exceeds the greater of 1 and nc, and c is the greatest of \a Q [a„\, . . ., 

I ««-]/«« |. 

(14.) Find the modulus of 

(2-3i)(3 + 4t ) 
(6+4i)(15-8i)" 

(15.) Find the modulus of 

{x+ J&+W- 

(16.) Find the modulus of 

bc(b - ci) + ca(c - ai) + ab(a - bi). 
(17.) Show that 

1 1 + ix + i~x- + iV + . . . ad » | = 1/ S !{1 + x 2 )', where as<l. 
(18.) Find the moduli of (x + yi) n and (x + yi) n /(x-yi) n . 

Express the following as complex numbers : — 
(19.) \/-7 + 24f. (20.) \ f Q + iJ13. 



(21.) *J - 7/36 + 2t/3. (22.) s' lab + 2(o a - "Fjl 

(23.) s!\+2x*]{x 2 -\)i. (24.) Vl + i ^(x 4 - 1 ). 

(25.) Find the 4th roots of - 119 + 120;'. 



252 EXERCISES XVI CHAP. 

(26.) Resolve x*- a 6 into factors of the 1st degree. 

(27.) Resolve a; 5 +l into real factors of the 1st or of the 2nd degree. 

(28.) Resolve x G + x 5 + x 4 +x 3 + x 2 +x + l into real factors of the 2nd 
degree. 

(29.) Resolve x 2m - 2 cos 0a m x m + a 2m into real factors of the 1st or of the 
2nd degree. 

(30.) If w be an imaginary nth root of +1, show that l + w + w 2 + . . . + 
co"- 1 = 0. 

(31.) Show that, if w be an imaginary cube root of +1, then 

x 3 + y 3 + z 3 - Sxyz = (x + y + z) (x + wy + w 2 z) (x + uPy + uz), 
and 

(x + wy + uhf + (x + w 2 y + wz) 3 = (2x-y -z) {2y -z~x) {2z-x- y). 

(32. ) Show that (x + y) m -x m -y m is divisible by x 2 + xy + y 1 for every odd 
value of m which is greater than 3 and not a multiple of 3. 
(33.) Show that 

U„ 2/-7T - T . 2nr\ („ . 2/7T „ 2rir\ . ^ n , v „.. 
Xcos Ysin ) + (Xsin + \ cos )i V = (\+\i) n . 
n n J \ n n J J 

(34.) Simplify 

(cos 2d - i sin 20) (cos <£ + i sin 4>) 2 (cos 20 + i sin 20) (co s < p - i sin <f>) 2 
cos (0 + <p) + i sin {0 + <p) cos (0 + 0) -i sin (0 + 0) 

(35.) If s /(a + bi)+ s /(c + di)= s /(x + yi), show that 

(x-a-c) 2 +(y-b-d) 2 = 4^J{(a 2 + b 2 )(e- + d 2 )}. 
(36.) Prove that one of the values of *J(a + bi) + £/(a - bi) is 

v /[ J \2a + 2V(« 2 + b 2 )} + 2 Z'(a 2 + b 2 )\ 
(37.) If w = cos tt/7 + i sin tt/7, prove that {x - w) {x + w 2 ) (x - it?) (x + w 4 ) 
(x - w 5 ) (x + w 6 ) = x 6 - x 5 + x 4 - x 3 + x 2 - x + 1 . 

(38.) Find the value of w r 1 + w;'' 2 4 . . . +w r n ; w lt w 2 . . . w„ being the 
nth roots of 1, and r a positive integer. What modification of the result is 
necessary if r is a negative integer ? 

Prove that 1/(1 +«)!«) -f 1/(1 + «>2 a; ) + - • • +l/(l+w„a:)=M/(l -a;"). 

(39.) Decompose 1/(1 +x + x 2 ) into partial fractions of the form aj(bx + c). 
Hence show that 

a:/(l + x + x 1 ) -x - x 2 + x 4 - x 5 + x 1 - x 8 + . . . + x 3n + l - x 3n+2 + R, 
where a^, x e , etc. , are wanting ; and find R. 

(40.) Find the equation of least degree, having real rational coefficients, 
one of whose roots is ^2 + i. 

One root of x 4 + 3a; 3 - 30a; 2 + 366.*:- 340 = is 3 + 5?:, find the other three 
roots. 

(41.) If a be a given complex number, and z a complex number whose 
affixe lies on a given straight line, find the locus of the affixe of a + z. 

(42.) Show that the area of the triangle whose vertices are the affixes of 
ssii s 2 , z 3 is 2 {(z 2 - z 8 ) | z x | 2 /Uz-l} . 

(43.) If 2 = (a + 7 cos 0) + i((i + y sin 0), where a, /3, y are constant and 
variable, find the maximum and minimum values of | z | ; and of amp z 
when such values exist. 



XII 



HISTORICAL NOTE 253 



(44. ) If the affixe of x + yi move on the line Sx + iy + 5 = 0, prove that the 
minimum value of\x + yi\ is 1. 

(45.) If u and v are two complex numbers such that u = v+l/v, show 
that, if the affixe of v describes a circle about the origin in Argand's diagram, 
then the aflixe of u describes an ellipse (x 2 /a 2 + ?/ 2 /& 2 = l) ; and, if the affixe of 
u describes a circle about the origin, then the affixe of v describes a quartic 
curve, which, in the particular case where the radius of the circle described by 
the affixe of u is 2, breaks up into two circles whose centres are on the i-axis. 

(46.) If a; and y be real, and x + y = \, show that the affixe of xz l + yz 2 lies 
on the line joining the affixes of z x and ».,. Hence show that the affixe of 
xz x + yz 2 lies on a fixed straight line provided lx + my—l, I and m being 
constants. 

(47.) If £ + ij£ be an imaginary root of iE 3 + 2cc + 1 = 0, prove that (£, ij) is 
one of the intersections of the graphs of ?j 2 = 3£ 2 +2 and ?? 2 = l/2 + 3/8£. 
Draw the graphs : and mark the intersections which correspond to the roots 
of the equation. 

If o be the real root of this cubic, show that the imaginary roots are 
H-a+*V(2-3/a)}. 

(48.) If £±r)i be a pair of imaginary roots of o?-px + q = 0, show that 
(if, tj) are co-ordinates of the real intersections of 3£ 2 - i) 2 -p = 0, 8^rj- + 2p^- 
3q = 0. Hence prove that the roots of the cubic are all real, or one real and 
two imaginary, according as 4p 3 < >27g a . What happens if 4p s =272 a ? 

(49.) If x 3 + qx + r = Q has imaginary roots, the real part of each is posi- 
tive or negative according as r is positive or negative. 

(50.) The cubic x 3 - 9a; 2 + 33a:- 65 = has an imaginary root whose 
modulus is N /13 ; find all its roots. 

(51.) Find the real quadratic factors of x 2n + x 2 "- 1 + ... +1 ; and hence 
prove that 

o„ • if ■ 27r . nir ,,_ , . 

2 ' Sm 2„ + 1 Sin S+l ' " ' Sm 2^Tl= ^ 2 " + 1 >- 



(52.) Find in rational integral form the equation which results by 
eliminating 6 from the equations x — a cos + b cos 30, y = a smd+b sin 36. 
(Use Demoivre's Theorem.) Give a geometrical interpretation of your 

analysis. 

Historical Note. — Imaginary quantities appear for the first time in the works 
of the Italian mathematicians of the 16th century. Cardano, in his Artie Magna 
sive de Rcgvlis Algebraicis Liber Units (1545), points out (cap. xxxvii., p. 66) 
that, if we solve in the usual way the problem to divide 10 into two parts whose 
product shall be 40, we arrive at two formula? which, in modern notation, may 
be written 5 + *J - 15, 5 - N ' - 15. He leaves his reader to imagine the meaning 
of these "sophistic" numbers, but shows that, if we add and multiply them 
in formal accordance with the ordinary algebraic rules, their sum and product 
do come out as required in the evidently impossible problem ; and he adds 
" hucusque progreditur Arithmetics subtilitas, cujus hoc extremum ut dixi adeo 
est subtile, ut sit inutile." Bombelli in his Algebra (1522), following Cardano, 
devoted considerable attention to the theory of complex numbers, more especially 
in connection with the solution of cubic equations. 



254 HISTORICAL NOTE 



CHAP XII. 



There is clear indication in the fragment De Arte Logistica (see above, p. 201) 
that Napier was in possession to some extent at least of the theory. He was 
fully cognisant of the independent existence of negative quantity ("quantitates 
defectives minores nihilo"), and draws a clear distinction between the roots of 
positive and of negative numbers. He points out (Napier's Ed., p. 85) that 
roots of even order have no real value, either positive or negative, when the 
radicand is negative. Such roots he calls " nugacia " ; and expressly warns 
against the error of supposing that J_| - 9= - M 9. In this passage there occurs 
the curious sentence, " Hujus arcani magni algebraici fundamentum superius, 
Lib. i. cap. 6, jecimus : quod (quamvis a nemine quod sciam revelatum sit) quan- 
tum tamen emolumeuti adferat huic arti, et cseteris mathematicis postea patebit." 
There is nothing farther in the fragment De Arte Logistica to show how deeply he 
had penetrated the secret which was to be hidden from mathematicians for 200 
years. 

The theory of imaginaries received little notice \mtil attention was drawn to 
it by the brilliant results to which the use of them led Euler (1707-1783) and his 
contemporaries and followers. Notwithstanding the use made by Euler and 
others of complex numbers in many important investigations, the fundamental 
principles of their logic were little attended to, if not entirely misunderstood. 
To Argand belongs the honour of first clearing up the matter in his Essai 
sur une maniere de representer les quantites imaginaires dans les constructions 
geometriques (1806). He there gives geometrical constructions for the sum and 
product of two complex numbers, and deduces a variety of conclusions therefrom. 
He also was one of the first to thoroughly understand and answer the question of 
§ 21 regarding the existence of a root of every integral function. Argand was an- 
ticipated to a considerable extent by a Danish mathematician, Caspar Wessel, who 
in 1797 presented to the Royal Academy of Denmark a remarkable memoir Om 
Direktionens ancdytiske Betegning, et Forslig, anvendt fwncmmelig til plane og 
sphaeriske Polygoners Opliisning , which was published by the Academy in 1799, 
but lay absolutely unknown to mathematicians, till it was republished by the 
same body in 1897. See an interesting address by Beman to Section A of the 
American Association for the Promotion of Science (1897). Even Argand's results 
appear to have been at first little noticed ; and, as a matter of history, it was 
Gauss who first initiated mathematicians into the true theory of the imaginaries of 
ordinary algebra. He first used the phrase com2>lex number, and introduced the 
use of the symbol i for the imaginary unit. He illustrated the twofold nature of 
a complex number by means of a diagram, as Argand had done ; gave a masterly 
discussion of the fundamental principles of the subject in his memoir on Bi- 
quadratic Residues (1831) (see his Works, vol. ii., pp. 101 and 171) ; and furnished 
three distinct proofs (the first published in 1799) of the proposition that every 
equation has a root. 

From the researches of Cauchy (1789-1857) and Riemann (1826-1866) on 
complex numbers has sprung a great branch of modern pure mathematics, called 
on the Continent function - theory. The student who wishes to attain a full 
comprehension of the generality of even the more elementary theorems of algebraic 
analysis will find a knowledge of the theory of complex quantity indispensable ; 
and without it he will find entrance into many parts of the higher mathematics 
impossible. 

For further information we may refer the reader to Peacock's Algebra, vol. ii. 
(1845) ; to De Morgan's Trigonometry and Double Algebra (1849), where a list 
of most of the English writings on the subject is given ; and to Hankel's 
Vorlesungen ilber die complexen Zahlen (1867), where a full historical account 
of Continental researches will be found. It may not be amiss to add that the 
theory of complex numbers is closely allied to Hamilton's theory of Quaternions, 
Grassmami's Ausdehuuugslehre, and their modern developments. 



CHAPTER XIII. 
Ratio and Proportion. 

RATIO AND PROPORTION OF ABSTRACT QUANTITIES. 

§ 1.] The ratio of the abstract quantity a to the abstract quantity 
b is simply the quotient of a by b. 

When the quotient a -s- b, or a/h, or y is spoken of as a ratio, 

it is often written a : b ; a is called the antecedent and b the con- 
sequent of the ratio. 

There is a certain convenience in introducing this new name, 
and even the new fourth notation, for a quotient. So far, how- 
ever, as mere abstract quantity is concerned, the propositions 
which we proceed to develop are simply results in the theory 
of algebraical quotients, arising from certain conditions to which 
we subject the quantities considered. 

If a > b, that is, if a - b be positive, a : b is said to be a ratio 
of greater inequality. 

If a < b, that is, if a - b be negative, a : b is said to be a ratio 
of less inequality. 

When two ratios are multiplied together, they arc said to be 

compounded. Thus, the ratio aa' : bb' is said to be compounded of 

the ratios a : b and a' : V. 

The compound of two equal ratios, a : b and a : b, namely, 

a 2 : b 2 , is called the duplicate of the ratio a : b. 

Similarly, of : b 3 is the triplicate of the ratio a : 5.* 

* Formerly a 2 : b- was spoken of as the double of the ratio a : b. Similarly 

.i * 
\Ja : \/b was called the half or subduplicate of a : b, and a 2 : b* the sesquipli- 

cate of a : b. 






256 PROPERTIES OF A RATIO chap. 

§ 2.] Four abstract numbers, a, b, c, d, are said to be proportional 
when the ratio a:b is equal to the ratio c : d. 
We then write 

a : b = c : d* 

a and d are called the extremes, and b and c the means, of the pro- 
portion, a and c are said to be homologues, and b and d to be 

homologu.es. 

If a, b, c, d, e, f, &c, be such that a : b = b : c = i : d = d : e = e :f 
= &c, a, b, c, d, e, f, &c, are said to be in continued proportion. 

If a, b, c be in continued proportion, b is said to be a mean 
proportional between a and c. 

If a, b, c, d be in continued proportion, b and c are said to be 
two mean proportionals between a and d ; and so on. 

§ 3.] If b be positive, and a>b, the ratio a:b is diminished by 
adding the same positive quantity to both antecedent and consequent ; 
and increased by subtracting the same positive quantity (<b) from 
both antecedent and consequent. 

If a<b, the words "increased" and "diminished" must be inter- 
changed in the above statement. 

a + x a b(a + x) - a(b + x) 



For, 



b + x b b(b + x) 

x(b - a) 



b(b + x)' 

Now, if a > b, b-a is negative ; and x, b, b + x are all positive 
by the conditions imposed ; hence x(b - a)jb(b + x) is negative. 

TT a + x a . 

Hence 7 — ■ 7 is negative, 

b + x b 

, a + x a 
that is, t < r. 

' b + x b 

a- x a x(a - b) 

Ac;ain, ■? t = 777 r- 

' b - x b b(b - x) 

But, since a > b, a-b is positive, and x and b are positive, and, 
since x<b, b-x is positive. Hence x(a - b)/b(b - x) is positive. 



* Formerly in writing proportions the sign : : (originally introduced by 
Oughtred) was used instead of the ordinary sign of equality. 



Xlll PERMUTATIONS OF A PROPORTION 257 

TT a — x a 

Hence -. > -. 

b — x b 

The rest of the proposition may be established in like manner. 

The reader will obtain an instructive view of this proposition 
by comparing it with Exercise 7, p. 267. 

§ 4.] Permutations of a Proportion. 

(1), 
(2), 
(3), 
(4). 



If 


a:b = c:d 


then 


b:a = d:c 




a :c = b:d 


and 


c :a = d :b 


For, from (1), we have 


a c 
b = d' 


Hence 


'b V 


that is, 


b_d 

a c ' 


that is, 


b : a-d: c, 


which establishes (2). 




Again, from (1), 


a c 
1 = 7v 



multiplying both sides by -, we have 

a b c b 
b c d c 

that is, - = - , ; 

c d 

that is, a:c = b: d, 

which proves (3). 

(4) follows from (3) in the same way as (2) from (1). 

§ 5.] The product of the extremes of a proportion is equal to the 
product of the means ; and, conversely, if the product of two quantities 
be equal to the product of two others, the four form a proportion, the 
extremes being tlie constituents of one of the products, the means the 
constituents of the other. 

VOL. J S 



258 


RULE OF THREE 


For, if 


a : b = c : d, 


that is, 


a c 
b = d' 


then 


- x bd = - : x bd, 
b d 


whence 


ad = be. 


Again, if 


ad = be, 


then 


ad/bd = bc/bd, 


whence 


a c 
b~d' 



CHAP. 



Cor. If three of the terms of a proportion be given, the remaining 
one is uniquely determined. 

For, when three of the quantities a, b, c, d are given, the equation 

ad = be, 

which results by the above from their being in proportion, be- 
comes an equation of the 1st degree (see chap, xvi.) to deter- 
mine the remaining one. 

Suppose, for example, that the 1st, 3rd, and 4th terms of the proportion 
are I, %, and f ; and let x denote the unknown 2nd term. 

Then *:<*=*:*; 

whence %y.x = \x%. 

Multiplying hy f , we have x=|xf xf, 

_ o 

— 3TT- 

§ 6.] Relations connecting quantities in continued proportion. 
If three quantities, a, b, c, be in continued proportion, then 

2 72 72 2 

a : c = a : b = b : c ; 
and b = \Z(ac). 

If four quantities, a, b, c, d, be in continued proportion, then 
a:d = a a : b' = b' : c 3 = c 3 : d 3 , 
and b = 'i/(a\l), c = t/(ad*). 

For the general proposition, see Exercise 12, p. 267. 



XIII 



DETERMINATION OF MEAN PROPORTIONALS 



259 



For, if 

then 

Therefore 
whence 



a : b 



b :c, 
a _b 
b~~c 



a b b b 

~ X - = - X -. 





(!)• 



c c c 

c~ c 2 ~ Ir 
Also ac = b 2 , 

whence b = \/(ac) (2). 

Equations (1) and (2) establish the first of the two proposi- 



tions above stated. 




Again, if 


a\b = b:c = c:d, 


then 


abac 
b c b d" 


Also 


a a 
b = b' 


hence 


a a a a b c 


that is, 


a 3 a 
b 3 = d' 


therefore 


a a 3 b 3 c a 
d~b 3 ~c 3 ~d 3 


Further, since 


a a 3 
d = b 3 ' 




b 3 — ci'd ; 


whence 


b = l/(a*d). 


Also, since 


a c 3 
d d 3 ' 




c 3 = ad 2 ; 


whence 


c = l/{ a d 2 ) 



(3). 



(4> 



(5). 

It should be noticed that the result (2) shows that the finding of a mean 
proportional between two given quantities a and c depends on the extraction 
of a square root. For example, the mean proportional between 1 and 2 is 

V(l x 2) = V2 = 1-4142 . . . 



260 DELTAN PROBLEM 






CHAP. 



Again, (i) and (5) enable us to insert two mean proportionals between two 
given quantities by extracting certain cube roots. For example, the two 
mean proportionals between 1 and 2 are 

^(lx 2) =4/2 = 1*2599 . . . 

and V / (lx2 2 ) = ^- =1-5874 . . . 

Conversely, of course, the finding of the cube root of 2, which again corre- 
sponds to the famous Delian problem of antiquity, the duplication of the 
cube, could be made to depend on the finding of two mean proportionals, a 
result well known to the Greek geometers of Plato's time. 

§ 7.] After what has been done, the student will have no 
difficulty in showing that 



(2). 



if 




a :b = c:d, 




then 




ma :mb = nc: nd 




and 




ma : nb = me : nd 




§8.] 
if 


Also that 


a l :b i = c l :d 1 , 
a ± \ o 2 = c 2 : « 2 > 

a n '. o n = e n : a n , 




then 


a,<i, 2 . . . a n 


: 6,6 S . . . b n = ca . . . c n 


■ d/h 


Cor. 








If 




a • b = c : d, 




then 




a n . ^ = c n : d n . 





.dn (1). 



(Here n, see chap, x., may he positive or negative, integral or 
fractional, provided a n , &c, be real, and of the same sign as a, 
&c.) 

§9.] // a:b = c:d, 

then a±b :b = c±d:d (1), 

a + b: a -b = c + d : c - d (2), 

la + mb : pa + qb = lc+ md \pc + qd (3), 

la r + mb r : pa r + qb r = lc r + md r :pc r + qd" (4), 

where I, m, p, q, r are any quantities, positive or negative. 



xiil CONSEQUENCES OF PROPORTION 261 

Also, if a, : b, = a 2 : b a = a z : b 3 = . . . = a n : i n , 

a, + a., + . . . + a n : 6 a + b 3 + . . . + &„ (5) ; 

and afeo fo 

;/(/,< + W + • • • + W) : W + U>/ + • ■ ■ + W) (6). 

Though outwardly somewhat different in appearance, these 
six results are in reality very much allied. Two different 
methods of proof are usually given. 



FIRST METHOD. 

Let us take, for example, (1) and (2). 

„. a c 

bince T = -, 

b d 

in a i c 

therefore =- ± 1 = •= ± 1 : 

b a 

a±b c ± d 
whence — -= — = — 5— ; 

b d 

this establishes the two results in (1). 
Writing these separately we have 

a + b c + d 







b d ' 






a - b c - d 






b d ' 


whence 


(a + b), 

h I 


' (a-b) (c + d)/ (c- d) 
b d 1 d ' 


that is, 




a+b c+d 
a-b c — d' 


which establishes (2). 




Similar 


treatment 


may be applied to the rest of the six 


results. 







2G2 EXPRESSION IN TERMS OF FEWEST VARIABLES chap. 



SECOND METHOD. 

Let us take, for example, (2). 

Since a/b = c/d, we may denote each of these ratios by the 
same symbol, p, say. We then have 









a 


c 






whence 






a = pb, 


c = pd 






Now, 


using (a), 


Ave have 














a + b 


pb + b 












a-b 


" pb-b' 
b( P +l) 

~b(p-iy 

p + i 
- p -v 






In exactly 


the i 


same way, we 


! have 












c + d 


pd + d 












c-d 


' pd- d' 

p+l 
-p-V 






Hence 






a + b p + 
a-b p- 


1 c + d 
1 " c - d' 






Again 


, let 


us take (5). 








We h: 


ive 


a, 


a 2 a 3 

Z h~b 3 ~ • • 


. = — , each 


i = P 


say, 


hence 


( 


h = p 


K <k = ph, 


. . ., a n 


= pbi 


l 5 


therefore 


«i 


+ do 


+ . . . +a n 


pb l + pb. 2 + 
b x + b. 2 + 


• • • 


+ pbn 


h 


+ K 


+ . . . + b n 


+ b n ' 










p(K + h + 


. . . 


+ b n ) 










b l + b i + . 




+ b» ' 


hence 








= P, 








a, 
6, 


a 2 


- &C - n - ffl 


+ a 2 + . . . 


+ a n 






— iVL, — p — , 


+ b 2 + . . . 


+ b n 





(a). 



XIII 



GENERAL THEOREM 2G3 



Finally, let us take (G). 

Since a/ = ( t AY = p'V, 

a/ = (ph 2 y = p >V, &c. 

we have 

y(ifi* + W+ ■ • • + W) = V(p r (iA r + W+ ■ ■ ■ +inb/)), 

(see chap, x., § 4). It follows that 

y(/,o, r + W + • ■ ■ + l n (hi r ) = _ a, = a* _ &(j 

;/(ZA r + w + ■ ■ • + W« r ) p ^ ^ 

Of the two methods there can be no doubt that the second 
is the clearer and more effective. The secret of its power lies 
in the following principle : — 

In establishing an equation, between conditioned quantities, if ice 
first express all the quantities involved in the equation in terms of the 
fewest quantities possible under the conditions, then the verification of 
the equation involves merely the establishment of an algebraical identity. 
In establishing (2), for instance, we expressed all the quantities 
involved in terms of the three b, d, p, so many being necessary, 
by § 5, to determine a proportion. 

A good deal of the art of algebraical manipulation consists 
in adroitly taking advantage of this principle, without at the 
same time destroying the symmetry of the functions involved. 

§ 10.] The following general theorem contains, directly or 
indirectly, all the results of last article as particular cases ; and 
will be found to be a compendium of a very large class of 
favourite exercises on the present subject, some of which will 
be found at the end of the present paragraph. 

If <M''i> ■''■_•> • • ., x„) be any homogeneous integral function of the 
variables it,, x. 2 , . . ., x a of the rth degree, or a homogeneous function 
of degree r, according to the extended notion of homogeneity and 
degree give<i at the foot of p. 73, and if 

«! : b l = a.,: b.,= . . . = a n : b n , 

then each of these ratios is equal to 



264 EXAMPLES chap. 

This theorem is an immediate consequence of the property 
of homogeneous functions given in chap, iv., p. 73. 

Example 1. 

AV Inch is the greater ratio, x 2 + y 2 : x + y, or x--y":x-y, x and y being 
each positive ? 

x 2 + y 2 x 2 -y 2 _ (x 2 + y' 2 ) (x-y)- {x 2 - y 2 ) (x + y) 
x + y ~ x-y ~ (x + y)(x-y) 

2xy 2 - 2x 2 y 
~{x + y)(x-y)' 

_ 2xy(x - y) 

{x + y)(x-y)' 
_ 2xy ^ 

x+y 

Now, if x and y be each positive, -2xyj(x + y) is essentially negative. 
Hence 

x 2 + y 2 : x + y < x 2 - y 2 : x - y. 

Example 2. 

If a:b=c;d, and A:B = C:D, then ci\/A - b\/B : c> s /C-d s /D = asJA 

+ 6VB:cVC+dVD- 

Let each of the ratios a : b and c :d = p, and each of the two A : B and 
C : D = <r, then a = pb, c — pd; A = <rB, C = crD. A\ 7 e then have 

as / A - b VB _ pb V(<rB) - b VB 

(pV<r-l)&VB _ K/B 

(p\/<r-l)d\/D dsJV 
In the same way we get 

a\/A + b K /B = ( P sJ<r + l)bsJB K/B 

cVC + ^V^ (pV<r + l)rfv'D rfV^ 
From (a) and (j8) the required result follows. 



(a). 

(/3). 



Example 3. 

If b be a mean proportional between a and c, show that 

(a+b + c)(a-b + c) = a 2 + b 2 + c 2 (a), 

and {a + b + c) 2 + d- + b 2 + c 3 = 2(a + b + c){a + c) (j8). 

Taking (a) we have 

(a+i+c)(a-6+c)=(a+c+6)(a+c-6), 
= (a+c) 2 -6 2 . 
Now, by data, a/b = b/c, and therefore b' 2 = ac ; hence 

(« + c) 2 - b 2 = (a + c) 2 - ac, 
= a 2 + OC + C 1 , 
= a- + b- + c-, 
since b 2 = ac. Hence (a) is proved. 



XIII 



EXAMPLES 265 



Taking now (/3), and, for variety, adopting the second method of § 9, let 

us put 

a b 

b = C =P ' 

Hence a — pb, b — pc; so that a = pipe) = p~c. 
We have to verify the identity 

(p-c + pc + cf + (j?cf + {pcf + c 2 = 2(p-c + pc + c){p-c + c) ; 
that is to say, 

{(ffi + p +l)* + (j,* + l ? + l)}<* = 2(? + p + l)((?+l)c> (7). 

Now 

{(p- + P + l) 2 + (^ + r^-l)^• 2 =(p 2 +/» + l){(p 2 + p + l) + ( / )•• ! -p + l)}c'• ! , 

= 2(p 2 + p + I)(p 2 + l)c 2 , 
which proves the truth of (7), and therefore establishes (j8). 

Example 4.* 

If x/(b +e-a) = yl(c + -b) = zfta + b-c), then (b-c)x + (c-a)y + (a-b)z = 0. 
Let us put 

a _ V _ z _ 
b+c-a c+a-b a+b -c 

then x = (b + c-a)p, 

y = (c + a-b)p, 
z = {a + b- c)p. 
Now, from the last three equations, we have — 
{b - c)x + (c - a)y + (a - b)z 

= (b -c)(b + c -a)p + (c-a)(c + a - b)p+ (a-b)(a + b-c)p, 
= {(S 8 - c 2 + c- - a 2 + a 2 - b' 2 ) - (a(b -c) + b(c - a) + e(a -b))}p, 

= {0-0}* 

= 0. 

Example 5. 

If bz + cy _cx + az _ay + bx 

b-c c-a a- b 
then (a + b + c)(x + y + z)=ax + by + cz (2). 

Let each of the ratios in (1) be equal to p, then 

bz + cy = p(b-c) (3), 

cx + az = p(c-a) (4), 

ay + bx — p(a-b) (5). 

From (3), (4), (5), by addition, 

(b + c)x+(c + a)y+ (a + b)z = p{ (b - c) + (c-a) + (« -b)}, 

= p0, 

= (6). 

If now we add ax + by + cz to both sides of (6) we obtain equation (2). 

* Examples 4, 5, and 6 illustrate a species of algebraical transformation 
which is very common in geometrical applications. In reality they are ex- 
amples of a process which is considered more fully in chap. xiv. 



(1), 



266 EXAMPLES 

Example 6. 

If cy + bz az + cx bx + ay 

qb + rc- pa re +}M - qb pa + qb- re 
show that 

x 
a {pa(a + b + c) - qb(a + b-c)- rc(a - b + c) } 

y 



CHAI\ 



(1), 



b{qb(a + b + c) -pa(a + b - c) -rc( -a + b + c)} 

z 



(2). 



c {rc(a + b + c)-qb(- a + b + c) -pa(a - b + c) } 
Let each of the fractions of (1) be = p; and observe that the three 

equations, 

cy + bz=(qb +rc -'pa)p (a) ~| 

az + ex = [re +}>a - qb)p (/3) J- (3), 

be + ay — {pa + qb - rc)p (7) J 

fa**) 

which thus arise are symmetrical in the triple set -J abc Y, so that the simul- 

[pqr) 

taneous interchange of the letters in two of the vertical columns simply 
changes each of the equations (3) into another of the same set. It follows, 
then, that a similar interchange made in any equation derived from (3) will 
derive therefrom another equation also derivable from (3). 

Now, if we multiply both sides of (j3) by b, and both sides of (7) by c, we 
obtain, by addition from the two equations thus derived, 

2bcx + a{cy + bz} =p{b(rc+2>a-qb) + c{pa + qb-rc)} (4). 

Now, using the value of cy + bz given by (a), we have 

2bcx + pa(qb + re - 2m) = p {pa(b + c) - qb(b - c) -rc( -b + c)} (5). 

Subtracting pa(qb + rc-pa) from both sides of (5), we have 

Ibex = p {pa{a + b + c)- qb(a + b-c)- rc(a - b + e) } (6). 

From (6), we have 

? =P. (7) . 

a{pa{a + b + c)-qb[a + b-c) -re(a- b + c)} 2abc 

/xap\ /xap\ 

We may in (7) make the interchange I into I, or I into I, and we shall 

\ ybq / \ zcr ' 

obtain two other equations derivable from (3) by a process like that used to 
derive (7) itself. These interchanges leave the right-hand side of (7) un- 
altered, but change the left-hand side into the second and third members of 
(2) respectively. Hence the three members of (2) are all equal, each being in 
fact equal to pj2abc. 

This is a good example of the use of the principle of symmetry in compli- 
cated algebraical calculations. 



xiii EXERCISES XVII 2G7 



Exercises XVII. 

(1.) Which is the greater ratio, 5 : 7 or 151 : 208 ? 

(2. ) If the ratio 3 : 4 be duplicated by subtracting x from both antecedent 
and consequent, show that a' = lf. 

(3.) What quantity x added to the antecedent and to the consequent of 
a : b will convert this ratio into c : d ? 

(4.) Find the fourth proportional to 3£, 5|, 6| ; also the third proportional 
to 1 + V2 and 3 + 2\/2. 

(5.) Insert a mean proportional between 11 anil 19 ; and also two mean 
proportionals between the same two numbers. 

(6.) Find a simple surd number which shall be a mean proportional be- 
tween \J7 - \/5 and 11V7+ 13\/5- 

(7.) If x and y be such that when they are added to the antecedent and 
consequent respectively of the ratio a : b its value is unaltered, show that 
x:y = a :b. 

(8. ) If x and y be such that when they are added respectively to the ante- 
cedent and consequent and to the consequent and antecedent of a : b the two 
resulting ratios are equal, show that either x = y or x + y = - a—b. 

(9.) Find a quantity x such that when it is added to the four given quan- 
tities a, b, c, d the result is four quantities in proportion. Exemplify with 
3, 4, 9, 13 ; and with 3, 4, 1£, 2. 

(10.) If four quantities be proportional, the sum of the greatest and least 
is always greater than the sum of the other two. 

(11.) If the ratio of the difference of the antecedents of two ratios to the 
sum of their consequents is equal to the difference of the two ratios, then the 
antecedents are in the duplicate ratio of the consequents. 

(12.) If the n quantities d\, a. 2 , . . ., o n be in continued proportion, then 
m : a n = a 1 "- 1 : a a n-1 =aa"~ 1 : a-i 11 ' 1 = &c. ; and 

a,= "'^(a 1 "--a n ), a s = VW-V,,), . . ., « P = "v / ("i"-'Vr, 1 '- 1 ). 
(13.) If (pa+qb+rc+sd)(pa-qb-rc+sd) 

= {pa-qb + rc- sd) (pa + qb-rc- sd), 
then be : ad =ps : qr ; 

and, if either of the two sets a, b, c, d or p, q, r, s form a proportion, the 
other will also. 

(14.) If a: b=c:d=e:f, 
then a 3 + 3a n -b + b*:c 3 + Zc-d + cP = a* + &' : c 3 + d 3 (a) ; 

//«V «V eV\ f(bd- b"P d-f-\ 

= a\lf+ c s b/+ c*bd : b 3 cc + tPae +J*ac (/3) ; 
pa - qc + re:pb-qd+ rf= ^/ace : y/bdf 

= vV - c °- + e " + 2ac ) : V(& 2 - cp +f + - hd ) M- 



268 EXERCISES XVII chap. 

t 

(15.) Ua:a'=b:b', 

then «'"+" + a'"b" + &"•+" : «'"'+" + a' m b'" + b >mr < " 

= («. + ft)»'+": («' + ft')" !+ ". 

(16.) If «:& = c : rf, and a 1/3 = 7: 5, 
then « 3 a 2 + (a 2 ft + aft 2 )a/3 + ft 3 /3 2 : (a 3 + ft 3 ) (a 2 + /3 2 ) 

= cV + (c 2 ^ + (5^)75 + d 3 8 2 : (c 3 + d 3 ) (y~ + 8 s ). 

(17.) If a: b = b :c = c:d, then 

[a? + ft 2 + e 2 ) (ft 2 + c 2 + d 2 ) = (ab + be + cdf (a) ; 

(ft - c) 2 + (c - a) 2 + (d - ft) 2 = (a - d) 2 (j8) ; 

«ft + crf + ft^=(« + ft + c)(ft-c + f?) (7); 

a + 6-c-tf = (a + ft)(ft-f0/ft (0); 

(a + 6 + c + a") (a- ft - 6 + 0") = 2(aft - erf) (ac - bd)/(ad + ftc) (e). 

(IS.) If a, ft, c be in continued proportion, 

then a 2 + aft + ft 2 : b 2 + be + c 2 = a : c (a); 

a 2 («-ft + c)(a + ft + c) = « 4 + a 2 ft 2 + ft 4 (/3) ; 

(ft + c) 2 /(ft-f) + (c + «) 2 /(c-«) + (a + ft) 2 /(«-ft) = 4ft(a + ft + c)/(«-c) (7). 

(19.) If a, ft, c, d he in continued proportion, 

then («-c)(ft-rf)-(«-^)(?'-c) = (&-c) 2 (a); 

V(a*)W(k)+V(«9=V{(« +*+«)(* +«+/>} (£)■ 

(20.) Uab=cd=ef, then 

(ac + ce + ea)/dbf(d + ft +/) = (a 2 + c 2 + e 2 )/(ft 2 d 2 + d 2 / 2 +/'-ft-'). 

(21.) If (a-J)/(rf-e)=(6-c)/(e-/) 1 then each of them 

= {b(f-d) + (cd-af)\/c(f-d). 

(22.) Ul;/yx=vli?=tfyz, thenx/^ = y/e = ^vi-. 
(23.) If 2a!+3y:3i/+4z:43+5aj=4a-56:3ft-o:26-3a J 
then 7a; + 6y+8«=0. 

(24.) If ax + cy :by + dz = ay + cz:bz + dx = az + ex : bx + dy, and if 
x + y + z + 0, ab-cd + 0, ad-bc + 0, then each of these ratios =a + c : ft + d ; 
and x 2 + y 2 + z 2 = yz + zx + xy. 

(25.) If (a - ny + mz)jl' = (b-7z + nx)/m'=(c - <mx + ly)/n', then 
/ m'c-n'b \n_f ii'a-l'c \, _f /'ft - m'a \ , » 

V X " K' + mm' + MwV ' ^ y ' U'+mm'+nn'J ' m ~[ z ~U' + m m' + n n'J' 71 ' 



RATIO AND PROPORTION OF CONCRETE QUANTITIES. 

§ 11.] AVe have now to consider how the theorems Ave have 
established regarding the ratio and proportion of abstract num- 
bers are to be applied to concrete quantities. We shall base 

* Important in the theory of the central axis of a system of forces, &c. 



xill CONCRETE RATIO AND PROPORTION 2G9 

this application on the theory of units. This, for practical pur- 
poses, is the most convenient course, but the student is not to 
suppose that it is the only one open to us. It may be well to 
recall once more that any theory may be expressed in algebraical 
symbols, provided the fundamental principles of its logic are in 
agreement with the fundamental laws of algebraical operation. 

§ 12.] If A and B be two concrete quantities of the same hind, 
which are expressible in terms of one and the same unit by the com- 
mensurable numbers a and b respectively, then the ratio of A to B is 
defined to be the ratio or quotient of these abstract numbers, namely, 
a : b, or ajb. 

It should be observed that, by properly choosing the unit, the ratio of 
two concrete quantities which are each commensurable with any finite unit at 
all can always be expressed as the ratio of two integral numbers. For ex- 
ample, if the quantities be lengths of Z\ feet and 4f feet respectively, then, 
by taking for unit |th of a foot, the quantities are expressible by 26 and 35 
respectively ; and the ratio is 26 : 35. This follows also from the algebraical 
theorem that (3 + £)/(4 + |) = 26/35. 

If A, B be two concrete quantities of the same kind, whose ratio is 
a : b, and C, D two other concrete quantities of the same kind {but not 
necessarily of the same kind as A and B), whose ratio is c:d, then 
A, B, C, D are said to be proportional when the ratio of A to B is 
equal to the ratio of C to D, that is, wht n 

a:b = c: d. 

We may speak of the ratio A : B, of the concrete magnitudes 
themselves, and of the proportion A : B = C : D, without alluding 
explicitly to the abstract numbers which measure the ratios ; but 
all conclusions regarding these ratios will, in our present manner 
of treating them, be interpretations of algebraical results such as 
we have been developing in the earlier part of this chapter, 
obtained by operating with a, b, c, d. The theory of the ratio 
and proportion of concrete quantity is thus brought under the 
theory of the ratio and proportion of abstract quantities. 

There are, however, several points which require a nearer 
examination. 

§ 13.] In the first place, it must be noticed that in a concrete 



270 



SPECIAL POINTS IN CONCRETE PROPORTION 



CHAP. 



ratio the antecedent and the consequent must be quantities of the 
same kind ; and in a concrete proportion the two first terms must 
be alike in kind, and the two last alike in kind. Thus, from the 
present point of view at least, there is no sense in speaking of 
the ratio of an area to a line, or of a ton of coals to a sum of 
money. Accordingly, some of the propositions proved above — 
those regarding the permutations of a proportion, for instance — 
could not be immediately cited as true regarding a proportion 
among four concrete magnitudes, unless all the four were of the 
same kind. 

This, however, is a mere matter of the interpretation of 
algebraical formulae — a matter, in short, regarding the putting of 
a problem into, and the removing of it from, the algebraical 
machine. 

§ 14.] A more important question arises from the considera- 
tion that, if we take two concrete 
magnitudes of the same kind at random, 
there is no reason to expect that there 
exists any unit in terms of which each 
is exactly expressible by means of com- 
mensurable numbers. 

Let us consider, for example, the 
historically famous case of the side AB 
and diagonal AC of a square ABCD. On the diagonal AC lay 
off AF = AB, and draw FE perpendicular to AC. It may be 
readily shown that 





3 




A 




E 






/ 


\ 






\ 




SF 




\ 








c 


\ 




D 



Hence 



BE = EF = FC. 
CF = AC - AB 
CE = CB - CF 



(2). 



Now, if AB and AC were each commensurably expressible in 
terms of any finite unit, each would, by the remark in § 1 2, be an 
integral multiple of a certain finite unit. But from (1) it follows 
that if this were so, CF would be an integral multiple of the 
same unit j and, again, from (2), that CE would be an integral 
multiple of the same unit. Now CF and CE are the side and 
diagonal of a square, CFEG-, whose side is less than half the side 



xiu COMMENSURABLES AND INCOMMENSURABLES 271 

of ABCD; and from CFEG could in turn be derived a still 
smaller square whose side and diagonal would be integral mul- 
tiples of our supposed unit ; and so on, until we had a square 
as small as we please, whose side and diagonal are integral 
multiples of a finite unit; which is absurd. Hence the side 
and diagonal of a square are not magnitudes such as A and B 
are supposed to be in our definition of concrete ratio. 

§ 15.] The difficulty which thus arises in the theory of con- 
crete ratio is surmounted as follows : — 

We assume, as axiomatic regarding concrete ratio, that if 
A' and A" be two quantities respectively less and greater than 
A, then the ratio A : B is greater than A' ; B and less than 
A" : B ; and we show that A' and A" can be found such that, 
while each is commensurable with B, they differ from each other, 
and therefore each differs from A by as little as we please. 

Suppose, in fact, that we take for our unit the nth part of B, 
then there will be two consecutive integral multiples of Bjn, say 
mB/n and (m + l)Bfn, between which A will lie. Take these 
for our values of A' and A" ; then 

A" - A' = (m + 1)B/tc - mBjn, 

= B>. 

Hence A" - A' can, by sufficiently increasing n, be made as small 
as Ave please. 

We thus obtain, in accordance with the definition of § 12, 
two ratios, m\n and (m + l)/n, between which the ratio A : B lies, 
each of which may be made to differ from A : B by as little as 
we please. 

Practically speaking, then, we can find for the ratio of two 
incommensurables an expression which shall be as accurate as 
we please. Regarding this matter, see vol. ii., chap, xxv., §§ 26-41. 

Example. 

If B be the side and A the diagonal of a square, to find a rational value 
of A : B which shall be correct to l/1000th. 

If we take for unit the l/1000th part of B, then B = 1000, and A 2 = 
2,000,000. Now 1414'-'= 1999396, and 1415 2 = 2002225. Hence 1414/I00O 
<A/B< 1415/1 000. But 1415/1000-1414/1000 = 1/1000. Hence we have 
A/B = 1-414, the error being < 1/1000. 



272 EUCLIDIAN THEORY OF PROPORTION chap. 

§ 16.] The theory of proportion given in Euclid's Elements gets over the 
difficulty of incommensurables in a very ingenious although indirect manner. 
No working definition of a ratio is attempted, but the proportionality of four 
magnitudes is defined substantially as follows : — 

If there be four magnitudes A, B, C, D, such that, always, 

mk>, = , or <«B, 
according as mC>, =, or <?;D, 

m and n being any integral numbers whatsoever, then A, B, C, D are said to 
be proportional. 

Here no use is made of the notion of a unit, so that the difficulty of in- 
commensurability is not raised. On the other hand, there is substituted a 
somewhat indirect and complicated method for testing the subsistence or non- 
subsistence of proportion ality. 

It is easy to see that, if A, B, 0, D be proportional according to the 

algebraical definition, they have the property of Euclid's definition. For, if 

a:b and c : d be the numerical measures of the ratios A :B and"C:D, we 

have 

a _c 

b~d' 

, ma mc 

hence — r = — ,, 

no nd 

from which it follows that ma>, —, or <nb, according as mc> , =, or <nd. 

The converse, namely, that, if A, B, C, D be proportional according to 

Euclid's definition, then 

a _c 

can be proved by means of the following lemma. 

Given any commensurable quantity ajb, another commensurable quantity 
can be found which shall exceed or fall short of a/b by as little as we please. 

Let n be an integral number, and let mb be the least multiple of b which 
exceeds na, so that 

na—mb-r, 
where r < b. 

Dividing both sides of this equation by rib, we have 

a _ in r 
b u nb ' 

, m a r 

whence T = _ i;» 

n b vb 

so that m/n exceeds a/b by r/nb. Now, since r never exceeds the given 
quantity b, by making n sufficiently great, we can make r/nb as small as we 
plea.se ; that is to say, we can make m/n exceed a/b by as little as we please. 

Similarly we may show that another commensurable quantity may be 
found falling short of a/b by as little as we please. 

From this it follows that, if two commensurable quantities differ by ever so 
little, we can always find another commensurable quantity which lies between 



xui COMPARISON OF THEORIES 273 

them ; for we can find another commensurable quantity which exceeds tlic 
less of the two by less than the difference between it and the greater. 
Suppose now that 

ma >, =, or <nb, 
according as mo, —, or <nd, 

m and n being any integers whatever, then we must have 

a _ c 
b~d' 
For, if these fractions (which we may suppose to be commensurable by 
virtue of § 15) dilfer by ever so little, it will be possible to find another 
fraction, n/m say, where n and m are integers, which lies between them. 
Hence, if a/b be the less of the two, we must have 

t< — , that is, ma < nb ; 
b m 

-> — . that is, mond. 
d m 

In other words we have found two integers, m and n, such that we have 
at once 

ma < nb 
and mond. 

But, by hypothesis, when ma<nb, we must have mc<nd. Hence the 
fractions a/b and c/d cannot be unequal. 



VARIATION. 

§ 17.] There are an infinite number of ways in which we 
may conceive one quantity y to depend upon, be calculable from, 
or, in technical mathematical language, be a function of, another 
quantity x. Thus we may have, for example, 

y = 3&, 

y =17/, 

y = ax + b, 

y = ax 2 + bx + c, 

y='2 \ f x, 
and so on. 

For convenience x is called the independent variable, and y the 
dependent variable ; because we imagine that any value we please 
is given to x, and the corresponding value of y derived from it 
by means of the functional relation. All the other symbols of 
quantity that occur in the above equations, such as 3, 17, a, b, c, 
VOL. I T 



274 INDEPENDENT AND DEPENDENT VARIABLE chap. 

2, &c, are supposed to remain fixed, and are therefore called 
constants. 

Here we attach meanings to the words variable and constant 
more in accordance with their use in popular language than 
those given above (chap, ii., § 6). 

The justification of the double usage, if not already apparent, 
will be more fully understood when Ave come to discuss the 
theory of equations, and to consider more fully the variations of 
functions of various kinds (see chaps, xv.-xviii.) 

§ 18.] In the meantime, we propose to discuss very briefly 
the simplest of all cases of the functional dependence of one 
quantity upon another, that, namely, which is characterised by 
the following property. 

Let the following scheme 



Values of 
the Independent Variable. 


Corresponding Values of the 
Dependent Variable. 


X 

x' 


V 

y 



denote any two corresponding pairs whatever of values of the 
independent and dependent variables, then the dependence is to 
be such that always 

y:y' = x:x' (1). 

It is obvious that this property completely determines the 
nature of the dependence of y upon x, as soon as any single cor- 
responding pair of values are given. Suppose, in fact, that, 
when x has the value x , y has the value y , then, by (1), 

y x 

y ~x ' 

whence y = ( — )x. 

Now we may keep x u and y„ as a fixed standard pair, for 
reference as it were ; their ratio y /x is therefore a given con- 



Xlll SIMPLEST CASE OF FUNCTIONAL DEPENDENCE 275 

stant quantity, which we may denote by a, say. We therefore 
have 

V = ax (2), 

that is to say, y is a given constant multiple of x ; or, in the 
language of chap, iv., § 17, a homogeneous integral function of x 
of the 1st degree. 

Example. Let us suppose that we have for any two corresponding pairs 
y, x and y', x' the relation y :x = y' :x' ; and that when x=Z, y = 6- Then 
since 6 and 3 are corresponding pairs y :x=6 :3. Hence y/x — 6/B = 2. 
Hence y = 2x. 

Conversely, of course, the property (2) leads to the property 
(1). For, from (2), 

y = ax ; 

hence, if x' and y' be other two corresponding values, 

y' = ax'. 

y ax x 
Hence '—~ — , = — • 

y ax x 

When y depends on x in the manner just explained it is said 
to vary directly as x, or, more shortly, to vary as x. 

A better * phrase, which is also in use, is " y is proportional 
to x." 

This particular connection between y and x is sometimes 

expressed by writing 

y<^x. 

§ 19.] In place of x, we might write in equation (2) x 2 , l/x, 
1/x 2 , x + b, and so on ; we should then have 

y = ax 2 (a), 

y = ajx ((3), 

y = a\i (y), 

y = a(x + b) (8). 

* The use of the word " Variation " in the present connection is unfortunate, 
because the qualifying particle " as " is all that indicates that we are here 
concerned not with variation in general, as explained in § 17, but merely with 
the simplest of all the possible kinds of it. There is a tendency in uneducated 
minds to suppose that this simplest of all kinds of functionality is the only 
one ; and this tendency is encouraged by the retention of the above piece of 
antiquated nomenclature. 



276 OTHER SIMPLE CASES chap. 

The corresponding forms of equation (1) would then be 

V : :'/ = X 2 : X" (a'), 

y:y' = llx:l/x' (ft'), 

T .y = 1/aM/i" (/), 

y.y' = x + b: x' + b (8'). 

y is then said to vary as, or be proportioned to, x 2 , l/'x, Ijx 2 , 
x + b. In cases (ft) and (y) y is sometimes said to vary inversely 
as x, and inversely as the square of x respectively. 

Still more generally, instead of supposing the dependent 
variable to depend on one independent variable, we may suppose 
the dependent variable u to depend on two or more independent 
variables, x, y, z, &c. 

For example, we may have, corresponding to (2), 



and, corresponding to (1), 



u = axy 


(4 


u = axyz 


(0, 


u = a(x + y) 


(v), 


u = ax/y 


{0); 


u : to' = xy : x'y' 


(0, 


u : m' = xyz : x'y'z' 


(0, 


u:u' = x + y. x' + y 


(v')> 


to : u' = xjy : x'jy' 


(d'\ 



In case (e) u is sometimes said to vary as x and y jointly ; 
in case (6) directly as x and inversely as y. 

§ 20.] The whole matter we are now discussing is to a large 
extent an affair of nomenclature and notation, and a little 
attention to these points is all that the student will require to 
prove the following propositions. We give the demonstrations 
in one or two specimen cases. 

(1 .) If z°=y and y ^ x, then z<=>=x. 

Proof. — By data z = ay, y = bx, where a and b are constants ; 
therefore z = dbx. Hence z^x, since ah is constant. 

(2.) If y i ' x x l and y 2 ^x 2 , then y x y 2 °= ;<•/.,. 

Proof. — By data y t = a x x lt y 2 = a^, where a, and a 2 are con- 



XIII 



PROPOSITIONS REGARDING VARIATION 



277 



stants. Hence y$ 9 = a^v^, which proves the proposition, since 
ai« 2 is constant. 

In general if y,^-x„ y,^x,, . . ., y n <=^x n , then y l y 2 ...y n 
oc #,£ 2 . . . x w And, in particular, if y cc x, then y n <^ x n . 

(3.) If y°zx, then zy°czx, whether z be variable or constant. 

(4.) If zozxy, then x^zjy, and y^zjx. 

(5.) If z depend on x and y, and on these alone, and if z^x 
when y is constant, and z°^y when x is constant, then z °= xy when 
both x and y wry. 

Proof. — Consider the following system of corresponding 
values of the variables involved. 



Dependent Variable. 


Independent Variables. 


Z 

z' 


x, y. 
x', y. 
x', y'. 



Then, since y lias the same value for both z and z u we have, 

by data, 

z x 



2, X 




Again, since x' is the same for both z l 


and z', we have, by 


data, 




»i y 




z' y- 




From these two equations we have 




z z x x y 

7 x I 7 = v x 7? ' 
Zi z x y 




1.1. l • z xy 

thatls ' ?-*?■ 





which proves that z^xy. 

A good example of this case is the dependence of the area of a triangle 
upon its base and altitude. 



278 EXAMPLES chap. 

"We have 

Area oc base (altitude constant) ; 
Area oc altitude (base constant). 
Hence area oc base x altitude, when both vary. 

(6.) In a similar manner we may prove that if z depend on 
x l , x 2 , . . . , x n , and on these alone, and vary as any one of these when 
the rest remain constant, then z oc x x x 2 . . . x n when all vary. 

(7.) If zee x (y constant) and z<^l/y (x constant), then z<>=x/y 
when both vary. 

For example, if V, P, T denote the volume, pressure, and absolute tem- 
perature of a given mass of a perfect gas, then 

V oc 1/P (T constant), V oc T (P constant). 
Hence in general V oc T/P. 

Example 1. 

If s oc t 2 when/ is constant, and s oc/ when t is constant, and 2s=/ when 
t = \, find the relation connecting s, f, t. 

It follows by a slight extension of § 20 (5) that, when/ and t both vary, 
s <xft 2 . Hence s = aft", where a is a constant, which we have to determine. 

Now, when t = l, s = \f, hence \f=afl 2 , that is, \f—af\ in other words, 
we must have a—h. The relation required is, therefore, s — \ffi. 

Example 2. 

The thickness of a grindstone is unaltered in the using, but its radius 
gradually diminishes. By how much must its radius diminish before the 
half of its mass is worn away ? Given that the mass varies directly as the 
square of the radius when the thickness remains unaltered. 

Let m denote the mass, r the radius, then by data, m = ar-, where a is 
constant. 

Let now r become ?•', and, in consequence, m become Jm, then b>i = ar' 2 , 
hence 



ar' 2 


_hn 


: 




ar 2 


~ m' 


r >2 

r- 


=h; 


r' 


l 


— 


^ 


r 


\A 



that is, 

whence 

It follows, therefore, that the radius of the stone must be diminished in 
the ratio 1 : V2. 

Example 3. 

A ami B are partners in a business in which their interests are in the 
ratio a : b. They admit C to the partnership, without altering the whole 
amount of capital, in such a way that the interests of the three partners in 
the business are then equal. C contributes £c to the capital of the firm. 



XIII 



EXERCISES XVIII 279 



How is the sum £c which is withdrawn from the capital to be divided between 
A and B ? and what capital had each in the business originally ? 

Solution. — Since what C pays in is his share of the capital, they each have 
finally £c in the business ; let now £x be A's share of C's payment, so that 
£{c-x) is B's share of the same. In effect, A takes £x and B £(c-x) out 
of the business. Hence they had originally £{c + x) and £(c + c-x) in the 
business. By data, then, we must have 

c + x _a 
2c-x~b' 
hence b(c + x) = a(2c-x) ; 

we have, therefore, bc + bx — 2ac - ax. 

From this last equation we derive, by adding ax - be to both sides, 

{a + b)x=(2a-b)c. 

Heuce, dividing by a + b, we have 

x J2a-V)c 

a + b v ' 

Hence c-x=c-- 



a + b 
(2b-a)c 
a + b 



(2). 



It appears, then, that A and B take £(2a - b)c/{a + b) and £{2b - a)c/(a + b) 
respectively out of the business. C's payment must be divided between them 
in the ratio of these sums, that is, in the ratio 2a- b :2b -a. They had in 
the business originally £3ac/(a + b) and £3bc/(a + b) respectively. 



Exercises XVIII. 

(1.) Ifyccx, and if y = 3§ when a; = 6i, find the value of y when a-=g. 

(2. ) y varies inversely as x 2 ; and z varies directly as ,<:-. When x= 2, y + z 
= 340 ; when ar=l, ?/-c = 1275. For what value of a; is y = zl 

(3.) zeeu-v; uxx; vecxr. When x = 2, s = 48 ; when x = 5, z = 30. 
For what values of x is c = ? 

(4.) If xy oc x 2 + y' 2 , and x = Z when y = 4, find the equation connecting 
y and x. 

(5.) If x + y<xx-y, then x 2 + y 2 <x xy and oP + y 3 <x .ri/(.>-±y). 

(6.) If {x + y + z)(x + y-z)(x-y + z)( -x + y + z) oc x'hf, then either x* + y 2 
oc zr or xr ■>- y — z- oc xy. 

(7.) Hxxy, then x 2 + y 2 cc xy. 

(8. ) If a* + \ « x 3 - i, then y <x. Ifx. 

(9. ) If x oc y 2 , y 3 oc z 4 , z 5 oc u e , tt 7 oc ^ then (x/v) (y/v) (z/v) (ujv) is constant. 
(10.) Two trains take 3 seconds to clear each other when passing in 
opposite directions, and 35 seconds when passing in the same direction : find 
the ratio of their velocities. 



280 EXERCISES XVIII chap, xiii 

(11.) A watch loses 2\ minutes per day. It is set right on the 15th March 
at 1 p.m. : what will the proper time be when it indicates 9 A.M. on the 20th 
April 1 

(12.) A small disc is placed between two infinitely small sources of radiant 
heat of equal intensity, at a point on the line joining them equidistant from 
the two. It is then moved parallel to itself through a distance aj2\/S towards 
one of the two sources, a being the distance between them : show that the 
whole radiation falling on the disc is trebled. 

(The radiation falling on the disc varies inversely as the square of the dis- 
tance from the source, when the disc is moved parallel to itself towards or 
from the source.) 

(13.) The radius of a cylinder is ?•, and its height h. It is found that by 
increasing either its radius or its height by x its volume is increased by the 
same amount. Show that x = r(r-2h),'h. "What condition is there upon r 
and h in order that the problem may be possible ? 

(Given that the volume of a cylinder varies directly as its height when 
its radius is constant, and directly as the square of its radius when its height 
is constant.) 

(14.) A solid spherical mass of glass, 1 inch in diameter, is blown into a 
shell bounded by two concentric spheres, the diameter of the outer one being 
3 inches. Calculate the thickness of the shell. (The volume of a sphere 
varies directly as the cube of its diameter. ) 

(15.) Find, the radius of a sphere whose volume is the sum of the volumes 
of two spheres whose radii are 3^ feet and 6 feet respectively. 

(16.) Two equal vessels contain spirits and water, the ratios of the amount 
of spirit to the amount of water being p : 1 and p : 1 respectively. The con- 
tents of the two are mixed : show that the ratio of the amount of spirit to the 
amount of water in the mixture is p + p' + 2pp' :2 + p + p'. 



CHAPTER XIV. 
On Conditional Equations in General. 

DEFINITIONS AND GENERAL NOTIONS. 

§ 1 .] It will be useful for the student at this stage to attempt 
to form a wider conception than we have hitherto presupposed of 
what is meant by an analytical function in general. Dividing the 
subjects of operation into variables (x, y, z, . . .) and constants 
(a, b, c, . . .), we have already seen what is meant by a rational 
integral algebraical function of the variables x, y, z, . . .; and we 
have also had occasion to consider rational fractional algebraical 
functions of x, y, z, . . . We saw that in distinguishing the 
nature of such functions attention was paid to the way in which 
the variables alone were involved in the function. We have already 
been led to consider functions like s/(x + Jy), or %/(x + >/y), 
or ax* + bx- + c, where the variables are involved by way of root 
extraction. Such functions as these are called irrational alge- 
braical functions. These varieties exhaust the category of what 
are usually called Ordinary * Algebraical Functions, In short, any 
intelligible concatenation of operations, in which the operands selected 
for notice and called the variables are involved in no other ways than 
by addition, subtraction, multiplication, division, and root extraction, is 
called an Ordinary Algebraical Function of these variables. 

Although Ave have thus exhausted the category of ordinary 
algebraical functions, we have by no means exhausted the possi- 

* The adjective " Ordinary " is introduced to distinguish the class of func- 
tions here defined from algebraical functions as more generally denned in 
chap, xxx., § 10. The word "Synthetic" is often used for "Ordinary" in 
the present connection. 



282 



CONDITIONAL AND IDENTICAL EQUATIONS 



CHAP. 



bilities of analytical expression. Consider for example a* where, 
as usual, x denotes a variable and a a constant. Here x is not 
involved in any of the Avays recognised in the definition of an 
algebraical function, but appears as an index or exponent. a x is 
therefore called an exponential function of x. It should be care- 
fully noted that the discrimination turns solely on the way in 
which the variable enters. Thus, while a x is an exponential 
function of x, x 0, is an algebraical function of x. There are other 
functions in ordinary use, — for example, sin a', logic, — and an 
infinity besides that might be imagined, which do not come 
under the category of algebraical ; all such, for the present, we 
class under the general title of transcendental functions, so that 
transcendental simply means non-algebraical. We use the term 
analytical function, or simply function, to include all functions, 
Avhether algebraical or transcendental, and we denote a function 
of the variables x, y, z, . . ., in which the constants a, b, c, . . . 
are also involved, by 

4>(x, y, z, . . . a, b, c, . . .); 
or, if explicit mention of the constants is unnecessary, by 

<f>(x, y, z, . . .). 

§ 2.] Consider any two functions whatever, say cf>(x, y, z, . . . 
a, b, c, . . .), and if(x, y, z, . . . a, b, c, . . .), of the variables 
x, y, z, . . ., involving the constants a, b, c, . . . 

If the equation 
<f>(z, y,z, . . . a, b, c, . . .) = f(z, y, z, . . . a, b, c, . . .) (1) 
be such that the left-hand side can, for all values of the variables 
x, y, z, . . ., be transformed into the right by merely apply- 
ing the fundamental laws of algebra, it is called an identity. 
With equations of this kind the student is already very familiar. 

If, on the other hand, the left-hand side of the equation (1) 
can be transformed into the right only when x, y, z, . . . have 
certain values, or are conditioned in some way, then it is said to be 
a Conditioned Equation, or an Equation of Condition.* Examples 

* "When it is necessary to distinguish between an equation of identity and 
an equation of condition, the sign = is used for the former, and the sign = 
for the latter. Thus, we should write [x + 1) (»- l)=u? - 1 j but 1c + 2 = 2. 



xiv CLASSIFICATION OF EQUATIONS 283 

of such equations have already occurred, more especially in chap, 
xiii. One of the earliest may be seen in chap, iv., § 24, where, 
inter alia, it was required to determine B so that we should 
have 2B + 2 = 2 ; in other words, to find a value of x to satisfy 
the equation 

2x+2 = 2 (2). 

Here 2x + 2 can be transformed into 2 when (and, as we 
shall hereafter see, only when) x = 0. 

Every determinate problem, wherein it is required to deter- 
mine certain unknown quantities in terms of certain other given 
or known quantities by means of certain given conditions, leads, 
when expressed in analytical language, to one or more equations 
of condition ; to as many equations, in fact, as there are condi- 
tions. The quantities involved are therefore divided into two 
classes, known and unknown. The known quantities are denoted 
by the so-called constant letters ; the unknown by the variable 
letters. Hence, in the present chapter, constant and known are 
convertible terms ; and so are variable and unknown. The con- 
stants may be actual numerical quantities, real and positive or 
negative (-4, -\, 0, +1, + f , &c), or imaginary or complex 
numbers ( - i, 1 + 2i, &c); or they may be letters standing for 
any such quantities in general. 

§ 3.] Equations are classified according to their form, and 
according to the number of variables that occur in them. 

If transcendental functions appear, as, for example, in 
2* = 2> x + 2, the equation is said to be transcendental. With 
such for the present we shall have little to do. 

If only the ordinary algebraical functions appear, as, for 
example, in «/(jc + y) + \/(x - y) = 1 , the equation is called an 
algebraical equation. Such an equation may, of course, be 
rational or irrational, and, if rational, either fractional or 
integral, according to circumstances. 

It will be shown presently that every algebraical equation 
can be connected with, or made to depend upon, an equation 
of the form 



284 MEANING OF SOLUTION chap. 

where <t> is a rational integral function. Such equations are 
therefore of great analytical importance ; and it is to them that 
the " Theory of Equations," as ordinarily developed, mainly 
applies. An integral equation of this kind is described by 
assigning its degree and the number of its variables. The degree 
of the equation is simply the degree of the function <f>. Thus, 
of + Ixy + if - 2 = is said to be an equation of the 2nd degree 
in two variables. 

§ 4.] Equations of condition may occur in sets of one or of 
more than one. In the latter case we speak of the set as a set 
or system of simultaneous equations. 

The main problem which arises in connection with every system of 
equations of condition is to find a set or sets of values of the variables 
which shall render every equation of the system an identity literal oi 
numerical. 

Such a set of values of the variables is said to satisfy the 
system, and is called a solution of the system of equations. If 
there be only one equation, and only one variable, a value of 
that variable which satisfies the equation is called a root. We 
also say that a solution of a system of equations satisfies the system, 
meaning that it renders each equation of the system an identity. 

It is important to distinguish between two very different 
kinds of solution. When the values of the variables which con- 
stitute the solution are closed expressions, that is, functions of 
known form of the constants in the given equations, we have 
what may be called a formal solution of the system of equations. 
In particular, if these values be ordinary algebraical functions 
of the constants, we have an algebraical solution. Such solutions 
cannot in general be found. In the case of integral algebraical 
equations of one variable, for example, if the degree exceed the 4th, 
it has been shown by Abel and others that algebraical solutions 
do not exist except in special cases, so that the formal solution, 
if it could be found, would involve transcendental functions. 

When the values of the variables which constitute the solution 
are given approximately as numbers, real or complex, the solution 
is said to be an approximate numerical solution. In this case the 



xiv EXAMPLES OF SOLUTION 285 

words " render the equation a numerical identity " are understood 
to mean "reduce the two sides of the equation to values 
which shall differ by less than some quantity which is assigned." 
For example, if real values of the two sides, say P and P', 
are in question, then these must be made to differ by less than 
some given small quantity, say 1/100,000; if complex values 
are in question, say P + Qi and P' + Q'i, then these must 
be so reduced that the modulus of their difference, namely, 
\/{(P - P') 2 + (Q - Q') 2 }> shall be less than some given small 
quantity, say 1/100,000. (Cf. chap, xii., § 21.) 

As a matter of fact, numerical solutions can often be obtained 
where formal solutions are out of the question. Integral alge- 
braical equations, for example, can always be solved numerically 
to any desired approximation, no matter what their degree. 

Example 1. 

2x + 2 = 2. 
jb=0 is a solution, for this value of x reduces the equation to 

2x0 + 2 = 2, 
which is a numerical identity. Strictly speaking, this is a case of algebraical 
solution. 

Example 2. 

ax — b 2 = 0. 

x = b-ja reduces the equation to 

a — b- = 0, 
a 

which is a literal identity ; hence x = b 2 /a is an algebraical solution. 

Example 3. 

x" - 2 = 0. 

Here x— + \/2 and x= - \J2 each reduce the equation to the identity 

2-2=0 ; 
these therefore are two algebraical solutions. 

On the other hand, a;=+l - 4142 and x = — 1*4142 are approximate 
numerical solutions, for each of them reduces x-~2 to - "00003836. which 
differs from by less than '00004. 

Example 4. 

0'-l) 2 + 2 = 0. 
x = \ + \/2i and x—l-\/2i are algebraical solutions, as the student will 
easily verify. 

.T = 10001 + l - 4142i and a;= 1*0001 - 1 - 4142i are approximate numerical 
solutions, for they reduce (.r- l) 2 + 2 to "00003837+ -00028284;:' and "00003837 
- , 00028284t respectively, complex numbers whose moduli are each less than 
"0003. 



286 CONDITIONAL EQUATION A HYPOTHETICAL IDENTITY chap. 

Example 5. 

x-y-1. 

Here se=l, y=0, is a solution ; so is a;=l"5, y = 5 ; so is x = 2, y — 1 ; and, 
in fact, so is x = a+ 1, y = a, where a is any quantity whatsoever. 

Here, then, there are an infinite number of solutions. 

Example 6. Consider the following system of two equations : — 

x — y=\, 2x + y—5. 
Here x=2, y = l is a solution ; and, as we shall show in chap, xvi., there is 
no other. 

The definition of the solution of a conditional equation 
suggests two remarks of some importance. 

1st. Every conditional equation is a hypothetical identity. In all 
operations with the equation ive suppose the variables to have such 
values as will render it an identity. 

2nd. The ultimate test of every solution is that the values which it 
assigns to the variables shall satisfy the equations when substituted therein. 

No matter how elaborate or ingenious the process by which 
the solution has been obtained, if it do not stand this test, it is 
no solution ; and, on the other hand, no matter how simply 
obtained, provided it do stand this test, it is a solution.* In 
fact, as good a Avay of solving equations as any other is to guess 
a solution and test its accuracy by substitution.! 

§ 5.] The consideration of particular cases, such as Examples 
1-6 of § 4, teaches us that the number of solutions of a system of 
one or more equations may be finite or infinite. If the number 
be finite, we say that the solution is determinate (singly determin- 
ate, or multiply determinate according as there are one or more 
solutions) ; if there be a continuous infinity of solutions, we say 
that the solution is indeterminate. 

The question thus arises, Under what circumstances is the 
solution of a system of equations determinate 1 Part at least of 
the answer is given by the following fundamental propositions. 

Proposition I. The solution of a system of equations is in general 
determinate (singly, or multiply according to circumstances) when the 
number of the equations is equal to the number of the variables. 



* A little attention to these self-evident truths would save the beginner 
from many a needless blunder. 

t This is called solving by "inspection." 



xiv PROPOSITIONS AS TO DETERMINATENESS OF SOLUTION 287 

Rightly considered, this is an ultimate logical principle which 
may be discussed, but not in any strictly general sense proved. 
Let us illustrate by a concrete example. The reader is aware 
that a rectilinear triangle is determinable in a variety of ways 
by means of three elements, and that consequently three condi- 
tions will in general determine the figure. To translate this into 
analytical language, let us take for the three determining elements 
the three sides, whose lengths, at present unknown, we denote 
by x, y, z respectively. Any three conditions upon the triangle 
may be translated into three equations connecting x, y, z with 
certain given or constant quantities ; and these three equations 
will in general be sufficient to determine the three variables, 
x, y, z. The general principle :VL common to this and like cases is 
simply Proposition I. The truth is that this proposition stands 
less in need of proof than of limitation. What is wanted is an 
indication of the circumstances under which it is liable to excep- 
tion. To return to our particular case : What would happen, 
for example, if one of the conditions imposed upon our triangle 
were that the sum of two of the sides should fall short of the 
third by a given positive quantity % This condition could be 
expressed quite well by an equation (namely, x + y = z - q, say), 
but it is fulfilled by no real triangle, f Again, it might chance 
that the last of the three given conditions was merely a con- 
sequence of the two first. We should then have in reality only two 
conditions — that is to say, analytically speaking, it might chance 
that the last of the three equations was merely one derivable from 
the two first, and then there would be an infinite number of 
solutions of the system of three variables. Such a system is 

x + y + z = 6, 

3.c+ 2y + z = 10, 

2x + y= 4, 

for example, for, as the reader may easily verify, it is satisfied 
by x = a - 2, y = 8 - 2a, z = a, where a is any quantity whatsoever. 



* A name seems to be required for thi.s all-pervading logical principle : 
the Law of Determinate Manifoldness might be suggested. 
+ See below, chap. xix. 



288 PROPOSITIONS AS TO DETEEMINATENESS OF SOLUTION chap. 

It will be seen in following chapters how these difficulties are 
met in particular cases. Meantime, let us observe that, if we 
admit Proposition I., two others follow very readily. 

Proposition II. If the number of equations be less than the number 
of variables, the solution is in general indeterminate. 

Proposition III. If the number of independent equations be 
greater than the number of variables, there is in general no solution, 
and the system of equations is said to be inconsistent. 

For, let the number of variables be n, and the number of 
equations to, say, where to < n. Let us assign to the first n - m 
variables any set of values we please, and regard these as constant. 
This we may do in an infinity of ways. If we substitute any such 
set of values in the to equations, we have now a set of to equa- 
tions to determine the last m variables ; and this, by Proposition 
I., they will do determinately. In other words, for every set of 
values we like to give to the first n - to variables, the to equations 
give us a determinate set of values for the last to. We thus get an 
infinite number of solutions ; that is, the solution is indeterminate. 

Next, let to be > n. If we take the first n equations, these 
will in general, by Proposition I., give a determinate set, or a 
finite number of determinate sets of values for all the n 
variables. If we now take one of these sets of values, and 
substitute it in one of the remaining to - n equations, that 
equation will not in general be satisfied ; for, if we take an 
equation at random, and a solution at random, the latter will 
not in general fit the former. The system of to equations will 
therefore in general be inconsistent. 

It may, of course, happen, in exceptional cases, that this 
proposition does not hold ; witness the following system of three 
equations in two variables : — 

SB-y=l, 2x + y=5, 3x"+2«/=8, 
which has the common solution x = 2, y = 1. 

§ 6.] We have also the further question, When the system 
is determinate, how many solutions are there ? The answer 
to this, in the case of integral equations, is furnished by the 
two following propositions : — 



xiv MULTIPLICITY OF DETERMINATE SOLUTIONS 289 

Proposition I. An integral equation of the nth degree in one 
variable has n roots and no more, which may be real or complex, and 
all unequal or not all unequal, according to circumstances. 

Proposition II. A determinate system ofm integral equations with 
m variables, whose degrees in these variables are p, q, r, . . . respect- 
ively, has, at most, pqr . . . solutions, and has, in general, just that 
number. 

Proposition I. was proved in the chapter on complex num- 
bers, where it was shown that for any given integral function 
of x of the nth degree there are just n values of x and no more 
that reduce that function to zero, these values being real or 
complex, and all unequal or not as the case may be. 

Proposition II. will not be proved in this work, except in 
particular cases which occur in chapters to follow. General 
proofs will be found in special treatises on the theory of equa- 
tions. We set it down here because it is a useful guide to the 
learner in teaching him how many solutions he is to expect. 
It will also enable him, occasionally, to detect when a system 
is indeterminate, for, if a number of solutions be found exceed- 
ing that indicated by Proposition II., then the system is certainly 
indeterminate, that is to say, has an infinite number of solu- 
tions. 

Example. The system x 2 + y 2 =l, x-y = \ has, by Proposition II. , 2x1 = 2 
solutions. As a matter of fact, these solutions are x = 0, y — — 1, and x=l, 

y=o. 

EQUIVALENCE OF SYSTEMS OF EQUATIONS. 

§ 7.] Two systems of equations, A and B (each of which may con- 
sist of one or more equations), are said to be equivalent when every 
solution of A is a solution of B, and every solution of B a solution 
of A. 

From any given system, A, of equations, we may in an in- 
finity of ways deduce another system, B; but it will not 
necessarily be the case that the two systems are equivalent. 
In other words, we may find in an infinity of ways a system, 
B, of equations which will be satisfied by all values of the 
VOL. I U 



290 DEFINITION OF EQUIVALENCE chap. 

variables for which A is satisfied ; but it will not follow con- 
versely that A will be satisfied for all values for which B is 
satisfied. To take a very simple example, x - 1 = is satisfied by 
the value x=l, and by no other; £(.t-1) = is satisfied by 
as=l, that is to say, x(x - 1) = is satisfied when x - 1 = is 
satisfied. On the other hand, x(x - 1) = is satisfied either by 
x = or by x = 1, therefore x — 1 = is not always satisfied when 
x(x - 1) = is so ; for x = reduces x - 1 to — 1, and not to 0. 
Briefly, x(x - 1) = may be derived from x- 1 = 0, but is not 
equivalent to x — 1 = 0. 

x(x - 1 ) = is, in fact, more than equivalent to x — 1 = 0, 
for it involves x - 1 = and x = as alternatives. It will be 
convenient in such cases to say that x(x - 1) = is equivalent to 

I -01 

\;c-l = 0j 

When by any step we derive from one system another which 
is exactly equivalent, we may call that step a reversible deriva- 
tion, because we can make it backwards without fallacy. If 
the derived system is not equivalent, we may call the step 
irreversible, meaning thereby that the backward step requires 
examination. 

There are few parts of algebra more important than the 
logic of the derivation of equations, and few, unhappily, that 
are treated in more slovenly fashion in elementary teaching. 
No mere blind adherence to set rules will avail in this matter ; 
while a little attention to a few simple principles will readily 
remove all difficulty. 

It must be borne in mind that in operating with conditional 
equations we always suppose, the variables to have such values 
as will render the equations identities, although we may not at 
the moment actually substitute such values, or even know them. 
We are therefore at every step, hypothetically at least, applying the 
fundamental laws of algebraical transformation just as in chap. i. 

The following general principle, already laid down for real 
quantities, and carefully discussed in chap, xii., § 12, for com- 
plex quantities, may be taken as the root of the whole matter. 



xiv ADDITION AND TRANSPOSITION OF TERMS 291 

Let P and Q be two functions of the variables x, y, z, . . ., vjhich 
do not become infinite * for any values of those variables that we have 
to consider. J/PxQ=0 and Q 4= 0, then mil P = 0, and ifP x Q = 
and P =t= 0, then will Q = 0. 

Otherwise, the only values of the variables which make P x Q = 
are such as make either P = 0, or Q = 0, or both P = and Q = 0. 

§ 8.] It follows by the fundamental laws of algebra that if 

P = Q • (l), 

then P ± R = Q ± R (2), 

where R is either constant or any function of the variables. 
We shall show that this derivation is reversible. 

For, if P ± R = Q ± R, 

then P±RtR = Q±RtR, 

that is, P = Q ; 

in other words, if (2) holds so does (1). 

Cor. 1 . If we transfer any term in an equation from the one side 
to the other, at the same time reversing its sign of addition or subtrac- 
tion, 07' if toe reverse all the signs on both sides of an equation, we 
deduce in each case an equivalent equation. 

For, if P + Q = R + S, say, 

then P + Q-S = R + S-S, 

that is, P + Q - S = R. 

Again, if P + Q = R + S, 

then P + Q-P-Q-R-S = R + S-P-Q-R-S, 

that is, - R - S = - P - Q, 

or -P-Q=-R-S. 

Cor. 2. Every equation can be reduced to an equivalent equation of 
the form — 

R=0. 

For, if the equation be P = Q, 

* In all that follows all functions of the variables that appear are supposed 
not to become infinite for any values of the variables contemplated. Cases 
where this understanding is violated must be considered separately. 



292 MULTIPLICATION BY A FACTOR chap. 

we have P - Q = Q - Q, 

that is P - Q = 0, 

which is of the form R = 0. 

Example. 

- Za? + 2x 2 + Sx = or - x - 3. 

Subtracting x- - x - 3 from both sides, we have the equivalent equation 

-3a; 3 + ar + 4a: + 3 = 0. 
Changing all the signs, we have 

3.« 3 - x 2 - 4x - 3 = 0. 
In this way an integral equation can always be arranged with all its terms on 
one side, so that the coefficient of the highest term is positive. 

§ 9.] It follows from the fundamental laws of algebra that 

if P = Q (1), 

then PR = QR (2), 

the step being reversible if R is a constant differing from 0, but not if 
R be a function of the variables* 

For, if PR = QR, 

an equivalent equation is, by § 8, 

PR - QR = (3), 

that is, (P - Q)R = (4). 

Now, if R be a constant 4= 0, it will follow from (4), by the 
general principle of § 7, that 

P-Q = o, 

which is equivalent to P = Q. 

But, if R be a function of the variables, (4) may also be satisfied 

by values of the variables that satisfy 

R = (5) ; 

and such values Avill not in general satisfy (1). 

In fact, (2) is equivalent, not to (1), but to (1) and (5) as 
alternatives. 

* This is spoken of as "multiplying the equation by R." Similarly the 
process of § 8 is spoken of as "adding or subtracting R to or from the equa- 
tion." This language is not strictly correct, but is so convenient that we 
shall use it where no confusion is to be feared. 



XIV DEDUCTION OF INTEGRAL FROM RATIONAL EQUATION 293 

Cor. 1. From the above it follows that dividing both sides of 
an equation by any function other than a constant not equal to zero is not 
a legitimate process of derivation, since we may thereby lose solutions. 

Thus PR = QR is equivalent to J _ Y ; 

whereas PR/R = QR/R* 

gives P = Q, 

which is equivalent merely to 

P-Q = 0. 

Example. If we divide both sides of the equation 

(x~l)x 2 = i{x-l) (a) 

by £- 1, we reduce it to x- = 4 (/3), 

which is equivalent to (x - 2) (x + 2) = 0. 
(a), on the other hand, is equivalent to 

(x-l)(w-2)(a;+2)=0. 

Hence (a) has the three solutions x-1, x = 2, x= - 2 ; while (/3) has only the 
two x = 2, x= - 2. 

Cor. 2. To multiply or divide both sides of an equation by 
any constant quantity differing from zero is a reversible process 
of derivation. Hence, if the coefficients of an integral equation be 
fractional either in the algebraical or in the arithmetical sense, we can 
always find an equivalent equation in which the coefficients are all 
integral, and have no common measure. 

Also, we can always so arrange an integral equation that the co- 
efficient of any term we please, say the highest, shall be + 1 . 

Example 1. 

3a + 2 6x + S_2x+_i 

~~4~~ + 5 - 8 

gives, on multiplying both sides by 40, 

10(3x + 2) + 8(6i- + 3) = 5(2x-+4), 

that is, 30^ + 20 + 48^ + 24 = 10x- + 20, 

whence, after subtracting lO.e + 20 from both sides, 

68a; + 24 = ; 

* As we are here merely establishing a negative proposition, the reader 
may, to fix his ideas, assume that all the letters stand for integral functions 
of a single variable. 



294 DEDUCTION OF INTEGRAL FROM RATIONAL EQUATION chap. 



whence again, after division of both sides by 68, 
Example 2. 



X+ Tf=°- 



K q p~q' J\ p P + 



- q y)=^y- 



If we multiply both sides by pq(p - q) {p + q), that is, by pq(2>~ ~ <T)i we derive 
the equivalent equation 

{{p 2 -q 2 )x+pqy} {{2r-q-)x+pqy)=2pq(p--q-)xy, 

that is, ( p 2 - q 2 ) 2 x 2 + 2pq{ p 2 - q 2 )xy +p 2 q-y 2 = 2pq{ p 2 - q 2 )xy, 

which is equivalent to [p 2 -q 2 ) 2 x 2j rp 2 q 2 y 2 = Q. 

Cor. 3. From every rational algebraical equation an integral equa- 
tion can be deduced ; but it is possible that extraneous solutions may 
be introduced in the process. 

Suppose we heave P = Q (a), 

where P and Q are rational, but not integral. Let L he the 
L.C.M. of the denominators of all the fractions that occur either 
in P or in Q, then LP and LQ are both integral. Hence, if we 
multiply both sides of (a) by L, we deduce the integral equation 

LP = LQ (/?). 

Since, however, the multiplier L contains the variables, it is 
possible that some of the solutions of L = may satisfy (ft), and 
such solutions would in general be extraneous to (a). We say 
possible ; in general, however, this will not hajjpp n, because P 
and Q contain fractions whose denominators are factors in L. 
Hence the solutions of L = will in general make either P or Q 
infinite, and therefore (P - Q)L not necessarily zero. The point 
at issue will be best understood by studying the two following 
examples : — 

Example 1. 

. z 2 -6a; + 8 x-2 
2 *' 3+ x-2 = x=S (a) - 

If we multiply both sides by (x- 2)(x- 3), we deduce the equation 

(2x-3)(x-2)(x-3) + (x 2 -6x+8)(x-3) = (x-2) 2 (/3), 

which is integral, and is satisfied by any solution of (a). "We must, however, 
examine whether any of the solutions of (x- 2)(x- 3) = satisfy (/3). These 
solutions are x=2 and x = 3. The second of these obviously does not satisfy 
(/3), and need not be considered; but x=2 does satisfy (/3), and we must 
examine (a) to see whether it satisfies that equation also. 



XIV RAISING BOTH SIDES TO SAME POWER 295 

Now, since x 2 - 6a: + 8 = (as— 2) (a; - 4), (a) may be written in the equivalent 

form 

x — 2 
2x - 3 + x - 4 : 



P = Q 


(1) 


P - Q = 


(2). 


+ P»- 2 Q + P' l " 3 Q 2 + . . 


. +Q n -\ ■ 



x-3 
which is obviously not satisfied by x = 2. 

It appears, therefore, that in the process of integralisation we have intro- 
duced the extraneous solution x=2. 

Example 2. 

2x-B + 2 ^^ 8 = X ~l («<). 

x - 2 x - 3 

Proceeding as before, we deduce 

{2x - 3) (a; - 2) (x - 3) + (2a: 2 - 6x + 8) (a; - 3) = (x - 2f iff). 

It will be found that neither of the values 33=2, a; = 3 satisfies (/3'). 
Hence no extraneous solutions have been introduced in this case. 

N.B. — The reason why a? = 2 satisfies (/3) in Example 1 is that the numer- 
ator a,- 2 - 6a: + 8 of the fraction on the left contains the factor x-2 which 
occurs in the denominator. 

Cor. 4. Raising both sides of an equation to the same integral 
power is a legitimate, but not a reversible, process of derivation. 

The equation 
is equivalent to 

If we multiply by P' 1 " 1 + P n ~ 2 Q + P»- 3 Q 2 + . . . + Q M ~\ we 

deduce from (2) 

P* _ Q» = (3), 

which is satisfied by any solution of (1); (3), however, is not 
equivalent to (1), but to 

P = Q\ 

P»-1 + P»-2Q + . . > + Q»-l = oj- 

It will be observed that, if we start with an equation in the 
standard form P - Q = 0, transfer the part Q to the right-hand 
side, and then raise both sides to the wth power, the result is the 
same as if we had multiplied both sides of the equation in its 
original form by a certain factor. To make the introduction of 
extraneous factors more evident we chose the latter process ; but 
in practice the former may happen to be the more convenient.* 

If the reader will reflect on the nature of the process described 
in chap. x. for rationalising an algebraical function by means 
of a rationalising factor, he will see that by repeated operations of 
this kind every algebraical equation can be reduced to a rational 

* See below, § 12, Example 3. 






296 EVERY ALGEBRAICAL EQUATION CAN BE INTEGRALISED chap. 

form ; but at each step extraneous solutions may be introduced. 
Hence 

Cor. 5. From every algebraical equation we can derive a rational 
integral equation, which will be satisfied by any solution of the given 
equation ; but it does not follow that every solution^ or even that any 
solution, of the derived equation will satisfy the original one. 
Example 1. Consider the equation 

V(z+l) + V(*-l) = l (a), 

where the radicands are supposed to be real and the square root to have the 
positive sign. * 

(a) is equivalent to \J(x + l) = l - \J{x- 1), 

whence we derive, by squaring, 

x+l = \+x-l-2'sJ{x-\), 
which is equivalent to 1 = - 2\J(x - 1). 

From this last again, by squaring, we derive 

1 = 4(35-1), 

which is equivalent to the integral equation 

Ax -5 = (/3), 

the only solution of which, as Ave shall see hereafter, is x = \. 

It happens here that x=\ is not a solution of (a), for V(f + 1) + V(i" - 1) 

— S i 1 _o 

— T + V — -■ 

Example 2. 

V(.c+i)-V(- >, -i) = i (o). 

Proceeding exactly as before we have 

a; + l = l + a:-l + 2V(*-l)i ' 
1=+2V(*-1), 
1 = 4(3-1), 
4a- 5 = (/3'), 

Here (/?') gives a:=f, which happens this time to be a solution of the 
original equation. 

We conclude this discussion by giving two propositions 
applicable to systems of equations containing more than one 
equation. These by no means exhaust the subject ; but, as our 
object here is merely to awaken the intelligence of the student, 
the rest may be left to himself in the meantime. 

§ 10.] From the system 

P, = 0, P 2 =0, . . ., P n = (A) 

we derive 

L 1 P 1 .+ IJP f +. . .+LnPn= 0, P 8 = 0, . . ., P n = (B), 
and the two will be equivalent if J Jl be a constant differing from 0. 



f When \Jx is imaginary, its "principal value" (see chap, xxix.) ought 
to be taken, unless it is otherwise indicated. 



XIV 



EXAMPLES OF DERIVATION 297 



Any solution of the system (A) reduces P 15 P 2 , . . ., P n all 
to 0, and therefore reduces L^ + LP, + . . . + L n P n to 0, and 
hence satisfies (B). 

Again, any solution of (B) reduces P 2 , P 3 , . . ., P n all to 0, 
and therefore reduces L,P, + L 2 P 2 + . . . + L n P„ = to L,P, = 0, 
that is to say, if L, he a constant 4= 0, to Pj = 0. Hence, in this 
case, any solution of (B) satisfies (A). 

If L, contain the variables, then (B) is equivalent, not to (A) 

simply, but to 

f P, = 0, P 2 = 0, . . ., P w = 0] 
tL.-O, P s = 0, . . , P w = 0j 

As a particular case of the above, we have that the two 

systems 

P = Q, R = S j 

and P + R = Q + S, R = S 

are equivalent. For these may be written 

P-Q = 0, R~^S=0; 



P-Q + R-S = 0, R-S = 0. 

If I, V, m, m! he constants, any one of which may he zero, hut 
which are such that lm' - I'm =t= 0, then the two systems 

U = 0, U' = 0, 

and IV + IV = 0, mV + niV = 

are equivalent. 

The proof is left to the reader. A special case is used and 
demonstrated in chap, xvi., § 4. 
§ 11.] Any solution of the system 

P = Q, R = S (A) 

is a solution of the system 

PR = QS, R = S (B); 

but the two systems are not equivalent. 
From P = Q, we derive 

PR = QR, 
which, since R = S, is equivalent to 

PR = QS. 
It follows therefore that any solution of (A) satisfies (B). 



298 EXAMPLES OF DERIVATION CHAP. 

Starting now with (B), we have 

PR = QS (1), 

R = S (2). 

Since II = S, (1) becomes 

PR = QR, 
which is equivalent to 

(P - Q)R = 0, 
that is, equivalent to 

fP-Q = 0\ 

Hence the system (B) is equivalent to 

fP = Q, R = S\ 

|R = 0, R = SJ" 
that is to say, to 

/P = Q,R = S\ 

\R = 0, S = 0J" 
In other words, (B) involves, besides (A), the alternative system, 

R = 0, S = 0. 

Example. From x-2 = l-y, x = l+y, 

a system which has the single solution x=2, y==l, we derive the system 

x(x-2) = l-y\ x=l+y, 
which, in addition to the solution x = 2, y—l, has also the solution x =Q,y= -1 
belonging to the system 

2=0, l + y = 0. 

§ 12.] In the process of solving systems of equations, one of 
the most commonly-occurring requirements is to deduce from two 
or more of the equations another that shall not contain certain 
assigned variables. This is called " eliminating the variables in 
question between the equations used for the purpose." In per- 
forming the elimination Ave may, of course, use any legitimate 
process of derivation, but strict attention must always be paid to 
the question of equivalence. 
Example. Given the system 

* 2 +r=i (i), 

x+y=l (2), 

it is required to eliminate y, that is, to deduce from (1) and (2) an equation 
involving x alone. 



xiv EXAMPLE OF ELIMINATION 299 

(2) is equivalent to 

y = l-x. 
Hence (1) is equivalent to 

a?+(l-xf=l, 
that is to sav, to 

2x 2 -2x=0, 
or, if we please, to 

x*--x = 0; 

and thus we have eliminated y, and obtained an equation in x alone. 

The method we have employed (simply substitution) is, of course, only 
one among many that might have been selected. 

Observe that, as a result of our reasoning, we have that the system (1) and 
(2) is equivalent to the system 

x 2 -x=0 (3), 

x + y=l (4), 

from which the reader will have no difficulty in deducing the solution of the 
given system. 

§ 13.] Although, as we have said, the solution of a system 
of equations is the main problem, yet the reader will learn, 
especially when he comes to apply algebra to geometry, that 
much information — very often indeed all the information that is 
required— may be derived from a system without solving it, 
but merely by throwing it into various equivalent forms. The 
derivation of equivalent systems, elimination, and other general 
operations with equations of condition have therefore an im- 
portance quite apart from their bearing on ultimate solution. 

We have appended to this chapter a large number of exercises 
in this branch of algebra, keeping exercises on actual solution for 
later chapters, which deal more particularly with that part of 
the subject. The student should work a sufficient number of the 
following sets to impress upon his memory the general principles 
of the foregoing chapter, and reserve such as he finds difficult for 
occasional future practice. 

The following are worked out as specimens of various artifices 
for saving labour in calculations of the present kind : — 

Example 1. Reduce the following equation to an integral form : — 

ax- + bx + c ax + b , . 

(a). 



px~ + qx + r px + q 
"We may write (a) in the form 

• x(ax + b) + c _ ax + b 
x{px + q) + r ~px + q 



(0). 



300 EXAMPLES OF INTEGRALISATION chap. 

Multiplying (/3) by (px + q){x(px + q) + r}, we obtain 

x(ax + b) (px + q)+ c{px + q) = x(ax + b) (px + q) + r(ax + b) (7). 

Now, (7) is equivalent to 

c(px + q) = r(ax + b) (5), 

which again is equivalent to 

(cp - ra)x + (cq - rb) = (e). 

The only possibly irreversible step here is that from (/3) to (7). 

Observe the use of the brackets in (/3) and (7) to save useless detail. 

Example 2. 
Integralise 

(a -x)(x + m)_(a + x) (x - m) 



x + n x-n 



(a). 



Since x + m = (x + n) + (m - n), x-m = (x-n)- (m-n), (a) may be written 
in the equivalent form, 

. ./, , m-n\ , , /, m-v\ ,_, 



whence the equivalent form 

{a-x)-(a + x) + (m-n)t + =0, 

\x+n x-n) 

that is, 



-2x + 2{m -y {n + a)x =0 (7). 

x- - n- 

Multiplying by -£(a; 2 -?i 2 ), we deduce from (7) the integral equation 

x{x 1 -n' i - (m - n) (n + a)) = (5). 

In this case the only extraneous solutions that could be introduced are 
those of x 1 - n- = 0. 

Note the preliminary transformation in (/3) ; and observe that the order 
in which the operations of collecting and distributing and of using any 
legitimate processes of derivation that may be necessary is quite unrestricted, 
and should be determined by considerations of analytical simplicity. Note 
also that, although we can remove the numerical factor 2 in (7), it is not 
legitimate to remove the factor x ; x = Q is, in fact, as the student will see by 
inspection, one of the solutions of (a). 

Example 3. 

X, Y, Z, U denoting rational functions, it is required to rationalise the 

equation 

X /X ± \/ Y ± y'Z ± VU = (a). 

We shall take + signs throughout ; but the reader will see, on looking 
through the work, that the final result would be the same whatever arrange- 
ment of signs be taken. 
From (a), 

VX+vy=-vz-\/u, 

whence, by squaring, 

X + Y + 2V(XY)=Z+U+2V(ZTJ) (/3). 



xiv EXAMPLES OF RATIONALISATION 301 

From (/3), 

X + Y-Z-U=- 2 V(XY) + 2 V(ZU), 
whence, by squaring, 

(X+Y-Z-U) 2 =4XY+4ZU-8V(XYZU) (7). 

We get from (7), 

X 2 + Y 3 + Z- + U 2 - 2XY - 2XZ - 2XU - 2YZ - 2YU - 2ZU= - 8 V(XYZU), 

whence, by squaring, 

J X s + Y 2 + Z 2 + U 2 - 2XY - 2XZ - 2XU - 2YZ - 2YU - 2ZU j 2 = 64X YZU (5). 

Since X, Y, Z, U are, by hypothesis, all rational, (5) is the required result. 
As a particular instance, consider the equation 

V(2k+3) + V(&g+2) - V(2as+5) - \/{&c)=0 [a'). 

Here X = 2a: + 3, Y=3a;+2, Z = 2a; + 5, U = Zx; and the student will find, 
from (5) above, as the rationalised equation, 

(48^ + 112a,- + 24) 2 =64(2a;+3)(3a: + 2)(2a; + 5)3a; (8'). 

After some reduction (5') will be found to be equivalent to 

(z-3) 2 = (Y). 

It may be verified that a; = 3 is a common solution of (a') and (e'). 

Although, for the sake of the theoretical insight it gives, we have worked 
out the general formula (5), and although, as a matter of fact, it contains as 
particular cases very many of the elementary examples usually given, yet 
it is by no means advisable that the student should work particular cases by 
merely substituting in (5) ; for, apart from the disciplinary advantage, it 
often happens that direct treatment is less laborious, owing to intervening 
simplifications. Witness the following treatment of the particular case (a') 
above given. 

From (a'), by transposition, 

V(2aJ+8)+V(3a!+2)=V(2a!+5) + 's/(3a5), 

whence, by squaring, 

5x + 5 + 2\/(6ar + 13a: + 6) = 5x + 5 + 2 y/(6x- + 15x), 
which reduces to the equivalent equation 

^(6x 2 + lZx+6)=s/(6x i +l5x) (£'). 

From (/3'), by squaring, 

6X 2 + lBx + 6 = 6a; 2 + 15s, 
which is equivalent to 

a;-3 = (5"). 

Thus, not only is the labour less than that involved in reducing (5), but 
(5") is itself somewhat simpler than (5'). 

Example 4. If 

x + y + z=0 (o), 

show that 

2(2/ 2 + yz + z*f = 3n(y 2 + yz + z") (/3j. 



302 EXAMPLES OF TRANSFORMATION chap. 

We have 

vf + yz + z^if + ziy + z), 

=(-z-x)*+z{-x),by(a), 

= Z 2 + ZX + X 2 , 

= ir + xy + y 2 , by symmetry. 
It follows then that 

2(f- + yz + z 2 ) 3 = 3(</ 2 + yz + z*)* (7), 

and m(y- + ijz + z'>) = 3(f- + yz + z*) 3 (5). 

From (7) and (5), (/3) follows at once. 

Example 5. If 

x + y+z = (a), 

show that 

(y + z)(z + x) J 

From (a), y + z=-x (7), 

whence, squaring and then transposing, we have 

y*+z*=a?-2yz (5). 

Similarly z + x=-y (7'), 

z n - + x- = y" i -2zx (5'). 

From the last four equations we have 

2 (y- + s 2 ) (z 2 + * 2 ) = s (a 2 - 2yz) (f- - 2sc) 
(y+2)(«+a:) a;?/ 

_ aPy 2 - 2x?z - 2y 3 z + ixyz 1 



Z\xy-2 



xy 

(3? + y 3 ls 



3 + 4; 2 | 



xyz 

= Xvy + 42a,- 2 - J- 2.r% 3 + ,3) (e) . 
Now, from (a), by squaring and transposing, 

2x' ! =-2?xy (f). 

Also 2x 2 (y 3 + z 3 )EE2a:y 2 (a: + y), 

= - ^x-rfz, by (a), 

= - xyzZxy (17). 

If we use (f) and (r?), (e) reduces to 

&+*)(*+*) = _ 5 
(y + z)(z + x) J ' 

which is equivalent to (/3). 

The use of the principles of symmetry in conjunction with the 2 notation 
in shortening the calculations in this example caunot fail to strike the 
reader. 



Example 6. If 



yz-x^zx-f 
y+z z+x 



xiv EXAMPLES OF TRANSFORMATION 303 

and if x, y, z be all unequal, show that each of these expressions is equal to 
(xy - z 2 )/(x + y), and also to x + y + z. 

Denote each of the sides of (a) by U. Then we have 

yz-x 2 



y + z 
zx-y 
~z + x 



U (/3), 

= U ( 7 ). 



Since y + z = and z + x = would render the two sides of (a) infinite, we 
may assume that values of a-, y, z fulfilling these conditions are not in ques- 
tion, and multiply (/3) and (7) by y + z and z+x respectively. We then 
deduce 

ye-x*-(y+z)\J=0 (5), 

zx-y n --(z + x)U = (e). 

From (5) and (e), by subtraction, we have 

z(x-y) + (x 2 -y 2 )-(x-y)V = 0, 
that is, (x + y + z-V)(x-y) = (f). 

Now x-y — is excluded by our data ; hence, by (f), we must have 

x + y + z-U = 0, (,), 

that is, V=x+y+z (6). 

We have thus established one of the desired conclusions. To obtain the 
other it is sufficient to observe that (77) is symmetrical in x, y, z. For, if we 
start with (77) and multiply by x - z (which, by hypothesis, 4= 0), we obtain 

y(x - z) + (x 2 - s«) - (x - Z )U = ; 

and, combining this by addition with (5), 

xy-i?-(z+y)U=Q; 

which gives (since x + y + 0) 

jj_ xy-z* 



x + y 

The reader should notice here the convenient artifice of introducing an 
auxiliary variable U. He should also study closely the logic of the process, 
and be sure that he sees clearly the necessity for the restrictions x-y + Q, 
x + y + 0. 

Example 7. To eliminate x, y, z between the equations 

y*+z*=ayz (a), 

z 2 + x 2 = bzx (£), 

x t +y*=cxy (7), 

where x + 0, w + 0, z + 0. 

In the first place, we observe that, although there are three variables, yet, 
since the equations are homogeneous, we are only concerned with the ratios 
of the three. We might, for example, divide each of the equations by x 2 ; 
we should then have to do merely with y/x and z/x, each of which might be 
regarded as a single variable. There are therefore enough equations for the 
purpose of the elimination. 



304 EXAMPLES OF ELIMINATION chap. 

From (a) and (j3) we deduce, by subtraction, 

x*-y 2 ={bx-ay)z (8). 

We remark that it follows from this equation that bx- ay + 0; for bx-ay — 
would give x 2 = y 2 , and hence, by (7), x = (at least if we suppose c=t=±2). 
This being so, we may multiply (/3) by (bx - ay) 2 . "We thus obtain 

z-(bx - ay) 2 + x 2 (bx — ay) 2 = bxz(bx - ay) 2 , 
whence, using (5), we have 

(a; 2 - y 2 ) 2 + x 2 (bx — ay) 2 = bx(bx -ay) (x 2 - y 2 ), 
which reduces, after transposition, to 

(x 2 - y 2 ) 2 = xy(ax - by) (bx - ay), 
that is to say, (x 2 + y 2 ) 2 - ±x 2 y 2 = xy(ax-by) (bx-ay) (e). 

Using (7), we deduce from (e) 

(c 2 - i)x 2 y 2 = xy(ax - by) (bx - ay), 
whence, bearing in mind that xy + 0, we get 

(c- 2 -i)xy = ab(x 2 + y 2 ) - (a 2 + b 2 )xy, 
which is equivalent to 

(a 2 + b 2 + c 2 - i)xy = ab(x 2 + y 2 ) (f). 

Using (7) once more, and transposing, we reach finally 

(a 2 + b 2 + c 2 - 4 - abc)xy - 0, 
whence, since xy + 0, we conclude that 

a 2 + b 2 + c 2 -4:-abc = ( v ), 

so that (77) is the required result of eliminating x, y, z between the equations 
(a), (/3), (7). Such an equation as (77) is often called the eliminant (or re- 
sultant) of the given system of equations. 

Example 8. Show that, if the two first of the following three equations be 
given, the third can be deduced, it being supposed that x =t= y + z + 0. 

a 2 (y 2 + yz + z 2 ) - ayz(y + z) + y 2 z 2 = (a), 

a 2 (z 2 + zx + x 2 ) - azx(z + x) + z 2 x 2 = (p), 

a 2 (x 2 + xy + y"-) - axy(x + y)+ x 2 y 2 = (7). 

This is equivalent to showing that, if we eliminate z between (a) and (/3), the 
result is (7). 

Arranging (a) and (/3) according to powers of z, we have 

aY -a(-ay + y 2 )z + (a 2 - ay + y 2 )z 2 = (5), 

a 2 x 2 -a(-ax + x 2 )z + (a 2 -ax + a?)z 2 = (e ). 

Multiplying (5) and (e) by x 2 and y 2 respectively, and subtracting, we get 

a 2 xy(x - y)z + {a 2 (x + y) - axy} (x - y)z 2 = 0, 
whence, rejecting the factor a(x-y)z, which is permissible since x + y, z#=0, 

axy+ {a(x + y)-xy}z = (f). 

Again, multiplying (5) and (e) by a 2 -ax + x 2 and a 2 -ay + y 2 respectively, 
and subtracting, we get, after rejecting the factor a 2 , 

a(x + y)-xy+ \a-(x + y)}z=0 (77). 



XIV 



EXERCISES XIX 305 



Finally, multiplying (f) and (v) by a{x + y)-xy and axy respectively, and 
subtracting, we get, since z=$=0, 

. {a{x + y)- xy) 2 - axy {a - (x + y)} = 0, 
which gives a 2 (x 2 + xy + y 2 ) - axy(x + y) + x 2 y 2 = 0, 

the required result. 

Exercises XIX. 
(On the Reduction of Equations to an Integral Form) 
Solve by inspection the following systems of equations : — 
(1.) ar-4-3(;r + 2) = 0. 

(2.) £±j ;=«-. 

x x—bx—a 

(3.) (a-b)x-a 2 + b- = 0. 

(4.) x(b-c)+y(c-a) + (a-b)=0, 

ax(b -c) + by(c -a) + c(a-b) = 0. 
(5.) x + y + z = a + b + c, 

ax + by + cz = a 2 + b'- + c 2 , 
bx + cy + az = be + ca + ab. 

(6.) For what values of a and b does the equation 

(a: - a) (3a: -2)= Bx 2 + bx + 10 
become an identity ? 

Integralise the following equations ; and discuss in each case the equiva- 
lence of the final equation to the given one. 

(7.) «±« + ?»±" 14 

x - 4 a; - 2 



(8.) 



l-8/ ( g+l) l + 3/(a;-l) 

a; + 1 + l/fc+l) as - 1 + l/(as - 1)' 



/n X 1111 

(9. ) + = + 

x+a x-c x-a x+c 

,,~ x flf 6" ar a- + b 

( 10 -) : — =+:r-i=;r-; + 



(11.) 
(12.) 



x-a x—b x-a x-b 

(3 -a-) (a; + 10) (8 + x) (x - 10) 

a;+ll a- 11 

x 2 +px + q _ x 2 +px + 1 

x 2 + rx + 2q ~ x 2 + rx + 2t 



(131 (x-a) 3 (x-b) 3 (x-c) 3 

K '' (c-a)(a-b) (a-b)(b-c) (b-c)(c-a) 

(u) x + T + U a; + T-U 

1 x 2 + (2-t)x + s(2-s-t) x 2 + (2-s)x + t(2-s-<) A 
when 2T=s + t-s 2 -st-t 2 . 

VOL. I 



(22.) 

(23.) 
(24.) 
(25.) 



306 EXERCISES XIX, XX chap. 

,,_ . x^ + ax + b x 2 + cx + d x" + ax + b' x" + cx + d' 

(15.) 1 — = 1 . 

x + a x + e x + a x + c 

Rationalise the following equations and reduce the resulting equation to 
as simple a form as possible : — 

(16.) \/X + \/Y + \JZ = 0, where X, Y, Z are rational functions of the 
variables. 

(17.) \/{x + a) + \/(x + b) + \J(x + c) = 0. 

(18.) \/Q.+x) + *J(i + x)-y/(9 + x) = 0. 

(19.) [x-c+ {(x-c) n - + f-' i i ]/[x + c+ {(x-c)- + y°-} h ] = m. 

(20. ) x - a = V {a- - vV* 2 - <*)} . 

(21.) VaB+V(a»-7) = 21/V(*-7). 

V» + 3" _ Va; + 29' 

\/{2+x) V(2-a;) 

V2 + V( 2 + ^O V2 - V( 2 - *) ' 
(24. ) \J{x + a) + V(se - a) + V(& + a") + V( 6 -«) = 0. 

V(l+a; + a; i! ) + V(l-a: + a; 2 ) _ 

V(i+*) + V(i-*) 

(26. ) (y - s) (OSB4- &)* + («-») (ay + ft) 4 + (a - ?/) (az + &)* = 0. 

(27.) 2V(y-~) = 0; and show that 2x= V( 32 2/~) (three variables a-, y, z). 

<28 -» v{»w(»--i)( + v^-v^-i)r vl ^ +1)1 - 

(29.) sb* +53!* -22=0. 

, Qn v v/(a + a;) \/(a + a:) Va; 

^ou. ; (-. 

a; a c 

(31. ) v/(a + V») + \/(a - vfc) = #fc 

(32.) a£+y*+z*=0, 

where a; + 2/ + 2=0. 



Exercises XX. 

(On t%e Transformation of Systems of Equations.) 

[In working this set the student should examine carefully the logic of every 
step he takes, and satisfy himself that it is consistent with his data. He 
should also make clear to himself whether each step is or is not reversible.] 

(1.) If y + s + z J^ + x ±y + 2 = 0, x + y + z*0, 

x y z 

41 1111 

then -H (-- = ■ 



x y z x+y+z 



xiv EXERCISES XX 307 

(2.) If a* + y*-=i?, 

then {{a? + z*)y\ - + {(a* - y s )z) 3 = {(if + a 8 )*} s ( Tait). 

(3.) If x, y, z be real, ami it' x\y - z) + y i (z- x) +z 4 {x-y) = 0, then two at 
least of the three must be equal. 

(4.) If (x + y + z) 3 = x* + y 3 + z\ 

then (x + y + z)-^ 1 = x 2n + l + y-»+ 1 + z 2 "+ ] . 

(5.) If 

(2> 2 x + 2pry + r 2 z) (q 2 x + 2qsy + s"z) — [pqx + (ps + qr)y + rsz) 2 , 

then either y- -zx = or ps - qr = 0. 



(6.) if jp^+*pr + *> =0i 

where r 2 = a; 2 + y- + z 2 , 

.1 « 2 V 2 z 2 

then -3-., 5 + -^ — a + -5Ki 5 = 0, 

b-cr - p" era- - p- a~b~ - p- 

where p 2 = a 2 x- + b 2 y 2 + c 2 z 2 . 

(Important in the theory of the wave surface. — Tait. 

(7.) If |±* = !±!! = *±£, andas+y+*=0i 

b-c c-a a-b 

show that each of them is equal to \/{2a; 2 /2(2<( 2 - S6c)}. 

a(&y + cz - ckc) = b(cz + ax- by) = c(oa; + by- cz), 
a + b + c = 0, 
x + y + z = 0. 
x + 2y _ y + 2z _ z + 2x 
2a + b~2b + c~2c~+a 



(8.) 


If 


and if 




then 




(9.) 


If 


then 




(10.) 


If 


then 




(11.) 


If 


then 





/ZxY- _Xxy _2x 2 
\Za) _ 2a6~2ft 2 " 
2ab + b 2 a 2 -b- 

X ~a 2 + ab + b 2 ' V ~a'- + ab + b 2 



x 3 + y = y- + x. 

(a -b) 2 a + b ab 

x = a + b + y. — -—, y = —— + —-, 
4(« + o) 4 a + b 

(x-a) 2 -(y-b) 2 = b 2 . 

(12.) If a — ax + by + cz + dw, 

(3 = bx + ay + dz + cw, 

y = ex + dy + az + bio, 

d = dx + cy + bz + aw, 
and if 

/{a, ft 7, 5) = (a + /3 + 7 + 5)(a- i 3 + 7-5)(a-/3-7 + 5)(a + /3-7-5), 

then 

/(«, /3, y, 5) =/(«, 6, c, d)f{x, y, z, w). 

(13.) If x+ y +2=0, then Zl/* 2 =(Zl/a;) 2 . 



308 EXERCISES XX, XXI chap. 



and 






Qin+n fyni-\-n gvi+n 



show that (2aj™ n/ (" , +»)) (2a,"" 2 /< m +">) = d m . 

(15.) If a _^ =& _J =c _^ a . +0} y=#O jZ *0 J 

., , w- - z- , z- - xr x* - y 

then a + J -. — b-] = c + f-. 

b-c c- a a-b 

(16.) If x + {yz - x 2 )j(x 2 + y 2 + z 2 ) be unaltered by interchanging x and y, it 

will be unaltered by interchanging x and z, provided x, y, z be all unequal ; 

and it will vanish if x + y + z=l. 

(17.) If zxy/(y 4 2) - 7? = xyz/(z + x) - y 2 , and x + y, then each of these is 

equal to xyz/(x + y) - z 2 and also to yz + zx + xy. 

(18.) Of the three equations 

__x y + z 

x 2 - w 2 ~ (m + 1 )w 2 - (n + 1 )yz ' 

V _ s + jg 

y 2 -w 2 (m+l)iv 2 -(7i+l)zx' 

z _ % + y 

z 2 -w 2 {m + l)vr~(n + l)xy' 
where x #=2/ + z, any two imply the third (Cayley). 
(19.) Given 

- 1 - V + ^— = 1, 



1 + x + xz l + y + xy l+z + yz 

x xy 1 

. + , ■ ... + , ■ . ... =h 



1+x + xz l+y + xy l+z + yz 
none of the denominators being zero, then x = y = z. 

(20.) Given ■2(y + z) 2 /x = 3Sx, 2a; *0, prove Z(y + z-x) 3 + II{y + z-x) = 0. 

(21.) Given 2a; =0, prove 2(a? + y 3 )/(a: + ?/) + 5a7/~2(l/ai) = 0. 

(22.) Given 2a; = 0, prove that Ske'Sa^/XB 6 is independent of a;, y, z, 

(23. ) If 2a; 5 = - 5xyzSxy, then 2a = 0, or 2a, 4 - Safy + 1.x 2 y 2 + 2Zx 2 yz = 0. 

(24.) If II(ar + l) = a 2 +l, n(a^-l)=a 2 -l, and 2a-y = 0, then x+y+z=0 
or = ±«. 

(25.) If x + y + z + u=0, then 42x* + 32,(y + z)(u + y)(u + z) = 0, where the 
2 refers to the four variables a;, y, z, u. 



Exercises XXI. 
(On Elimination.) 
(1.) Eliminate x between the equations 

x + l/x = y, x n +l/x 5 — z. 
(2.) If z= \/{ay--d 2 ly), y = \J(ax 2 -a 2 /x), express \J(az 2 - a 2 [z) in terms 
of a;. 



XIV 








(3.) 


If 


the i 


I 






(i.) 


Given 


prove that 



EXERCISES XXI 309 

(p(x) = (a x - ar x )l(a x + ar x )\ 
F(x)=2/{a*+ar*), 

4>{x + y) = (<p(x) + <p(y) )/(l + <p{x)4>{y) ), 
¥{x + y) = F(x)¥(y)/(l+<p(x)cl>(y)). 
x(y + z-x) _ y(z + x-y) _ z(x + y -z) 

a b c 

a(b + c-a) _b{c + a- b) _c(a + b-c) 
x y z 

(5.) Given bz + cy = cx + az — ay + bx, x 2 + y 2 + z 2 =2yz + 2zx + 2xy, prove 
that one of the functions «±£>±e = 0. 

(6.) Show that the result of eliminating x and y between the equations 
x y , x 2 y" „ „ 

a + b = 1 > S + ^= 1 ' Xy =r> 
is (b"c"- + a"-cr 2 )-p* + 2abc 2 cf-(b-c" + a"-d n - - 2a 2 b"-)p 2 + arb^d^a 2 - c 2 ) (b 2 - d?) = 0. 
(7.) Eliminate x, y, x', y' from 

ax + by = c 2 , x- + y 2 = c~, , , _ 

a'x' + by = c' 2 , x' 2 + y' 2 = c' 2 , Xy + X V ~ 
(8.) If l/(x + a) + l/(y + a) + l/(z + a) = l/a, with two similar equations it 
which b and c take the place of a, show that Z(l/«) = 0, provided a, b, c be all 
different. 

(9.) Show that any two of the following equations can be deduced from 
the other three : — 

ax + be = zu, by + ca = nv, cz + db = vx, du + ec = xy, ev + ad = yz. 
(10.) Eliminate x, y, z from the three equations 
(z + x-y){x + y-z) = ay~, 
(■'' s y-z){y + z-x) = bzx, [y + z- x) {z + x- y) = cxy ; 
and show that the result is abc — {a + b + c- 4) 2 . 



CHAPTER XV. 
Variation of Functions. 

§ 1.] The view which we took of the theory of conditional 
equations in last chapter led us to the problem of finding a set 
of values of the variables which should render a given conditional 
equation an identity. There is another order of ideas of at least 
equal analytical importance, and of wider practical utility, which 
we now proceed to explain. Instead of looking merely at the 
values of the variables x, y, z, . . . which satisfy the equation 

f(x, y, z, . . .) = 0, 

that is, which render tlie function /(x, y, z, . . .) zero, we consider 
all possible values of the variables, and all possible corresponding 
values of the function ; or, at least, we consider a number of such 
values sufficient to give us a clear idea of the whole ; then, among 
the rest, we discover those values of the variables which render 
the function zero. The two methods might be illustrated by the 
two possible ways of finding a particular man in a line of soldiers. 
We might either go straight to some part of the ranks where 
a preconceived theory would indicate his presence ; or we might 
walk along from one end of the line to the other looking till we 
found him. In this new Avay of looking at analytical functions, 
the graphical method, as it is called, is of great importance. 
This consists in representing the properties of the function in 
some way by means of a geometrical figure, so that we can with 
the bodily eye take a comprehensive view of the peculiarities of 
any individual case. 



CHAP. XV 



THE GRAPHICAL METHOD 



311 



GENERAL PROPOSITIONS REGARDING FUNCTIONS OF ONE 

REAL VARIABLE. 

§ 2.] For the present we confine ourselves to the case of a 
function of a single variable, fix)', and we suppose that all the 
constants in the function are real numbers, and that only real 
values are given to the variable x. "We denote, as in chap, 
xiii., § 1 7, f(x) by y, so that 

V =/(■'■) (IX 

and we shall, as in the place alluded to, speak of x and y as 
the independent and dependent variables ; we are now, in fact, 
merely following out more generally the ideas broached there. 

Y 




Fig. 1. 



To obtain a graphical representation of the variation of the 
function /(.»•) we take two lines X'OX, Y'OY, at right angles to 
each other (co-ordinate axes). To represent the values of x we 
measure x units of length, according to any convenient scale, 
from the intersection along X'OX to the right if x have a 
positive value, to the left if a negative value. To represent the 
values of y we measure lengths of as many units, according to 
the same or, it may be, some other fixed scale, from X'OX 
parallel to Y'OY, upwards or downwards according as these 
values are positive or negative. 

For example, suppose that, when Ave put x = - 20, x = — 7, 
x= + IS, x= + 37, the corresponding values of 



312 



are 



CONTINUITY OF FUNCTION AND OF GRAPH 

/(-20), /(-7), /( + 18), /(+37) 
+ 4, - 10, +7, - 6 



CHAP. 



respectively ; so that we have the following scheme of corre- 
sponding values : — 



X 


y 


-20 

- 7 
+ 18 
+ 37 


+ 4 

- 10 

+ 7 

- 6 

l 



Then we measure off OM 2 (left) =20, OM 4 (left) = 7, OM 7 (right) 
= 18, OM 9 (right) = 37 ; and M 2 P 2 (up) = 4, M 4 P 4 (down) = 10, 
M 7 P 7 (up) = 7, M 9 P 9 (down) = 6. 

To every value of the function, therefore, corresponds a re- 
presentative point, P, whose abscissa (OM) and ordinate (MP) 
represent the values of the independent and dependent variables; 
that is to say, the value of x and the corresponding value of /(./:). 
Now, when we give x in succession all real values from — oo to 
+ qo , y will in general * pass through a succession of real values 
without at any stage making a sudden jump, or, as it is put, without 
becoming discontinuous. The representative point will therefore 
trace out a continuous curve, such as Ave have drawn in Fig. 1. 
This curve Ave may call the graph of the function. 

§ 3.] It is obvious that when Ave Icuoav the graph of a 
function we may find the value of the function corresponding to any 
value of the independent variable x Avith an accuracy that depends 
merely on the scale of our diagram and on the precision of our 
drawing instruments. All Ave have to do is to measure off the 
value of x in the proper direction, OM 7 say ; then draw a 
parallel through M 7 to the axis of y, and find the point P 7 Avhere 
this parallel meets the graph ; then apply the compasses to M 7 P 7 , 
and read off the number of units in M 7 P 7 by means of the scale 
of ordinates. This number, taken positive if P 7 be above the 



We shall retina to the exceptional eases immediately. 



xv GRAPHICAL SOLUTION OF AX EQUATION 313 

axis of x, negative if below, will be the required value of the 
function. 

The graph also enables us to the same extent to solve the 
converse problem, Given the value of the function, to find the corre- 
sponding value or values of the independent variable. 

Suppose, for example, that Fig. 1 gives the graph of f(x), 
and Ave wish to find the values of x for which f(x) = + 7. All 
we have to do is to measure ON 7 = 7 upwards from on the axis 
of y ; then draw aline (dotted in the figure) through N 7 parallel 
to the axis of x, and mark the points where this line meets the 
graph. If P 7 be one of them, we measure N 7 P 7 (obviously = OM 7 ) 
by means of the scale of abscissae, and the number thus read off 
is one of the values of x for which f(x) = + 7 ; the others are 
found by taking the other points of intersection, if such there be. 

Observe that the process we have just described is equivalent 
to solving the equation 

f{x) = + 7. 

In particular we might look for the values of x for which /(./•) 
reduces to zero. When/(a;) becomes zero, that is, when the ordin- 
ate of the graphic point is zero, the graph meets the axis of x. 
The axis of x, then, in this case acts the part formerly played by 
the dotted parallel, and the values of x required are - OMj, 
-OM 3 , +OM, +OM 8 , +OM 10 , where OM„ OM 3 , Sec, stand 
merely for the respective numbers of units in these lengths when 
read off upon the scale of abscissae. Hence 

By means of the graph of the function /(.c) we can solve the 
equation 

/(■*) = (2). 

The roots of this equation are, in point of fact, simply the values 
of x which render the function f(x) zero ; we may therefore, 
when it is convenient to do so, speak of them as the roots of the 
function itself. 

§ 4.] The connection between the general discussion of a 
function by means of the graphical or any other method and the 
problem of solving a conditional equation will now be apparent 
to the reader, and he will naturally ask himself how the graph 



314 



EXAMPLE 



CHAP. 



is to be obtained. We cannot, of course, lay down all the 
infinity of points on the graph, but we can in various ways infer 
its form. In particular, we can assume as many values of the 
independent variable as we please, and, from the known form of 
the function f(x), calculate the corresponding values of y. We 
can thus lay down as many graphic points as we please. If care 
be taken to get these points close enough where the form of the 
curve appears to be changing rapidly, we can draw with a free 
hand a curve through the isolated points which will approach 
the actual graph sufficiently closely for most practical purposes. 

When the form of the function is unknown, and has to be 
determined by observation — as, for example, in the case of the 
curve which represents the height of the barometer at different 
times during the day — the course we have described is the one 
actually followed, only that the value of y is observed and not 
calculated. 

Before going further into details it will be well to illustrate 
by a simple example the above process, Avhich may be unfamiliar 
to many readers. 

Example. 

Let the function to be discussed be l-x", then the equation (1) which 
determines the graph is y= 1 - a?. 

We shall assume, for the present without proof, what will probably be at 
once admitted by the reader, that, as x increases without break from up to 
+ oo, x 2 increases without break from up to + oo ; and that a ,2 > = <1, 
according as x> = < 1. 

Consider, in the first place, merely positive values of x. When x = 0, 
y = l ; and, so long as x<l, l-x- is positive. When x=l, y = l -1 = 0. 
When x>l, then » ,2 >1 and 1 -x 2 is negative. Hence from x = until x=l, 
l-x"- continually decreases numerically, but remains always positive. When 
x = l, l-x 2 becomes zero, and when x is further increased l-x 2 becomes 
negative, and remains so, but continually increases in numerical value. 

We may represent these results by the following scheme of corresponding 
values : — 



X 


y 





i 


<+l 


+ 


+ 1 





>+l 


- 


+ 0O 


— CO 



XV 



EXAMPLE 



315 




The general form of the graph, so far as the right-hand side of the axis of y 
is concerned, will be as in Fig. 2. 

As regards negative values of 
x and the left-hand side of the 
axis of }i, in the present case, 
it is merely necessary to notice 
that, if we put a; = -a, the re- 
sult, so far as 1 - x 2 is concerned, 
is the same as if we put x = +a; 
for 1 - ( - af= 1 - ( + a)-. Hence 
for every point P on the curve, 
whose abscissa and ordinate are 
+ OM and + MP, there will be 
a point P', whose abscissa and 
ordinate are - OM and + M P. P 
and P' are the images of each 
other with respect to Y'Y ; and the part AP'B' of the graph is merely an 
image of the part APB with respect to the line Y'Y. 

Let us see what the graph tells us regarding the function 1 -x". 

First we see that the graph crosses the ar-axis at two points and no more, 
those, namely, for which x= + 1 and x= - 1. Hence the function 1 -x- has 
only two roots, +1 and - 1 ; in other words, the equation 

1-^ = 

has two real roots, x — -hi, x= - 1, and no more. 

Secondly. Since the part BAB' of the graph lies wholly above, and the 
parts C'B', CB wholly below the a:-axis, we see that, for all real values of x 
lying between -1 and +1, the function 1 -ar is positive, and for all other 
real values of x negative. 

Thirdly. We see that the greatest positive value of 1 -x 2 is 1, correspond- 
ing to x—0 ; and that, by making x sufficiently great (numerically), we can 
give 1 -x- a negative value as large, numerically, as we please. 

All these results could be obtained by direct discussion of the function, 
but the graph indicates them all to the eye at a glance. 

§ 5.] Hitherto we have assumed that there are no breaks or 
discontinuities in the graph of the function. Such may, how- 
ever, occur ; and, as it is necessary, when we set to work to 
discuss by considering all possible cases, above all to be sure 
that no possible case has escaped our notice, we proceed now to 
consider the exceptions to the statement that the graph is in 
general a continuous curve. 



f. 



The function f(x) may become infinite for a finite value of x. 



316 



DIFFERENT KINDS OF DISCONTINUITY 



CHAP. 



Example 1. 

Consider the function 1/(1 -a;). "When a; is a very little less than + 1, 
say x= -99999, then y = \j(\-x) gives y = + 1/-00001 = +100000 ; that is to 
say, y is positive and very large ; and it is obvious that, by bringing x suffi- 
ciently nearly up to + 1, we can give y as large a positive value as we please. 
On the other hand, if a: be a very little greater than +1, say x= +1 '00001, 
then 2/=l/(- "00001)= -100000 ; and it is obvious that, by making x ex- 
ceed 1 by a sufficiently small quantity, y can be made as large a negative 
quantity as we please. 

The graph of the function 1/(1 - x) for values of x near + 1 is therefore as 
follows : — 

The branch BC ascends to 
an infinite distance along 
KAK' (a line parallel to the 
2/-axis at a distance from it 
= +1), continually coming 
nearer to KAK', but never 
reaching it at any finite dis- 
tance from the a;-axis. The 
branch DE comes up from an 
infinite distance along the 
other side of KAK' in a similar 
manner. 

Here, if we cause x to in- 
crease from a value OL very 
little less than + 1 to a value 
OM very little greater, the 
value of y will jump from a very large positive value + LC to a very large 
negative value - WD ; and, in fact, the smaller we make the increase of x, 
provided always we pass from the one side of + 1 to the other, the larger will 
be the jump in the value of y. 

It appears then that, for x= +\, 1/(1— a) is both infinite and discon- 
tinuous. 

Example 2. 

y= 1/(1 -x)~. 

We leave the discussion to the 
reader. The graph is as in Fig. 4. 

The function becomes infinite 
when x= +1 ; and, for a very small 
increment of x near this value, the 
increment of y is very large. In fact, 
if we increase or diminish x from the 
value + 1 by an infinitely small 
amount, y will diminish by an in- 
finitely great amount. 

Here again we have infinite value of the function, and accompanying 
discontinuity. 



Y 


B 




K 

M 


O 


I 


k' 


\ 


X 

r 



Fig. 3. 




Fio. 1. 



XV 



DIFFERENT KINDS OF DISCONTINUITY 



31 



Fio. 5. 



II. The value of the function may make a jump without becoming 
infinite. 

The graph for the neighbourhood of such a value would be 
of the nature indicated in Fig. 5, where, while x passes through 
the value (DM, y jumps from MP 
to MQ. 

Such a case cannot, as we shall 
immediately prove, occur with in- 
tegral functions of x. In fact it ol M 
cannot occur with any algebraical 
function, so that we need not 
further consider it here. 

The cases we have just considered lead us to give the follow- 
ing formal definition. 

A function is said to he continuous when for an infinitely small 
change in the value of the independent variable the change in the value 
of the function is also infinitely small ; and to be discontinuous when 
for an infinitely small change of the independent variable the change in 
the value of the function is either finite or infinitely great. 

III. It may happen that the value of a function, all of whose con- 
stants are real, becomes imaginary for a real value of its variable. 

Example. 

This happens with the function + \J{1 - x"). If we confine ourselves to the 
positive value of the square root, so that we have a single-valued function to 
deal with, the graph is as in Fig. 6 : — 

a semicircle, in fact, whose centre is at the 
origin. 

For all values of a;>+l, or <— 1, the 
value of y= + \/(l - ar) is imaginary ; and the 
graphic points for them cannot be constructed 

- in the kind of diagram we are now using. 

The continuity of the function at A can- 
not, strictly speaking, be tested ; since, if we 
attempt to increase x beyond + 1, y becomes 
imaginary, and there can be no question of the 
magnitude of the increment, from our present point of view at least.* 

No such case as this can arise so long as /(.?•) is a rational 

algebraical function. 




Fio. 6. 



See below, § 18. 



318 LIMITING CASES chap. 

We have now enumerated the exceptional cases of functional 
variation, so far at least as is necessary for present purposes. 
Graphic points, at which any of the peculiarities just discussed 
occur, may be generally referred to as critical points. 



ON CERTAIN LIMITING CASES OF ALGEBRAICAL OPERATION. 

§ 6.] We next lay down systematically the following propo- 
sitions, some of which we have incidentally used already. The 
reader may, if he choose, take them as axiomatic, although, as 
we shall see, they are not all independent. The important 
matter is that they be thoroughly understood. To secure that 
they be so we shall illustrate some of them by examples. In 
the meantime we caution the reader that by " infinitely small " 
or "infinitely great" we mean, in mathematics, " smaller than 
any assignable fraction of unity," or " as small as we please," and 
"greater than any assignable multiple of unity," or "as great as 
Ave please." He must be specially on his guard against treating 
the symbol <x> , which is simply an abbreviation for " greater 
than any assignable magnitude," as a definite quantity. There 
is no justification for applying to it any of the laws of algebra, 
or for operating with it as we do with an ordinary symbol of 
quantity. 

I. If P be constant or variable, provided it does not become 
infinitely great when Q becomes infinitely small, then when Q becomes 
infinitely small PQ becomes infinitely small. 

Observe that nothing can be inferred without further examin- 
ation in the case where P becomes infinitely great when Q 
becomes infinitely small. This case leads to the so-called inde- 
terminate form oo x 0.* 

Example 1. 

Let us suppose, for example, that P is constant, =100000, say. Then, if 
we make Q = l/100000, we reduce PQ to 1 ; if we make Q = 1/100000000000, 
we reduce PQ to 1/1000000 ; and so on. It is abundantly evident, therefore, 
that by making Q sufficiently small PQ can be made as small as we please. 



* Indeterminate forms are discussed in chap, xxv 



ZV LIMITING CASES 319 

Example 2. 

LetP = x + l, Q = a:-1. 

Here, when x is made to approach the value + 1, P approaches the finite 
value + 2, while Q approaches the value 0. Suppose, for example, we put x — 1 
+ 1/100000, then 

PQ = (2 + 1/100000) x 1/100000, 
^2/100000 + 1/10 10 , 
and so on. Obviously, therefore, by sufficiently diminishing Q, we can make 
PQ as small as we please. 

Example 3. 

Y = l/(x--l), Q=jb-1. 

Here we have the peculiarity that, when Q is made infinitely small, P (see 
below, Proposition III.) becomes infinitely great. We can therefore no longer 
infer that PQ becomes infinitely small because Q does so. In point of fact, 
PQ=(«- l)/(x-- l)=l/(aj+l), which becomes 1/2 when x=l. 

II. If ¥ be either constant or variable, provided it do not become 
infinitely small when Q becomes infinitely great, then when, Q becomes 
infinitely great PQ becomes infinitely great. 

The case where P becomes infinitely small when Q becomes 
infinitely great must be further examined ; it is usually referred 
to as the indeterminate form x oo . 

Example 1. 

Suppose P = l/100000. Then, by making Q = 100000, we reduce TQ to 1 ; 
by making Q = 100000000000 we reduce PQ to 1000000 ; and so on. It is 
clear, therefore, that by sufficiently increasing Q we could make PQ exceed 
any number, however great. 

The student should discuss the following for himself : — 

Example 2. 

Y=x + \, Q = l/(z-l). 
PQ = oo whena;=l. 
Example 3. 

P=(a;-l) a , Q = l/(.*-l). 
PQ = 0when x=l. 

III. If P be either constant or variable, provided it do not become 
infinitely small when Q becomes infinitely small, then when Q becomes 
infinitely small P/Q becomes infinitely great. 

The case where P and Q become infinitely small for the same 
value of the variable requires further examination. This gives 

the so-called indeterminate form -• 



320 LIMITING CASES 



CHAP. 



Example 1. 

Suppose P constant = 1/100000. If we make Q = 1/100000, P/Q becomes 
1 ; if we make Q = 1/100000000000, P/Q becomes 1000000 ; and so on. Hence 
we see that, if only we make Q small enough, we can make P/Q as large as 
we please. 

The student should examine arithmetically the two following cases : — 

Example 2. 

T=x + 1, Q=a-1. 

P/Q = oo when x—1. 
Example 3. 

P=*-l, Q=«-l. 
P/Q=l whencc=l. 

IV. If P be either constant or variable, provided it do not become 
infinitely great when Q becomes infinitely great, then when Q becomes 
infinitely great P/Q becomes infinitely small. 

The case where P and Q become infinitely great together re- 
quires further examination. This gives the indeterminate form 

00 
00 

Example 1. 

Suppose P constant = 100000. If we make Q = 100000, P/Q becomes 1; 
if we make Q = 100000000000, P/Q becomes 1/1000000 ; and so on. Hence by 
sufficiently increasing Q we can make P/Q less than any assignable quantity. 

Example 2. 

P=aj+1, Q=l/(as-l). 

p/Q = when x—1. 
Example 3. 

P = l/(a;-l) a , Q = l/(.z-l). 

P/Q = oo when sc=l. 

V. If P and Q each become infinitely small, then P + Q becomes 
infinitely small. 

For, let P be the numerically greater of the two for any 
value of the variable. Then, if the two have the same sign, and, 
a fortiori, if they have opposite signs, numerically 

P + Q < 2P. 

Now 2 is finite, and, by hypothesis, P can be made as small as 
we please. Hence, by I. above, 2P can be made as small as 
we please. Hence P + Q can be made as small as we please. 

VI. If either P or Q become infinitely great, or if P and Q each 



XV 



LIMITING CASES 321 



become infinitely great and both have finally the same sign, then P + Q 
becomes infinitely great. 

Proof similar to last. 

The inference is not certain if the two have not ultimately 
the same sign. In this case there arises the indeterminate 
form oo - co . 

Example 1. 

?=x*/(x-l)*, Q = (2x-\)/(x-ir-. 
When x= 1, we ha ve P = 1/0 = + oo , Q , = 1 /0 = + oo . Also 

F + Q = rA^+ 2X ~ l 3S + 2X ~ l 



(x-i)-'(z-iy (x-iy 

Example 2. 



2 i i 

= -=oo, when x=l. 



P=a?/(a!-l) 2 , q=-(2x-l)J(x-lf. 

Here x=l makes P=+oo, Q=-oo, so that we cannot infer r + Q = <». 
In fact, in this case, 

p. __£! 2*-l_( a :-l) a 

L+K -(;c-lr< (x-lf-ix-l)*- 1 

for all values of x, or, say, for any value of x as nearly = + 1 as we please. In 
this case, therefore, by bringing x as near to +1 as we please, we cause the 
value of P + Q to approach as near to +1 as we please. 

§ 7.] The propositions stated in last paragraph are the funda- 
mental principles of the theory of the limiting cases of algebraical 
operation. This subject will be further developed in the chapter 
on Limits in the second part of this work. 

In the meantime we draw the following conclusions, which 
will be found useful in what follows : — 

I. If P = P^ . . . P Jt , then P will remain finite if P,, P 2 , . . ., 
P n all remain finite. 

P will become infinitely small if one or more of the functions 
P n P 2 , . . ., ~P n become infinitely small, provided none of the remain- 
ing ones become infinitely great. 

P will become infinitely great if one or more of the functions P n 
P 2 , . . ., P n become infinitely great, provided none of the remaining 
ones become infinitely small. 

II. If S = P, + P, + . . . + P n , then S will remain finite if P,, 
P 2 , . . ., P,j, each remain finite. 

VOL. I Y 



322 LIMITING CASES chap. 

S will become infinitely small if P u P 2 , . . ., P n each become in- 
finitely small. 

S vMl become infinitely great if one or more of the functions P,, 
P 3 , . . ., P n become infinitely great, provided all those that become 
infinitely great have the same sign. 

III. Consider the quotient P/Q. 

p 

j- xoill certainly be finite if both P and Q be finite, 

may be finite if P = 0, Q = 0, 

or if P — oo , Q = oo . 

P 

pr icill certainly = if P = 0, Q. 4= 0, 

or ?/ P 4= co , Q = oo • 

may =0i/P = 0, Q = 0, 

or if P = co , Q = co , 

P 

pr will certainly = co (/* P = oo , Q 4= oo , 

^ on/P + O, Q = ; 

may = oo if P = 0, Q = 0, 
or ?/ P = oo , Q = co . 



ON THE CONTINUITY OF FUNCTIONS, MORE ESPECIALLY OF 
RATIONAL FUNCTIONS. 

§ 8.] We return now to the question of the continuity of 
functions. 

By the increment of a function f(x) corresponding to an increment 

h of the independent variable x we mean f(x + h) - f {•>')■ 

For example, HfixJ—x 1 , the increment is (x + h) 2 -x i —2xh + Jr. 
lff(x) = l/x, the increment is l/{x + h)-l/x= -h/x(x + h). 

The increments may be either positive or negative, according 
partly to choice and partly to circumstance. The increment of 
the independent variable x is of course entirely at our disposal ; 
but when any value is given to it, and when x itself is also 
assigned, the increment of the function or dependent variable 
is determined. 



xv CONTINUITY OF A SUM OR PRODUCT 323 

Example. 

Let the function be 1/x, then if x=l, h = 3, the corresponding increment 
ofl/.ris -3/1(1+3)= -3/4. Ifz=2, h = 3, the increment of 1/x is -3/2(2 + 3) 
~ - 3/10, and so on. 

If P be a function of x, and p denote its increment when x 
is increased from x to x + h, then, by the definition of p, P +p is 
the value of P when x is altered from x to x + A. 

"We can now prove the following propositions : — 

I. The algebraic sum of any finite number of continuous functions 
is a continuous function. 

Let us consider S = P - Q + R, say. If the increments of P, 
Q, R, when x is increased by h, be p, q, r, then the value of S, 
when x is changed to x + h, is (P + p) - (Q + q) + (R + r) ; and the 
increment of S corresponding to h is p - q + r. Now, since P, Q, 
R are continuous functions, each of the increments, p, q, r, becomes 
infinitely small when h becomes infinitely small. Hence, by § 7, 
I., p — q + r becomes infinitely small when h becomes infinitely 
small. Hence S is a continuous function. 

The argument evidently holds for a sum of any number of 
terms, provided there be not an infinite number of terms. 

II. The p-oduct of a finite number of continuous functions is a 
continuous function so long as all factors remain finite. 

Consider, in the first place, PQ. Let the increments of P 
and Q, corresponding to the increment h of the independent vari- 
able x, be p and q respective^. Then when x is changed to x + h 
PQ is changed to (P + p) (Q, + q), that is, to VQ, + pQ, + q~P + pq. 
Hence the increment of PQ corresponding to h is 

jjQ + ^P +pq. 

Xow, since P and Q are continuous, p and q each become in- 
finitely small when h becomes infinitely smalL Hence by § 7, I. 
and II., it follows that pQ + qP + pq becomes infinitely small 
when h is made infinitely small ; at least this will certainly be 
so, provided P and Q remain finite for the value of x in question, 
which we assume to be the case. 

It follows then that PQ is a continuous function. 

Consider now a product of three continuous functions, say 
PQR. By what has just been established, PQ is a continuous 



324 ANY INTEGRAL FUNCTION CONTINUOUS chap. 

function, which we may denote by the single letter S ; then 
PQR = SR where S and R are continuous. But, by last case, SR 
is a continuous function. Hence PQR is a continuous function. 
Proceeding in this way, we establish the proposition for any 
finite number of factors. 

Cor. 1. If A be constant, and P a continuous function, then AP 
is a continuous function. 

This can either be established independently, or considered 
as a particular case of the main proposition, it being remembered 
that the increment of a constant is zero under all circumstances. 

Cor 2. Ax m , where A is constant, and in a positive integer, is a 
continuous function. 

For x m = x x xx . . . x x (m factors), and x is continuous, being 
the independent variable itself. Hence, by the main proposi- 
tion, x m is continuous. Hence, by Cor. 1, At'" is a continuous 
function. 

Cor. 3. Every integral function of x is continuous ; and cannot 
become infinite for a finite value of x. 

For every integral function of x is a sum of a finite number 
of terms such as Ax" 1 . Now each of these terms is a continuous 
function by Cor. 2. Hence, by Proposition I., the integral func- 
tion is continuous. That an integral function is always finite 
for a finite value of its variable follows at once from § 7, I. 

III. If P and Q be integral functions of x, then P/Q is finite and 
continuous for all finite values of x, except such as render Q = 0. 

In the first place, if Q 4= 0, then (see § 7, III.) P/Q can only 
become infinite if either P, or both P and Q, become infinite ; but 
neither P nor Q can become infinite for a finite value of x, because 
both are integral functions of x. Hence P/Q can only become 
infinite, if at all, for values of x which make Q = 0. 

If a value which makes Q = makes P 4= 0, then P/Q certainly 
becomes infinite for that A-alue. But, if such a value makes both 
Q = and also P = 0, then the matter requires further investi- 
gation. 

Next, as to continuity, let the increments of P and Q corre 



XV CONTINUITY OF A RATIONAL FUNCTION 325 

sponding to h, the increment of x, be p and q as heretofore. 
Then the increment of P Q is 

Y+p _ P _ pQ - gP 

Q + 'l Q~Q(Q + ?)" 
Now, by hypothesis, ^ and q each become infinitely small 
when h does so. Also P and Q remain finite. Hence pQ - qP 
becomes infinitely small. It follows then that (pQ - #P)/Q(Q + q) 
also becomes infinitely small when h does so, provided always 
(see § 6) that Q does not vanish for the value of x in question. 

Example. 

The increment of l/(se— 1) corresponding to the increment, h, of a: is 
l/(x + h-l)-l/(x-l)= - h/(x - 1) (a*+ h - 1). N ow, if x = 2, say, this becomes 
- hftl+h), which clearly becomes infinitely small when h is made infinitely 
small. On the other hand, if x—1, the increment is -h/Oh, which is 
infinitely great so long as h has any value differing from by ever so little. 

§ 9.] When a function is finite and continuous between two 
values of its independent variable x = a and x = b, its graph forms 
a continuous curve between the two graphic points whose 
abscissas are a and b ; that is to say, the graph passes from the 
one point to the other without break, and without passing any- 
where to an infinite distance. 

From this we can deduce the following important pro- 
position : — - 

Jff(x) be continuous from x - a to x = b, and iff(a) =p, f(b) = q, 
then, as x passes through every algebraical value between a and b, fix) 
passes at least once, and, if more than once, an odd number of times 
through every algebraical value between p and q. 

Let P and Q be the graphic points corresponding to x = a and 
x = b, AP and BQ their ordinates ; then AP = p, BQ = q. We 
have supposed p and q both positive ; but, if either were negative, 
we should simply have the graphic point below the a'-axis, and 
the student will easily see by drawing the corresponding figure 
that this would alter nothing in the following reasoning. 

Suppose now r to be any number between p and q, and 
draw a parallel UV to the z-axis at a distance from it equal 
to r units of the scale of ordinates, above the axis if r be 
positive, below if r be negative. The analytical fact that r is 



326 



LIMITS FOR THE ROOTS OF AN EQUATION 



CHAP. 



intermediate to p and q is represented by the geometrical fact 
that the points P and Q lie on opposite sides of UV. 



Y 










Q 




u 


f? 




R R 





V 




p 











A M, M 2 Nlj B X 

Fig. 7. 



Now, since the graph passes continuously from P to Q, it 
must cross the intermediate line UV ; and, since it begins on one 
side and ends on the other, it must do so either once, or thrice, 
or five times, or some odd number of times. 

Every time the graph crosses UV the ordinate becomes equal 
to r ; hence the proposition is proved. 

Cor. 1. If f(a) be negative andf(b) be positive, or vice versa, then 
f(x) has at least one root, and, if more than one, an odd number of 
roots, between x = a and x = b, provided f{x) be continuous from x = a 
to x-b. 

This is merely a particular case of the main proposition, for 
is intermediate to any two values, one of which is positive and 
the other negative. Hence as x passes from a to bf(.r) must pass 
at least once, and, if more than once, an odd number of times 

through the value 0. 

In fact, in this case, 

the axis of x plays the 

/ \ part of the parallel UV. 

Observe, however, in 

regard to the converse of 



12 



Fig. 8. 



Fio. 9. 



this proposition, that a 
function may pass through the value without changing its sign. 
For the graph may just graze the re-axis as in Figs. 8 and 9. 



XV VALUES FOR WHICH A FUNCTION CHANGES SIGN 327 

Cor. 2. Iff((i) and f(b) have like signs, then, if there be any real 
roots of f(x) between x = a and x — b, there must be an even number, 
provided f(x) be continuous between x = a and x = b. 

Since an integral function is always finite and continuous for 
a finite value of its variable, the restriction in Cor. 1 is always 
satisfied, and we see that 

Cor. 3. An integral function can change sign only by passing 
through the value 0. 

Cor. 4. If P and Q be integral functions of x algebraically prime 
to each other, P/Q can only change sign by passing through the values 

or oo . 

"With the hint that the theorem of remainders will enable 
him to exclude the ambiguous case 0/0, we leave the reader to 
deduce Cor. 4 from Cor. 3. 

Example 1. 

When sb=0, l-a?=+l; and when x= +2, l-ar= -3. Hence, since 

1 -a; 2 is continuous, for some value of x lying between and +2 1 — a? must 
become 0; for is between +1 and - 3. In point of fact, it becomes once 
between the limits in question. 

Example 2. 

y=a?-fa?+llx-6. 

Whenai=0, y — -6; and when x= + 4, y= +6. Hence, between x = and 
x= +4 there must lie an odd number of roots of the equation 

a- 3 -6a; 2 +lla;-6 = 0. 
It is easy to verify in the present case that this is really so; for x 3 -6x i 
+ llx - 6 = (x - 1) (x - 2) (x - 3) ; so that the roots in question are x — 1, x — 2, 
se=3. 

The general form of the graph in the present case is as follows : — 




Fig. io. 

Example 3. 

When x = 0, l/(l-a;)=+l; and when x= +2, l/(l-a:)=-] : but since 
1/(1 — x) becomes infinite and discontinuous between x = and x= +2, namely, 
when X=l, we cannot infer that, for some value of x between and +2, 



328 SIGN OF /(0) AND /( ± oo ) chap. 

1/(1 -x) will become 0, although is intermediate to + 1 and -1. In fact, 
1/(1 — x) does not pass through the value between x=0 and x= +2. 

§ 10.] It will be convenient to give here the following pro- 
position, which is often useful in connection with the methods 
Ave are now explaining. 

If fix) be an integral function of x, then by making x small 
enough we can always cause f(x) to have the same sign as its lowest 
term, and by making x large enough we can always cause f(x) to have 
the same sign as its highest term. 

Let us take, for simplicity, a function of the 3rd degree, say 
y = px 3 + qx 2 + rx + s. 
If we suppose s 4= 0, then it is clear, since by making x small 
enough we can (see § 7, II.) make px 3 + qx 2 + rx as small as we 
please, that we can, by making x small enough, cause y to have 
the same sign as s. 

If s = 0, 

then we have y =px* + qx 2 + rx, 

= (px 2 + qx + r)x. 
Here by making x small enough we can cause px 2 + qx + r to have 
the same sign as r, and hence y to have the same sign as rx, 
which is the lowest existing term in y. 

Again, we may write 




ir- 

Here by making x large enough we may make q/x + rjx 2 + s/x 3 as 
small as we please (see § 6, IV., and § 7, II.), that is to say, 
cause p + q/x + rjx 2 + s/x 3 to have the same sign as p. Hence by 
making x large enough we can cause y to have the same sign 



as px 3 



If we observe that, by chap, xiv., § 9, we can reduce every 
integral equation to the equivalent form 

f{x) = x n + p n _ , x"- * + . . .+^ = 0, 
and further notice that, in this case, if n be odd, 

/( + oo ) = + oo , /( - oo ) = - oo , 
ami, if n be even, 

/( + oo ) = + oo , /( - oo ) = + oo , 
we have the following important conclusions. 



XV 



MINIMUM NUMBER OF REAL ROOTS 



329 



Cor. 1. Every integral equation of odd degree with real co- 
efficients has at least one real root, and if it has more than one it 
has an odd number. 

Cor. 2. If an integral equation of even degree with real coefficients 
has any real roots at all, it lias an even number of such. 

Cor. 3. Every integral equation with real coefficients, if it has any 
complex roots, has an even number of such. 

The student should see that he recognises what are the cor- 
responding peculiarities in the graphs of integral functions of 
odd or of even degree. 

Example. 
„ Show that the equation 

x i -6, 3 + llx--r-l = () 
has at least two real roots. 
Let y = x i -6x' i + llx--x-4. 

We have the following scheme of corresponding values : — 



X 

— OO 



+ oo 


V 


+ oo 
- 4 

+ oo 



Hence one root at least lies between - oo and 0, and one at least between 
and + oo . In other words, there are at least two real roots, one negative 
the other positive. 

We can also infer that, if the remaining two of the possible four be also 
real, then they must be either both positive or both negative. 

When the real roots of an integral equation are not very 
close together the propositions we have just established enable 
us very readily to assign upper and lower limits for each of 
them ; and in fact to calculate them by successive approxima- 
tion. The reader will thus see that the numerical solution of 
integral equations rests merely on considerations regarding con- 
tinuity, and may be considered quite apart from the question 
of their formal solution by means of algebraical functions or 
otherwise. The application of this idea to the approximate 
determination of the real roots of an integral equation will be 
found at the end of the present chapter. 



330 



MAXIMA AND MINIMA ALTERNATE 



CHAP. 



GENERAL PROPOSITIONS REGARDING MAXIMA AND MINIMA 
VALUES OF FUNCTIONS OF ONE VARIABLE. 

§11.] When fix) in passing through any value, f(a) say, ceases to 
increase and begins to decrease, f(a) is called a maximum value of f{x). 

When fix) in passing through the value f (a) ceases to decrease and 
begins to increase, f(a) is called a minimum value of fix). 

The points corresponding to maxima and minima values of 
the function are obviously superior and inferior culminating 
points on its graph, such as P 2 and P 9 in Fig. 1. They are 
also points where, in general, the tangent to the graph is parallel 
to the axis of x. It should be noticed, however, that points 
such as P and Q in Fig. 1 1 are maxima and minima points, 
according to our present definition, although it is not true in 
any proper sense that at them the tangent is parallel to OX. It 





Fig. 11. 



Fio. 12. 



should also be observed that the tangent may be parallel to OX 
and yet the point may not be a true maximum or minimum 
point. Witness Fig. 12. 

We shall include both maximum and minimum values as at 
present defined under the obviously appropriate name of turning 
values. 

§ 12.] By considering an unbroken curve having maxima 
and minima points (see Fig. 1) the reader will convince himself 
graphically of the truth of the following propositions : — 

I. So long as f(x) remains continuous its maxima and minima 
values succeed each other alternately. 

II. If x = a, x = b be two roots of f(x) (a alg.<b), then, if f(x) be 
not constant, but vary continuously between x-a and x = b, there must 



XV 



CRITERION FOR TURNING VALUES 331 



be either at least one maximum or at least one minimum value of f(x) 
between x - a and x = b. 

In particular, if f(x) become positive immediately after x 
passes through the value a, then there must he at least one 
maximum before x reaches the value b ; and, in like manner, if 
f(x) become negative, at least one minimum. 

§ 1 3.] It is obvious, from the definition of a turning value, 
and also from the nature of the graph in the neighbourhood of 
a culminating point, that we can always find two values of the func- 
tion, on opposite sides of a turning value, which shall be as nearly 
equal as we please. These two values will be each less or each greater 
than the turning value according as the turning value is a maximum 
or minimum. 

Hence, if p be infinitely near a turning value of fix) (less in 
the case of a maximum, greater in the case of a minimum), then 
two roots oi fix) -p will be infinitely nearly equal to one another. 
It follows, therefore, that, if p be actually equal to a turning value 
of f(x), the function f(x) -p will have two of its roots equal. This 
criterion may be used for finding turning values, as will be seen 
in a later chapter. 

CONTINUITY AND GRAPHICAL REPRESENTATION OF A 
FUNCTION OF TWO INDEPENDENT VARIABLES. 

§ 14.] Let the function be denoted by f(x, y), and let us 
denote the dependent variable b} r z ; so that 

*=/(«, y)- 

We confine ourselves entirely to the case where fix, y) is an integral 
function, and we suppose all the constants to be real, and consider mily 
real values of x and y. The value of z will therefore be always real. 

Since there are now two independent variables, x and //, 
there are two independent increments, say h and k, to consider. 
Hence the increment of z, that is, f(x + h, y + k) -f(x, y), now 
depends on four quantities, x, y, h, k. Since, however, f(x, y) 
consists of a sum of terms such as Ax m f n , it can easily be shown 
by reasoning, like that used in the case oif(x), that the increment 
of z always becomes infinitely small when h and k are made infinitely 



332 



GRAPHIC SURFACE FOR Z =/(», y) 



CHAP. 



small. Hence, as x and y pass continuously from one given pair of 
values, say (a, b), to another given pair, say (a', b'), z passes continu- 
ously from one value, say c, to another, say c'. 

§ 15.] There is, however a distinct peculiarity in the case 
now in hand, inasmuch as there are an infinity of different ways 
in which (x, y) may pass from (a, b) to (a', b'). In fact we re- 
quire now a two-dimensional diagram to represent the variations of 
the independent variables. Let X'OX, Y'OY be two lines in a 




Fio. 13. 

horizontal plane drawn from west to east and from south to 
north respectively. Consider any point P in that plane, whose 
abscissa and ordinate, with the usual understanding as to sign, 
are x and y. Then P, which we may call the variable point, gives 
us a graphic representation of the variables (x, y). 

Let us suppose that for P x = a, y = b, and that for another 
point P' x = a', y = b'. Then it is obvious that, if Ave pass 
along any continuous curve whatever from P to P', x will vary 
continuously from a to a', and y will vary continuously from 
b to 1/ ; and, conversely, that any imaginable combination of a 
continuous variation of x from a to a' with a continuous varia- 
tion of y from b to V will correspond to the passage of a point 
from P to P' along some continuous curve. 

It is obvious, therefore, that the continuous variation of 



XV CONTOUR LINES 333 

(x, y) from (a, h) to (a\ b') may be accomplished in an infinity 
of ways. We may call the path in which the point which repre- 
sents the variables travels the graph of the variables. 

To represent the value of the function z = f(x, y) we draw 
through P, the variable point representing (x, y), a vertical line 
PQ, containing .: units of any fixed scale of length that may be 
convenient, upwards if z be positive, downwards if z be negative. 
Q is then the graphic point which represents the value of the 
function z =/(•'", y). 

To every variable point in the plane XOY there corresponds 
a graphic point, such as Q ; and the assemblage of graphic points 
constitutes a surface which we call the graphic surface of the 
function f{x, y) 

When the variable point travels along any particular curve S 
in the plane XOY, the graphic point of the function travels along 
a particular curve 2 on the graphic surface ; and it is obvious 
that S is the orthogonal projection of 2 on the plane XOY. 

§ 1 6.] If we seek for values of the variables which correspond 
to a given value c of the function, we have to draw a horizontal 
plane U, c units above or below XOY according as c is positive 
or negative ; and find the curve 2 where this plane U meets the 
graphic surface. This line 2 is what is usually called a contour 
line of the graphic surface. In this case the orthogonal projection 
S of 2 upon XOY will be simply 2 itself transferred to XOY, 
and may be called the contour line of the function for the value c. 
All the variable points upon 8 correspond to pairs of values of 
(x, y), for which f(x, y) has the given value c. 

If we take a number of different values, c„ c 2 , c 3 , . . ., c,„ we 
get a system of as many contour lines. Suppose, for example, 
that the graph of the function were a rounded conical peak, then 
the system of contour lines would be like Fig. 14, where the 
successive curves narrow in towards a point which corresponds 
to a maximum value of the function. 

Any reader who possesses a one -inch contoured Ordnance 
Survey map has to hand an excellent example of the graphic 
representation of a function. In this case x and y are the dis- 



334 



f(x, y)=0 REPRESENTS A PLANE CURVE 



CHAP. 



tances east from the left-hand side of the map, and north from 
the lower side ; and the function z is the elevation of the land 




Fig. 14. 



at any point above the sea level. The study of such a map from 
the present point of view will be an excellent exercise both in 
geometry and in analysis. 

An important particular case is that where we seek the 
values of x and y which make f(x, y) = 0. In this case the plane 
U is the plane XOY. This plane cuts the graphic surface in a 
continuous curve S {zero contour line), every point on which 
has for its abscissa and ordinate a pair of values that satisfy 
f{x, y) = 0. 

The curve S in this case divides the plane into regions, such that 
in any region f(x, y) has always either the sign + or the sign — , 
and S always forms the boundary between two regions in which f(x, y) 
has opposite signs. 

If we draw a continuous curve from a point in a + region 
to a point in a - region, it must cross the boundary S an odd 
number of times. This corresponds to the analytical statement 
that iff(a, b) be positive and f(a, b') be negative, then, if (x, y) vary 
continuously from {a, b) to (a', b'), f(x, y) will pass through the value 
an odd number of times. 

The fact just established, that all the " variable points " for 
which f(x, y) — lie on a continuous curve, gives us a beautiful 
geometrical illustration of the fact established in last chapter, that 
the equation f(x, y) = has an infinite number of solutions, and 
gives us the fundamental idea of co-ordinate geometry, namely, that 



XV 



EXAMFLE 



335 



a plane curve can be analytically represented by means of a single 
equation connecting two variables. 

Example. 

Consider the function z = x 2 + y 2 - 1. If we describe, with as centre, a 
circle whose radius is unity, it will be seen that for all points inside this circle 
z is negative, and for all points outside z is positive. Hence this circle is the 
zero contour line, and for all points on it we have 

x 2 + if--l=0. 
Y 




Fig. 15. 



INTEGRAL FUNCTIONS OF A SINGLE COMPLEX VARIABLE. 

§ 17.] Here we confine ourselves to integral functions, but no longer 
restrict either the constants of the function or its independent variable 
x to be real. 

Let us suppose that x = £ + ?ii, and let us adopt Argand's 
method of representing £ + yi H 

graphically, so that, if 12 M = £, 
MP = ij, in the diagram of Fig. 
1 6, then P represents £ + tji. 

If P move continuously from 

any position P to another P', the 

complex variable is said to vary 

continuously. If the values of 

(£, rj) at P and P' be (a, ft) and 

(a', ft') respectively, this is the Fro. 16. 

same as saying that £ + iji is said to vary continuously from the 

value a + fti to the value a + /3'i, when £ varies continuously from a 

to a, and tj varies continuously from ft to ft'. There are of course 



P 



Q 



M 



M' 2 



336 



CONTINUITY OF COMPLEX FUNCTION 



CHAP. 



an infinite number of ways in which this variation may be 
accomplished. 

§ 1 8.] Suppose now we have any integral function of x whose 
constants may or may not be real. Then we have f(x) =f($ + rji) ; 
but this last can, by the rules of chap, xii., always be reduced 
to the form £' + rji, Avhere £' and rj are integral functions of £ 
and ?; whose constants are real (say real integral functions of 
(£, rj) ). Now, by § 14, £' and rj are finite and continuous so 
long as (£, ?/) are finite. Hence /(£ + rfi) varies continuously when 
£ + r/i varies continuously. 

A graphic representation of the function /(£ + rji) can be 
obtained by constructing another diagram for the complex 
number £' + iji. Then the continuity of /(£ + rji) is ex- 
pressed by saying that, when the graph of the independent vari- 
able is a continuous curve S, the graph of the dependent variable is 
another continuous curve S'. 



Example. 
Let 



2/=+V(l-^). 



H 



12 



H 






$ 








1 


J 








B' 


C, 




0' 


A' 


c 


»— * 



Fig. 17. 



Fig. 18. 



For simplicity, we shall confine ourselves to a variation of x which admits 
only real values ; in other words, we suppose r\ always =0. 

The path of the independent variable is then IACBJ, the whole extent 
of the £-axis. In the diagram we have taken CA = CB = 1 ; so that A and 
B mark the points in the path for which the function begins to have, and 
ceases to have, a real value. 

Let Fig. 18 be the diagram of the dependent variable, ;/ = £' + t?7- I f 
A'C'-l (A', B', and W are all coincident), then the path of the dependent 



XV 



COMTLEX ROOTS DETERMINED GRAPHICALLY 



337 



variable is the whole of the 77' -axis above fi, together with A'C, each 
reckoned twice over. The pieces of the two paths correspond as follows : — 



Independent 
Variable. 


Dependent Variable. 


IA 

AC 
CB 
BJ 


FA' 
A'C 
CB' 
B'J' 



§ 19.] £' and rj being functions of £ and >;, we may represent 
this fact to the eye by writing 

r = </>(£ v), v' = M v)- 

If we seek for values of (£, rj) that make £' = 0, that is the same 
as seeking for values of (£, rj) that make <£(£, rj) = 0. All the 
points in the diagram of the independent variable corresponding 
to these will lie (by § 16) on a curve S. 

Similarly all the points that correspond to rj = 0, that is, to 
^(£, rj) = 0, lie on another curve T. 

The points for which both £' = and ?/ = 0, — in other words, 
the points corresponding to roots of /(£ + ?;?), — must therefore be 
the intersections of the two curves S and T. 



Example. 
If we jmtx=a+rri, and y=s' + 7 ?'i wu have 

=m-$v) + (e-v-)i. 

Hence £'=2(4 -£»;), v ' = ^- v ". 

Hence the S and T curves, above spoken of, are given by the equations 

2(4 -fr) = (S), 

?-v-=Q (T). 



These are equivalent to 



VOL. I 



4 






(S), 

(T). 



338 



HORNER S METHOD 



CHAP. 



The student should have no difficulty in constructing these. The diagram 
that results is 

H 






\ 



N 





M 



Fir,. 19. 



The S curve (a rectangular hyperbola as it happens) is drawn thick. The 
T curve (two straight lines bisecting the angles between the axes) is dotted. 
The intersections are P and Q. 

Corresponding to P we have if = 2, ?? = 2; corresponding to Q, £= -2, 



V- 



_ o 



It appears therefore that the roots of the function are + 2 + 2i and - 2 - 2i. 
The student may verify that these values do in fact satisfy the equation 

ix 2 + 8 = 0. 



HORNER'S METHOD FOR APPROXIMATING TO THE VALUES OF THE 
REAL ROOTS OF AN INTEGRAL EQUATION. 

§ 20.] In the following paragraphs we shall show how the 
ideas of §§ 8-10 lead to a method for calculating digit by digit 
the numerical value of any real root of an integral equation. It 
will be convenient in the first place to clear the way by estab- 
lishing a few preliminary results upon which the method more 
immediately depends. 

§ 21.] To deduce from the equation 

p^ n + p x X n ~ l + . . . + p n _ iX + p n =0 (1 ) 

another equation each of wlwse roots is m times a corresponding root 






xv DIMINUTION OF ROOTS 339 

of (1). Let x be any root of (1) ; and let £ = mz. Then x = gjm. 
Hence, from (1), we have 

Pod m) n +p 1 (g/m) n - 1 + . . . +p n - l ($/m) +p n = 0. 

If we multiply by the constant in' 1 , we deduce the equivalent 
equation 

pg 1 + pM n - 1 + ■ ■ ■+ Pn - .»" " *£ +JW»" = (2), 
which is the equation required. 

Cor. The equation whose roots are those of (1) with the signs 
changed is 

Po$" -Pi?- 1 + • • ■ + (- r^Pn-^ + ( - )"Pn = (3). 

This follows at once by putting m = - 1 in (2). We thus 
see that the calculation of a negative real root of any equation 
can always be reduced to the calculation of a positive real root 
of a slightly different equation. 

Example. The equation whose roots are 10 times the respective roots of 

Sa?-15x i +5x + 6 = 

is 3^ -150x 2 + 500.c+ 6000 = 0. 

§ 22.] To deduce from the equation (1) of § 21 another, each of 
xohose roots is less by a than a corresponding root of (I). 

Let x denote any root of (1); £ the corresponding root of 
the required equation ; so that g = x — a, and x = £ + a. Then 
we deduce at once from (1) 

P& + a) n +;>,(£ + a)"" 1 + . . . +p n -i($ + «) +Pn = (4). 
If we arrange (4) according to powers of £, we get 

ft£* + $£*"* + : • - + q n -i€ + qn = o (5), 

which is the equation required. 

It is important to have a simple systematic process for 
calculating the coefficients of (5). This may be obtained as 
follows. 

Since £ = x - a, we have, by comparing the left-hand sides of 
(1) and (5), 

p x n +p l x n ~ 1 + . . . +p n _ l x +p n 

=p Q (x-a) n + q l (z-a) n - 1 + . . . + ?»-i(a:-a) + ?»• 



340 APPROXIMATION TO ROOT chap. 

The problem before us is, therefore, simply to expand the 
function/(.T) =p x n +p 1 x n ~' 1 + . . . +p n -iX +p n m powers of (x - a). 
Hence, as we have already seen in chap, v., § 21, q n is the 
remainder when f{x) is divided by x - a ; q n _^ the remainder 
when the integral quotient of the last division is divided by 
x- a ; and so on. The calculation of the remainder in any 
particular case is always carried out by means of the synthetic 
process of chap, v., § 13. 

It should be observed that the last coefficient, q n , is the value 
of f(a). 

Example. To diminish the roots of 

by 1. 

AVc simply reproduce the calculation of § 21, Example 1, in a slightly 

modified form ; thus — 



5 -11 


+ 10 


-2 (1 


5 


-6 


4 


-6 


4 


1 2 


5 


-1 




-1 


1 3 




5 






1 4 






Hence the rerpiired equation is 







5f + 4£ 2 +3£ + 2 = 0. 

§ 23.] If one of the roots of the equation (1) of § 21 be small, 

say between and + 1, then an approximate value of that root is 

-Pn/Pn-v For, if x denote the root in question, we have, 

by (i), 



x = - 



Pn 



(Pn-2+Pn- 3 V + - ■ .+iV'"" 2 ) (6). 



Pa- i Pn-i 

Hence, if x be small, we have approximately z = - p n ;Pn-\- 

It is easy to assign an upper limit to the error. We have, 
in fact, 

x = -PnfPn-i -5 

■II _/• 

where mode<^— (1 + x + . . . +o: n ~"), 

l'll-X 

p r being the numerically greatest among the coefficients p . . ., 
Pn-i, Pn-2- Hence, since x^>l, we have 

mode<(?i- l)p,/p n -i- 



XV iiokxek's PROCESS 341 

It would be easy to assign a closer limit for the error ; hut 
in the applications which we shall make of the theorem we have 
indirect means of estimating the sufficiency of the approxima- 
tion ; all that is really wanted for our purpose is a suggestion of 
the approximation. 

Example. The equation 

x" + 8192ar + 16036288a; - 5969856 = 
has a root between and 1, find a first approximation to that root. 
By the above rule, we have for the root in question 
a: = 59S9856/160362S8-e 
= "37227 - e 
where e<2 x 8192/16036288 < -00103. 

Hence o;= *372, with an error of not more than 1 in the last digit. 
In point of fact, since x< '4, we have 

e< |(-4) 3 + 8192 x (-4) 2 }/16036288, 
< 1311/16036288 < 1600/16000000, 
<-0001 ; 
so that the approximation is really correct to the 4th place of decimals. 

§ 24.] Horner s Method. Suppose that we have an equation 
/(.'•) = 0, having a positive root 235 - 67 . . . This root would be 
calculated, according to Horner's method, as follows : — First we 
determine, by examining the sign of f(x) } that /(.<•) = has one 
root, and only one,* lying between 200 and 300 : the first digit 
is therefore 2. Then we diminish the roots of f(z) = by 200, 
and thus obtain the first subsidiary equation, say /,(.>•) = 0. Then 
fi(x)-0 has a root lying between and 100. Also since the 
absolute term of f t (x) is /(200), and no root of /(.») = lies 
between and 200, the absolute term of f(x) (that is, /(0)) and 
the absolute term oif^x) must have the same sign. By examin- 
ing the sign of /,(«) for x = 0, 10, 20, . . ., 90, we determine 
that this root lies between 30 and 40 : the next digit of the 
root of the original equation is therefore 3. The labour of this 
last process is, in practice, shortened by using the rule of § 23. 
Let us suppose that 30 is thus suggested ; to test whether this 

* For a discussion of the precautions necessary when an equation has two 
roots which commence with one or more like digits, see Burnside and Pcmton's 
Theory of Equations, § 104. 



3-42 HORNER'S PROCESS chap. 

is correct we proceed to diminish the roots of /,(•>') = by 30, — 
to deduce, in fact, the second subsidiary equation f s (x) = 0. Since 
the roots of /(.?:) have now been diminished by 230, the absolute 
term off a (x) is/(230). Hence the absolute term of f,(.r) must 
have the same sign as the absolute term of /,(#), unless the digit 
3 is too large. In other words, if the digit 3 is too large, we 
shall be made aware of the fact by a change of sign in the 
absolute term. In practice, it does not usually occur (at all 
events in the later stages of the calculation) that the digit 
suggested by the rule of § 23 is too small ; but, if that were so, 
we should become aware of the error on proceeding to calculate 
the next digit which would exceed 9. 

The second subsidiary equation is now used as before to 
find the third digit 5. 

The third subsidiary equation would give "6. To avoid the 
trouble and possible confusion arising from decimal points, we 
multiply the roots of the third (and of every following) subsidiary 
equation by 10; or, what is equivalent, we multiply the second 
coefficient of the equation in question by 10, the third coefficient 
by 100, and so on; and then proceed as before, observing, how- 
ever, that, if the trial division for the next digit be made after 
this modification of the subsidiary equation, that digit will 
appear as 6, and not as "6, because the last coefficient has been 
multiplied by 10 n and the second last by 10 B-1 . 

The fundamental idea of Horner's method is therefore simply 
to deduce a series of subsidiary equations, each of which is used 
to determine one digit of the root. The calculation of the 
coefficients of each of these subsidiary equations is accomplished 
by the method of § 22. After a certain number of the digits of 
the root have been found, a number more may be obtained by a 
contraction of the process above described, the nature of which 
will be easily understood from the following particular case. 

Example. Find an approximation to the least positive root of 

/(.-•) = .," + 2.v 2 - 5z-7 = (1). 

Since /(0)=-7, /(l)=-9, /(2)=-l, /(3)=+23, 

the root in cjuestion lies between 2 and 3. The first digit is therefore 2. 



XV EXAMPLE 34.°, 

We now diminish the roots of (1) by 2. The calculation of the coefficients 
runs thus : — 

1 + 2 -5 -7 (2 

_2 _8 _6 

4 3 "\ - 1 

2 12 

~6 Tu 
2 



where the prefix -| is used to mark coefficients of the first subsidiary equation. 
The first subsidiary equation is therefore 

a? + 8x* + l5x-l = 0. 

Since the next digit follows the decimal point, we multiply the roots of 
this equation by 10. The resuming equation is then 

a? + 8(k- 2 + 1500.*! -1000 = (2). 

Since 1000/1500 <1, it is suggested that the next digit is 0. "We there- 
fore multiply the roots of (2) by 10, and deduce 

ar ? + 800x- 2 + 150000.>;- 1000000 = (3). 

Since 1000000/150000 = 6- . . ., the next digit suggested is C. We now 
diminish the roots of (3) by 6. 



+ 800 


+ 150000 


-1000000 (06 


6 


4836 


929016 


806 


154836 


4 I -70984000 


6 


4872 




812 


4 i 15970800 




6 






8180 







The resulting subsidiary equation, after the multiplication of its roots 
by 10, is 

a? + 8180a 3 + 15970800a - 70984000 (4). 

Since 70984000/15970800 = 4- . . ., the next digit suggested is 4. The 
reader should notice that, owing to the continual multiplication of the roots 
by 10, the coefficients towards the right increase in magnitude much more 
rapidly than those towards the left : it is for this reason that the rule of § 23 
becomes more and more accurate as the operation goes on. Thus, even at the 
present stage, the quotient 70984000/15970800 would give correctly more than 
one of the following digits, as may be readily verified. 

We now diminish the roots of (4) by 4 ; and add the zeros to the coefficients 
as before. 



344 EXAMPLE chap 





EXAMPLE 




+ 8180 
4 


+ 15970800 
32736 


-70984000 (4 
64014144 


8184 
4 

8188 
4 

81920 


16003536 
32752 
s l 1603628800 


b |- 6969856000 







Then we have the subsidiary equation 

a; 3 + 81920a- 2 + 1603628800a - 6969856000 (5). 

It will be observed that throughout the operation, so far as it has gone, 
the two essential conditions for its accuracy have been fulfilled, namely, that 
the last coefficient shall retain the same sign, and that each digit shall come 
out not greater than 9. It will also be observed that the number of the 
figures in the working columns increases much more rapidly than their utility 
in determining the digits of the root. All that is actually necessary for the 
suggestion of the next digit at any step, and to make sure of the accuracy of 
the suggestion, is to know the first two or three figures of the last two 
coefficients. 

Unless, therefore, a very large number of additional digits of the root is 
required, we may shorten the operation by neglecting some of the figures in 
(5). If, for example, we divide all the coefficients of (5) by 1000, we get the 
equivalent equation * 

•001a 3 + 81 -92a 2 + 1603628 "8a - 6969S56 = (5'). 

Hence, retaining only the integral parts of the coefficients, we have 

0a- 3 + 81a 2 + 1603628a - 6969856 = (5"). 

It will be noticed that the result is the same as if, instead of adding zeros, 
as heretofore, we had cut off one figure from the second last coefficient, two 
from the third last, and so on.t 

Since 6969856/1603628 = 4- . . ., we have for the next digit 4. We then 
diminish the roots of (5") by 4. In the necessary calculation the first working 
column now disappears owing to the disappearance of the coefficient of a 3 ; we 
have in its place simply 81 standing unaltered. It is advisable, however, in 
multiplying the contracted coefficients by 4 to carry the nearest number of 
tens from the last figure cut off (just as in ordinary contracted multiplication 
and division and for the same reason). 

* If the reader find any difficulty in following the above explanation of 
the contracted process, he can satisfy himself of its validity by working out 
the above calculation to the end in full and then running his pen through the 
unnecessary figures. 

t In many cases it may not be advisable to carry the contraction so far at 
each step as is here done. We might, for instance, divide the coefficients of 
5 by 100 only. The resulting subsidiary equation would then be 

Oa 3 + 81 9a 2 + 1 6036288a - 09698560, 
with which we should proceed as before. 



XV 



EXAMPLE 



345 



The next step, therefore, runs thus : 

81 +1603628 

328 

1603957 

328 



6969856 

64 15828 

- 554028 



(4 



(6); 
(6'). 



"I 1604285 
The corresponding subsidiary equation is 

81a 2 + 16042850a- - 55402800 = 
or, contracted, Ox- + 160428.1- - 554028 = 

The next digit is 3 ; and, as the coefficient of x 2 , namely - 81, still has a 
slight effect on the second working column, the calculation runs thus: — 
+160428 -554028 (3 

2 481293 

71- 



160431 
2 



-72735 



'I 160433 
The resulting subsidiary equation after contraction is 

16043a; -72735 = (7). 

The rest of the operation now coincides with the ordinary process of 
contracted division ; it represents, in fact, the solution of the linear equation 
(7), that is (see chap, xvi., § 1), the division of 72735 by 16043. 

The whole calculation may be arranged in practice as below. But the 
prefixes 2 |, 3 |, &c, which indicate the coefficients of the various equations, 
may be omitted. Also the record may be still farther shortened by performing 
the multiplications and additions or subtractions mentally, and only recording 
the figures immediately below the horizontal lines in the following scheme. 
The advisability of this last contraction depends of course on the arithmetical 
power of the calculator. 



+ 2 


-5 


-7 (2-064434533 


2 


8 


6 


4 


3 


8 | - 1000000 


2 


12 


929016 


6 


" 3 | 150000 


4 | -70984000 


2 


4836 


64014144 


3 | 800 


154836 


1 - 6969856 


6 


4872 


6415828 


806 


1 15970800 


"1 - 554028 


6 


32736 


481293 


812 


16003536 


'1-72735 


6 


32752 


64173 


4 | 8180 


b l 1603628^ 


"1 - 8562 


4 


328 


8022 


8184 


1603957 


9 |-5l0 


4 


328 


481 


8188 


"1 160428$ 


10 l-59 


4 


2 


48 


'I^N 


160431 
2 


-11 



16fc(^Sf 



346 EXERCISES XXII 



i'H U\ 



The number of additional digits obtained by the contracted process is less 
by two than the number of digits in the second last coefficient at the beginning 
of the contraction. Owing to the uncertainty of the carriages the last digit 
is uncertain, but the next last will in such a case as the present be abso- 
lutely correct. In fact, by substituting in the original equation, it is easily 
verified that the root lies between 2-064434534 and 2-064434535 ; so that the 
last digit given above errs in defect by 1 only. The number of accurate 
figures obtained by the contracted process will occasionally be considerably 
less than in this example ; and the calculator must be on his guard against 
error in this respect (see Horner's Memoir, cited below). 

§ 25.] Since the extraction of the square, cube, fourth, . . . 
roots of any number, say 7, is equivalent to finding the positive 
real root of the equations, x 2 + Ox - 7 = 0, x a + Ox 2 + Ox - 7 = 0, 
x* + Ox 3 + (bf + Oo: - 7 = 0, . . . respectively, it is obvious that 
by Horner's method we can find to any desired degree of 
approximation the root of any order of any given number 
"whatsoever. In fact, the process, given in chap, xi., § 13, for 
extracting the square root, and the process, very commonly 
given in arithmetical text-books, for extracting the cube root will 
be found to be contained in the scheme of calculation described 
in § 24.* 

* Horner's method was first published in the Transactions of the Philoso- 
phical Society of London for 1819. Considering the remarkable elegance, 
generality, and simplicity of the method, it is not a little surprising that it 
has not taken a more prominent place in current mathematical text-books. 
Although it has been well expounded by several English writers (for example, 
De Morgan, Todhunter, Burnside and Panton), it has scarcely as yet found a 
recognised place in English curricula. Out of five standard Continental text- 
books where one would have expected to find it we found it mentioned in only 
one, and there it was expounded in a way that showed little insight into its 
true character. This probably arises from the mistaken notion that there is 
in the method some algebraic profundity. As a matter of fact, its spirit is 
purely arithmetical ; and its beauty, which can only be appreciated after one 
has used it in particular cases, is of that indescribably simple kind which 
distinguishes the use of position in the decimal notation and the arrangement 
of the simple rules of arithmetic. It is, in short, one of those things whose 
invention was the creation of a commonplace. For interesting historical 
details on this subject, see De Morgan — Companion to British Almanack, for 
1839; Article "Involution and Evolution," Penny Cyclopsedia ; and Budget 
of Paradoxes, pp. 292, 374. 



XV 



EXERCISES XXII 34" 



Exercises XXII. 



[The student should trace some at least of the curves required in the 
following graphic exercises by laying them down correctly to some convenient 
scale. He will find this process much facilitated by using paper ruled into 
small squares, which is sold under the name of Plotting Paper.] 

Discuss graphically the following functions : — 

(40 y=rz 1 Tv i - (5.) y=^i- (6.) y=- x 



"(sb-1) 3 ' v ' J se-2 v "' * J' 2 -9 

(7.) Construct to scale the graph of y = - x- + 8x - 9 ; and obtain graphic- 
ally the roots of the equation a- 2 - 8a: + 9 = to at least three places of 
decimals. 

(8.) Solve graphically the equation 

a? - 16a; 2 + 7Lc- 129 = 0. 
(9.) Discuss graphically the following question. Given that y is a con- 
tinuous function of x, does it follow that a; is a continuous function of y ? 
(10.) Show that when h, the increment of x, is very small, the increment of 

p„x" +p n -ix"- 1 + . . . +pxx +p 
is (npnX"- 1 + (n - 1 )p n -ix"- 2 + . . . + 1 .pi)h. 

(11. ) If h be very small, and x=l, find the increment of 2a-* 3 - 9ar + 12a 1 + 5. 

(12.) If an equation of even degree have its last term negative, it has at 
least two real roots which are of opposite signs. 

(13.) Indicate roughly the values of the real roots of 

10a.- 3 -17ar + a; + 3 = 0. 

(14.) What can you infer regarding the roots of 

a,- 3 -5a; + 8 = 0? 

(15.) Show by considerations of continuity alone that a;" -1=0 cannot 
have more than one real root, if n be odd. 

(16.) If f{x) be an integral function of a;, and if/(a)= -p,f{b)= +q, where 
p and q are both small, show that x — (qa+2^b)/{p + q) is an approximation to 
a root of the equation /(a:) = 0. 

Draw a series of contour lines for the following functions, including in 
each case the zero contour line : — 

(17.) z = xy. (18.) z = ~. (19.) z = x--f. (20.) z = ^±£ '. 

y x 

Is the proposition of § 16 true for the last of these ? 

Draw the Argand diagram of the dependent variable in the following 
cases, the path of the independent variable being in each case a circle of radius 
unity whose centre is fi : — 

<21.) y=|. (22.)y= + sjx. (23.) y= ^x. (24.) y = \ -a- 2 . 



348 



EXERCISES XXII 



CHAP. XV 



Find by Horner's method the positive real roots of the following equations 
in each case to at least seven places of decimals : — * 



(25.) a? -2 = 0. 

(27.) a 3 + x -1000 = 0. 

(29.) a 4 + x 3 + x" + x- 127694 = 0. 

(30. ) a 4 - 80a 3 + 24a: 2 - 6a: - 80379639 = 0. 



(26.) x 3 - 2a- 5 = 0. 

(28. ) x 3 - 46a; 2 - 36a + 18 = 0. 

(31.) x 5 - 4a- 4 + 7a 3 -863 = 0. 
(33.) x 5 -4a -2000 = 0. 



(32.) a 5 -7 = 0. 

(34. ) 4a 6 + 7a 5 + 9a: 4 + 6a 3 + 5a 2 + 3a - 792 = 0. 

(35.) Find to twenty decimal places the negative root of 2a 4 + 3a 3 - 6x - 8 
= 0. 

(36.) Continue the calculation on p. 344 two stages farther in its uncon- 
tracted form ; and then estimate how many more digits of the root could 
be obtained by means of the trial division alone. 

* Most of these exercises are taken from a large selection given in De 
Morgan's Elements of Arithmetic (1854). 



CHAPTEE XVI. 
Equations and Functions of the First Degree. 

EQUATIONS WITH ONE VARIABLE. 

§ 1.] It follows by the principles of chap. xiv. that every 
integral equation of the 1st degree can be reduced to an equiva- 
lent equation of the form 

ax + b = (I); 

this may therefore be regarded as a general form, including all 
such equations. As a particular case b may be zero ; but we 
suppose, for the present at least, that a is neither infinitely great 
nor infinitely small. 

Since a 4= 0, we may write (1) in the form 

"{— ("«)}"° (2>: 

whence we see that one solution is x = - bja. We know already, 
by the principles of chap, xiv., § 6, that an integral equation of 
the 1st degree in one variable has one and only one solution. Hence 
we have completely solved the given equation (1). 

It may be well to add another proof that the solution is unique. 
Let us suppose that there are two distinct solutions, x — a and x = §, of (1). 
Then we must have 

aa + b = 0, 

ap + b=0. 

From these, by subtraction, we derive 

a(a - jS) = 0. 

Now, by hypothesis, «=t=0, therefore we must have a-/S = 0, that is, a = /3; 
in other words, the two solutions are not distinct. 



350 TWO LINEAR EQUATIONS IN ONE VARIABLE chap. 

§ 2.] Two equations of the 1st degree in one variable will in 
general be inconsistent. 

If the equations be ax + b = (1), 

a'x + b' = (2), 
the necessary and sufficient condition for consistency is 

aV - a'b = (3). 

The solution of (1) is x = -b/a, and the solution of (2) is 
x = - b'/a'. These will not in general be the same ; hence the 
equations (1) and (2) will in general be inconsistent. 

The necessary and sufficient condition that (1) and (2) be 
consistent is 

b V 

Since a 4= 0, a 4= 0, (4) is equivalent to 

a'b - ab', 
or ab' - a'b = 0. 

Obs. 1. If b = and b' = 0, then the condition of consistency 
is satisfied. In this case the equations become ax - 0, a'x = ; 
and these have in fact the common solution x = 0. 

Obs. 2. When two equations of the 1st degree in one vari- 
able are consistent, the one is derivable by multiplying the other 
by a constant. In fact, since a 4= 0, if we also suppose b 4= 0, we 
derive from (3), by dividing by ab and then transposing, 

— = -r, each = I; say ; 

a b ' J ' 

hence a' = ka, b' = kb, 

so that a'x + b' = hax + kb, 

= k(ax + b). 

If, then, (3) be satisfied, (2) is nothing more or less than 

k(ax + b) = 
where k is a constant. 

This might have been expected, for, transpositions apart, the 
only way of deriving from a single equation another perfectly 
equivalent is to multiply the given equation by a constant. 



XVI 



EXERCISES XXIII 35 i 



Exercises XXIII. 



Solve the following equations : — 

l + (l-g)/2 
(2.) 3 =1 



(3.) 



51 62 



5 + 



3a; -1 12a; + 5 
2 29 



(4.) 3- 4 -i-~ 5 " 

i-X 

(5.) -68(-32a;--5) + p—=3-694.r, 

find x to three places of decimals. 
(6.) a/(l-bx)=b/(l-ax). 

(7. ) (" + x) (b + x) - a{b +c) = (ca? + bx 2 )jb. 

x-a x-c _ 

(8-) i, +i = 2 - 

K ' b- a b-c 

x 2 -a" x--b 2 %*-<? 

(9.) - + r- + = a + b + c-3x. 

N ' x-a x - o x-c 

(10.) (a 3 + b 3 )x + a 3 - b x = ft 4 -b* + a 6(« 3 + b"). 

O 9 O 9 

a;^ - or ar - c- 

(11.) -i i =c-a. 

^ > b+a b+c 

(12.) (a?-l)(aJ+2)(2jj-2)=(2a;-l)(2a5+l)(a!/2+I). 



(13.) 

(14.) 
(15.) 
(16.) 
(17.) 
(18.) 



12 3 4 



as+1 


X + 2 


x + 3 


a;+4* 


x- 


1 


x-B 


x-2 


a;- 4 


x- 


2 


x-4 


x- 3 


a;-5" 


11 

12B + 11 


5 
6a; +~5~ 


7 
'4a; + 7 




3- 
1- 


x 

X 


5-a: 
'T^x~ 


ar>-2 

1-Zx + x 2 


2 


; + 


14 


10 

r + 


6 



aH-2 a-+10 a; + 6 a-+14' 

a; 2 - 4a; 4- 5 
a;- + 6a; +10 



\x+Zj 



352 ax + hj + c = HAS do 1 solutions chap. 

rim J- J- 3(^+5) 6.r+i7 

1 ; sc + l .r + 2 (s+l)(as+2) tf + 2 ' 

a; + 2a a:-2a_ 4a& 
(20-) 2b~^x + 2b + x~ ib 2 -x 2 ' 

(a + b)x + c ( a-b)x + e _ iab 
( ' (a-b)x + d {a + b)x+f~(a + b){a-b) 

a+b a-b a b 

(22.) + ,= ;. 

v ' x-a x-b x-a x-b 

(23.) -5-.+ * " + * 



x-a x-b x(x-a-b) + ab' 



(24-) --r^+ ~rr- = 2 - 

' x+a-bx+b-c 
(85 ) X _- ? 1 



(a; - a) (x - b) (x -a){x- c) (x -b)(x- c) 

1 2 



+ ; 



{x + a)(x + b) (x + a)(x + c) (x + b)(x + c) 



EQUATIONS WITH TWO VARIABLES. 

§ 3.] A single equation of the 1st degree in two variables has a 
one-fold infinity of solutions. 
Consider the equation 

ax + by + c = (1). 

Assign to y any constant value we please, say (3, then (1) 
becomes 

ax + b/3 + c = (2). 

We have now an equation of the 1st degree in one variable, 
which, as we have seen, has one and only one solution, namely, 
x = - (bp + c)ja. 

We have thus obtained for (1) the solution x= - (b(3 + c)ja, 
y = f3, where ft may have any value we please. In other words, 
we have found an infinite number of solutions of (1). 

Since the solution involves the one arbitrary constant /?, 
we say that the equation (1) has a one-fold infinity (sometimes 
symbolised by oo x ) of solutions. 
Example. 

the solutions are given by 

2/3-1 . 



XVI 



TWO LINEAR EQUATIONS IN TWO VARIABLES 



353 



we have, for example, for /3= -2, 0= - 1, j3 = 0, /3= +-, /3= + 1, /3= + 2, the 
following solutions : — 



/3 


_2 


-1 





1 
+ 2 


+ 1 


+ 2 




5 




1 




1 




a; 


~3 


-1 


~3 





+ 3 


+ 1 


y 


-2 


-' 





1 

+ 2 


+ 1 


+ 2 



And so on. 

§ 4.] We should expect, in accordance with the principles of 
chap, xiv., § 5, that a system of two equations each of the 1st 
degree in two variables admits of definite solution. 

The process of solution consists in deducing from the given 
system an equivalent system of two equations in which the 
variables are separated ; that is to say, a system such that x 
alone appears in one of the equations and y alone in the 
other. 

We may arrive at this result by any method logically con- 
sistent with the general principles we have laid down in chap 
xiv., for the derivation of equations. The following proposition 
affords one such method : — 

If I, V, m, m' be constants, any one of ichich may be zero, but 
which are such that lm! - I'm 4= 0, then the two systems 

ax + by + c = (1), 

a'x + b'y + c' = Q (2), 

and 

J(ax + by + c)+ l'(a'x + b'y + c') = (3), 

m(ax + by + c) + m'ia'x + b'y + c) = (4), 

are equivalent. 

It is obvious that any solution of (1) and (2) will satisfy (3) 
and (4) ; for any such solution reduces both ax + by + c and 
a'x + b'y + c' to zero, and therefore also reduces the left-hand 
sides of both (3) and (4) to zero. 

Again, any solution of (3) and (4) is obviously a solution of 
VOL. I 2 a 



354 SOLUTION BY CEOSS MULTIPLICATION chap. 

m'{ l(ax + by + c) + l'(a'x + b'y + c')} 

- I' {m(ax + by + c) + m'(a'x + b'y + c')} = (5), 

- m { l(ax + by + c) + l'(a'x + b'y + c')} 

+ 1 {m(ax + by + c) + m'(a'x + b'y + c')} = (6). 

Now (5) and (6) reduce to 

(lm' - I'm) (ax + by + c ) = (7), 

{lm! - I'm) {ax + b'y + c') = (8), 

and, provided lm' - l'm^¥ 0, (7) and (8) are equivalent to 

ax +by + c - 0, 
a'x + b'y + c' - 0. 

We have therefore shown that every solution of (1) and (2) is a 
solution of (3) and (4) ; and that every solution of (3) and (4) is 
a solution of (1) and (2). 

All we have now to do is to give such values to I, V, m, m' 
as shall cause y to disappear from (3), and x to disappear from 
(4). This will be accomplished if we make 

I = + V, V = -b, 
m = - a', m' = + a ; 
so that lm' — I'm — ab' - a'b. 

The system (3) and (4) then reduces to 

(ab' - a'h)x + cb' - c'b = (3'), 

(ab' - a'b)y + c'a - ca' = (4') ; 

and this new system (3'), (4') will be equivalent to (1), (2), 
provided 

ah' -a'b^O (9). 

But (3') and (4') are each equations of the 1st degree in one 
variable, and, since ab' - a'b 4= 0, they each have one and only 
one solution, namely — 

cb' - c'b " 



X ab' - a'b 

y 



ac' - a'c 
" ab'- a'b J 



(10). 



XVI 



MEMORIA TECHNICA 355 



It therefore follows that the system 

ax + by + c = (1), 

a'x + b'y + c'=0 (2) 

has one and only one definite solution, namely, (10), provided 

ab' - a'b * (9). 

The method of solution just discussed goes by the name of 
cross multiplication, because it consists in taking the coefficient 
of y from the second equation, multiplying the first equation 
therewith ; then taking the coefficient of y from the first equation, 
multiplying the second therewith ; and finally subtracting the 
two equations, Avith the result that a new equation appears not 
containing y. 

The following memoria tcchnica for the values of x and y will enable the 
student to recollect the values in (10). 

The denominators are the same, namely, ab' - a'b, formed from the co- 
efficients of x and y thus 

a\ yb 




a'y \6' 



the line sloping down from left to right indicating a positive product, that 
from right to left a negative product. 

The numerator of x is formed from its denominator by putting c and c' in 
place of a and a' respectively. 

The numerator of y by putting c and c' in place of b and b'. 

Finally, negative signs must be affixed to the two fractions. 

Another way which the reader may prefer is as follows : — 

Observe that we may write (10) thus, 

_bc'-b'c ca' - c'a 

X ~ab r -a'b' y ~ab'-a'b (11) ' 

where the common denominator and the two numerators are formed according 
to the scheme 




\ 





^a . 
It is very important to remark that (1) and (2) are col 

laterally symmetrical with respect to I a, b , see chap, iv., § 20. 

V, v) 

Hence, if we know the value of x, we can derive the value of y 
by putting everywhere b for a, a for b, b' for a', and a for b'. In 






356 



EXAMPLES 



CHAP. 



fact the value of y thus derived from the value of x in (10) is 
- (ca - c'a)/(ba' - b'a) ; and this is equal to - (ac' - a'c)/(ab' - a'b), 
which is the value of y given in (10). 



Example 1. 



3x+22/-3 = 
9x + iy + 5 = 



(a), 
08). 



Proceeding by direct application of (11), we have 

+ 3\/+2 X /-3\/+3 



10 + 12 11 



9 



y=- 



27-15 2 



12 + 18 15' " 12 + 18 5' 

Or thus : multiply (a) by 2, and we have the equivalent system 

Sx + iy - 6 = 0, 
- 9a; + 4y + 5 = ; 
whence, by subtraction, 

15a;-ll=0, 



which gives 



11 
* = I5- 



Again multiplying (a) by 3, and then adding Q3), we have 

10y-4 = 0, 
which gives 

4 2 

Example 2. 

a/3 *' 

a'V 7 

Multiplying the first of these equations by -, and subtracting the second, we 
obtain 

i i\ _i i 

a 2 /3 * J87 ' 

<x 2 (7-/3) 



that is, 
whence 



# = 



7(«"/3) 



Since the equations are symmetrical in f ' ' ) we get the value of y by 
interchanging a and /3, namely, 



y= 



7 (/3-a) 



xvi SPECIAL CASES 357 

Sometimes, before proceeding to apply the above method, it 
is convenient to replace the given system by another which is 
equivalent to it but simpler. 

Example 3. 

a?x+b-y = 2ab(a + b) (a), 

b(2a + b)x + a(a + 2% = a 3 + a?b + ab 2 + b* (/3). 

By adding, we deduce from (a) and (/3) 

(a + b)°~x + (a + b) 2 y = (a + b) 3 , 

which is equivalent to 

x + y = a + b (y). 

It is obvious that (a) and (7) are equivalent to (a) and (/3). Multiplying 

(7) by b' 2 and subtracting, we have 

(a 2 -b 2 )x = 2a 2 b + ab 2 -b 3 , 

= b(2a-b){a + b). 

Hence #=-— — =— . 

a- b 

Since the original system is symmetrical in { '"' J, we have 

a(2b - a) 

§ 5.] Under the theory of last paragraph a variety of par- 
ticular cases in which one or more of the constants a, b, c, a\ b', c' 
involved in the two equations 

ax + by + c = 0, 
a'x + b'y + c' = 

become zero are admissible ; all cases, in short, which do not 
violate the condition ab' - a'b =t= 0. 

Thus we have the following admissible cases : — 



'.-> 



a = (1), b' = (4), 

b = (2), a = and b' = (5), 

a' - (3), a' = and 6 = (G). 

The following are exceptional cases, because they involve ab' - a'b 

= 0:— 

a = and a' = (I.), a, b, a', V all different 

a = and b = (H.), from 0, but such that 

V = and a' = (III.), aJ' - a'J = (V.) 

b' = and 6 = (IV.), 



358 HOMOGENEOUS SYSTEM chap. 

We shall return again to the consideration of the exceptional 
cases. In the meantime the reader should verify that the 
formulae (10) do really give the correct solution in cases (1) to 
(6), as by theory they ought to do. 

Take case (1), for example. The equations in this case reduce to 
bi/ + e-0, a'x + b'y + c' = 0. 
The first gives y — - c/b, and this value of y reduces the second to 

a x - -r + c = 0, 

. - , • b'c-bc' 

which gives x= n — ■• 

a b 

It will be found that (10) gives the same result, if we put a = 0. 

There is one special case that deserves particular notice, that, 

namely, where c = and c' = ; so that the two equations are 

homogeneous, namely, 

ax + by = (a), 

a'x + b'y = (ft). 

If ab' - a'b 4= 0, these formula? (10) give x = 0, y = as the only 
possible solution. If ab' - a'b = 0, these formulae are no longer 
applicable ; what then happens will be understood if we reflect 
that, provided y 4= 0, (a) and (/S) may be written 

az + b = (a'), 

a'z + b' = ((3'), 

where z - x/y. 

We now have two equations of the 1st degree in z, which 
are consistent (see § 2), since ab' — a'b = 0. Each of them gives 
the same value of z, namely, z = - b/a, or z = - b'/a' (these two 
being equal by the condition ab' - a'b = 0). 

If then ab' — a'b 4= 0, the only solution of (a) and ((3) is x = 0, 
y = ; if ab' - a'b = 0, x and y may have any values such that the 
ratio x/y = -b/a= - b'/a'. 

§ 6.] There is another way of arranging the process of solu- 
tion, commonly called Bezoitt's method* which is in reality merely 
a variety of the method of § 4. 

* For an account of Bezout's methods, properly so called, see Mini's 
papers on the " History of Determinants ;" Proc. R.S.E., 1886. 



&vi USE OF UNDETERMINED MULTIPLIER 359 

If X be any finite constant quantity whatever* then any solution of the 
system 

ax + by + c=0, a'x + b'y + c' = (1) 

is a solution of the equation 

[ax + by + c) + \(a'x + b'y + c') = (2), 

that is to say, of (a + \a')x + (b + \b')y + (e + \c') = (3). 
Now, since X is at our disposal, we may so choose it that y shall disappear 
from (3) ; then must 

\b' + b = Q (4), 

and (3) will reduce to (a + \a')x + (c + \c') = 0' (5). 

From (4) we have \= - bjb', and, using this value of X, we deduce from (5) 

c_+\c'_ b'c - be' 
a + \a' ab'-a'b' 
which agrees with (10). 

The value of y may next he obtained by so determining X that x shall 
disappear from (3). We thus get 

\a' + a = (6), 

(b + \b')y + (c + \c') = (7), 

and so on. 

To make this method independent and complete, theoretically, it would 

of course be necessary to add a proof that the values of x and y obtained do 

in general actually satisfy (1) and (2); and to point out the exceptional case. 

§ 7.] There is another way of proceeding, which is inter- 
esting and sometimes practically useful. 
The systems 

ax + by + c = ] . > 

a'x + b'y +. d = J { ' 

-i ax + c x 

aml »-— J" } (2) 

a'x + b'y + c' = J 

are equivalent, provided b 4= 0, for the first equation of (2) is 
derived from the first of (1) by the reversible processes of trans- 
position and multiplication by a constant factor. 

Also, since any solution of (2) makes y identically equal 
to - (ax + c)/b, we may replace y by this value in the second 
equation of (2). We thus deduce the equivalent system, 

* So far as logic is concerned- X might be a function of the variables, but 
for present purposes it is taken to be constant. A letter introduced in this 
way is usually called an "indeterminate multiplier " ; more properly it should 
be called an "undetermined multiplier." 



360 SOLUTION BY SUBSTITUTION CHAP. 

ax + c ' 

b'iax + c) , „ 
a'a; - -^—7 — '- + c' = 


Now, since b 4= 0, the second of the equations (3) gives 

(a'b - aJ> + (6c' - b'c) = (4). 

If a'b - aft' 4= 0, (4) has one and only one solution, namely, 

be' -b'c . . 

x = aY^7b (5) ' 

this value of x reduces the first of the equations (3) to 

1 f a(bc' - b'c) } 

qj= ~n-w^b- +c \' 

abc' - a'bc 



b(ab' - a'b)' 

that is, to y = ca ~ C f (6). 

* ab - ao 

The equations (5) and (6) are equivalent to the system (3), 
and therefore to the original system (1). Hence we have proved 
that, if ab' - a'b =1= and b 4= 0, the system (1) has one and only 
one solution. 

We can remove the restriction b 4= ; for if b = the first of 

the equations (1) reduces to ax + c = 0. Hence (if a 4= 0, which 

must be, since, if both a = and b = 0, then ab' - a'b = 0) we have 

x = - c/a, and this value of x reduces the second of equations 

(l)to 

a 'c ,, , „ 
+ b'y + c = 0, 

which gives (since V cannot in the present case be without 
making ab' - a'b = 0) y = (ca' - c'a)/ab'. Now these values of x 
and y are precisely those given by (5) and (6) when b = 0. 

The excepted case b = is therefore included ; and the only 
exceptional cases excluded are those that come under the condi- 
tion ab' - a'b = 0. 



xvi SYSTEM OF THREE EQUATIONS IN X AND y 3G1 

The method of this paragraph may be called solution -by 
substitution. The above discussion forms a complete and 
independent logical treatment of the problem in hand. The 
student may, on account of its apparent straightforwardness 
and theoretical simplicity, prefer it to the method of § 4. 
The defect of the method lies in its want of symmetry ; the 
practical result of which is that it often introduces needless 
detail into the calculations. 

Example. 

3«+2y- 3 = (a), 

-9ai + 4i/ + 5 = (/3). 

From (a) we have y= — ^ (y)- 

a 

Using (y), we reduce (/3) to 

-9a!+2(-3a;+3) + 5=0, 
that is, -15b+11=0; 

whence 

This value of x reduces (7) to 



15 



-3x^ + 3 
15 

y= — 2 — 

2 

= 5' 
The solution of the system (a) and (J3) is therefore 

11 2 

X= 15' y = 5 

§ 8.] Three equations of the 1st degree in two variables, say 

ax + by + c = 0, a'x + b'y + c' - 0, a"x + b"y + c" = (1 ), 

will not be consistent unless 

a" {be' - b'c) + b"(ca' - c'a) + c"(ab' - a'b) = (2) ; 

and they will in general be consistent if this condition be satisfied. 

We suppose, for the present, that none of the three functions 
ab' — a'b, a'b-ab", a'b" - a'b' vanishes.* This is equivalent to 
supposing that every pair of the three equations has a deter- 
minate finite solution. 

If we take the first two equations as a system, they have 
the definite solution 

* See below, § 25. 



362 CONDITION OF CONSISTENCY chap. 

be' - b'c ca' - c'a 

X = ^b'~^ r b' V = ab' - a'b' 

The necessary and sufficient condition for the consistency of the 
three equations is that this solution should satisfy the third 
equation ; in other words, that 

be' -b' c erf -ca 

a -n - + b -j-, + e = 0. 

ab - ab ab - ab 

Since ab' - a'b 4= 0, this is equivalent to 

a" {be' - b'c) + b"(ca' - c'a) + c"(ab' - a'b) = (3). 

The reader should notice that this condition may be written in 
any one of the following forms by merely rearranging the 
terms : — 

a(b'c" - b"c') + b(c'a" - c"a') + c(a'b" - a'b') = (4), 

a'(bc" - b"c) + b'(ca" - c"a) + c'(ab" - a"b) = (5), 

a(b'c" - b"c') + a'(b"c - be") + a"(bc' - b'c) = (6), 

b(c'a" - c'a) + b'(c"a - ca") + b"(ca' - c'a) = (7), 

c(a'b" - a"b') + c'(a"b - ab") + c"(ab' - a'b) = (8), 

ab'c"-ab"c' + bc'a" -bc"a' + ca'b" -ca"b' = (9). 

The forms (4) and (5) could have been obtained directly by 
taking the solution of the two last equations and substituting in 
the first, and by solving the first and last and substituting in the 
second, respectively. Each of these processes is obviously logic- 
ally equivalent to the one actually adopted above. 

The forms (6), (7), (8) would result as the condition of the 
consistency of the three equations 

ax + a'y + a" = 0, bx + b'y+b" = Q, cx + c'y + c" = (10). 

We have therefore the following interesting side result : — 

Cor. If the three equations (I) be consistent, then the three equa- 
tions (10) are consistent. 

If the reader will now compare the present paragraph with § 2, he will 
see that the function 

ab' - a'b 

plays the same part for the system 

ax + b = 0, a'.r + b' = 



XVI 



DETERMINANT OF THE SYSTEM 



363 



as does the function 

a(b'c" - b"c') + b{c'a" - c"a') + c(a'b" - a"b') 
for the system 

ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = 0. 
These functions are called the determinants of the respective systems of equa- 
tions. The}' are often denoted by the notations 

a b 
a! V 



for ab' - a'b 



\a b c 
\a' V c' 



for ab'c" - ab"c' + bc'a" - bc"a' + ca'b" - ca"b' 



(ii); 

(12). 



The reader should notice— 

1st. That the determinant is of the 1st degree in the constituents of any 
one row or of any one column of the square symbol above introduced. 

2nd. That, if all the constituents be considered, its degree is equal to the 
number of equations in the system. 

A special branch of algebra is nowadays devoted to the theory of deter- 
minants, so that it is unnecessary to pursue the matter in this treatise. For 
the sake of more advanced students we have here and there introduced results 
of this theory, but always in such a way as not to interfere with the progress 
of such as may be unacquainted with them. 

The reader may find the following memoriae technicse, useful in enabling 

him to remember the determinant of a system of three equations : — 

For the form (4), 

a b c 



b' 



b" 






V 



b\ 



to be interpreted like the similar scheme in § 4. 
For the form (9), 

b 




or &"/ no- 

where the letters in the diagonal lines are to form products with the signs -f 
or - , according as the diagonals slope downwards from left to right or from 
right to left. 

Example. 

To show that the equations 

3.r + 5t/-2 = 0, 4ic + 6?/-l=0, 2.r + 4?/ - 3 = 

are consistent. 

Solving the first two equations, we have x= -7/2, j/ = 5/2. These values 



364 



EXERCISES XXIV 



CHAP. 



reduce 2x + iy-B to -7 + 10-3, which is zero. Hence the solution of the 
first two equations satisfies the third ; that is, the three are consistent. 

We might also use the general results of the above paragraph. 

Since 3x6-5x4= -2, 5x2-3x4=-2, 4 x 4- 2 x 6= +4, each pair 
of equations has by itself a definite solution. Again, calculating the deter- 
minant of the system by the rule given above, we have, for the value of the 
determinant, - 54 - 10 - 32 + 24 + 12 + 60 = 0. Hence the system is consistent. 
+ 3+5 2+3+5 




+ 2 + 



Exercises XXIV. 
Solve the following : — 

(1.) £*+&=«, |aj+Jy=6J. 

(2.) &B+3y=18, 3x-2y = 9. 

(3.) -123.?+ -685?/ = 3 -34, -893a!- '59% =8 '71, 

find x and y to five places of decimals. 



(4.) 

(5.) 
(6.) 
(7.) 

(8.) 

(9.) 
(10.) 

(11.) 

(12.) 
(13.) 



x + y:x-y = 5 : 3, x + 5y=Z6. 
Sx + l = 2y + l = Zy + 2x. 
{x+S)(y+S) = (x-l)(y+2), 8x + 5 = 9y + 2. 

x + y = a + b, (x + a)l(y + b) = b/a. 



x y 
la mb 



2via~ r Zlb 



ax + by = 0, (a-b)x+(a + b)y = 2c 
(a + b)x- (a-b)y~c, (a-b)x + (a + h)y=e. 

(a + b)x + (a -b)y = a- + 2ab - b 2 , (a - by + [a - b)y = a 2 + b 2 . 



y 



-ab, 



V 



a- -b 2 a 2 + ab + b 2 ~""'' a 2 +b 2 ' a 2 -ab + b 2 
(ap m + bq'")x + {ii]i mJ r l + bq m + x )y = «/>"'+- + bq m+2 , 
[np n + bq")x + {«2>"+ l + bq"+^ )y = «^"+- + bq"+ 2 . 



a{2a+b). 



(14.) Find \ and p. so that x 3 + Xx 2 + /xx + abc may be exactly divisible by 
x - b and by x - c. 

(15.) If X 4=0, and if x-y = a-b, -^— + -^—=1, - X —+-!L=l, be con- 

J a + X b + X ' a - X b-\ 

sistent, show that X= ±\/aL 

(16.) If the system (b + c)x+{c + a)y + (a + b) = 0, (c + a)x+(a + b)y+ (b + c) 
— 0, (a + b)x+{b + c)y + (c + a)~0, be consistent, then os 8 + &*+<?- 3a&c=0. 

(17.) Find the condition that ux-\-by = c, a 2 x+b-i/ = c'-, a*x + b 3 y = c 3 be 
consistent. 



xvi SYSTEMS OF ONE AND OF TWO EQUATIONS IN X, 1J, Z 365 

(18.) Find an integral function of a; of the 1st degree whose values shall 
be +9 and +10 when x has the values -3 and + 2 respectively. 

(19.) Find an integral function of x of the 2nd degree, such that the 
coefficient of its highest term is 1, and that it vanishes when x=2 and when 
x= -3. 

(20.) Find an integral function of x of the 2nd degree which vanishes 
when .r=0, and has the values -1 and + 2 when x— - 1 and x =+3 
respectively. 



EQUATIONS WITH THREE OR MORE VARIABLES. 

§ 9.] A single equation of the 1st degree in three variables admits 
of a two-fold infinity of solutions. 
For in any such equation, say 

ax + by + cz + d = 0, 

we may assign to two of the variables any constant values we 
please, say y = ft, z — y, then the equation becomes an equation 
of the 1st degree in one variable, which has one and only one 
solution, namely, 

b(3 + cy + d 

x = — • 

a 

We thus have the solution 

bB + cy + d 

x = f — ■. y = P, * = y- 

Since there are here two arbitrary constants, to each of 
which an infinity of values may be given, we say that there is 
a two-fold infinity (oo 2 ) of solutions. A symmetric form is given 
for this doubly indeterminate solution in Exercises xxv., 27. 

§ 10.] A system of two equations of the 1st degree in three vari- 
ables admits in general of a one-fold infinity of solutions. 

Consider the equations 

ax + by + cz + d = 0, a'x + b'y + c'z + d' = (1). 

We suppose that the functions be' - b'c, ca - c'a, ab' - a'b do not 
all vanish, say ab' — a'b 4= 0. 

If we give to z any arbitrary constant value whatever, say 
z = y, then the two given equations give definite values for x and 
y. We thus obtain the solution 



366 HOMOGENEOUS SYSTEM OF TWO EQUATIONS chap. 

(bc'-b'c)y + (bd' -b'd) (ca' - c'a)y + (da' - d'a) _ 

X = ~~; ab r ^7b ,V ~ ab'-ab »* 7 W- 

Since we have here one arbitrary constant, there is a one-fold 
infinity of solutions. 

Cor. There is an important particular case of the above that 
often occurs in practice, that, namely, where d = and d' = 0. 
We then have, from (2), 

be' - b'c ca' - c'a 

ab - ab ' * ah - ab' ' 

This result can be written as follows : — 

x y 



be' - b'c ab' - aV 

V = 7 
ca' - c'a ab' - a'b' 

z = 7 
ab' - a'b ab' - a'b 

Now, y being entirely at our disposal, we can so determine 
it that y/(ab' - a'b) shall have any value we please, say p. Hence, 
p being entirely arbitrary, we have, as the solution of the system, 



ix +by + cz =0 ) . v 

i'x + b'v + c'z = 0) ^ '* 



ax 

ax + by 
x - p(bc' - b'c), y = p(ca' - c'a), z = p(ab' - a'b) (4). 

It will be observed that, although the individual values of 
x, y, z depend on the arbitrary constant p, the ratios of x, y, z 
are perfectly determined, namely, we have from (4) 

x : y : z = (be' - b'c) : (ca' - c'a) : (ab' - a'b). 

Example 1. 

2x + 3y + 4z=0, 

3x-2y-6z-0, 
x y z 






-2 -6 3 -2 

give x:y:z- -10 : 24 : -13; 



xvi SYSTEM OF THREE EQUATIONS IN X, If, Z 367 

or, which is the same thing, 

a;=-10p, ?/ = 24/>, z — - ISp, 
p being any quantity whatsoever. 

Example 2. 

ax + by + cz — 0, 
a 2 x + b 2 y + c 2 z=0, 
give x = {be 2 - b 2 c)p = - bcp(b - c), 

y = {ca 2 -c 2 a)p = - cap(c - a) , 
z= {ab 2 - a 2 b)p — - abp(a - b): 

If we choose, we may replace - abep by <r, say, and we then have 

x-a(b-c)/a, y = <r(c-a)/b, z — a{a-b)\c, 

where a is arbitrary. 

In other words, we have 

x :y :z = (b- c)/a : (c - a)/b : (a - b)/c. 
§ 11.] A system of three equations of the 1st degree in three 
variables, say 

ax +by + cz + d =0 (1), 

a'x + b'y + c'z + 6! = (2), 

a"x + b"y + c"z + d" = (3), 

has one and only one solution, provided 

ab'c" - ab"c' + be' a" - bc"a' + ca'b" - cab' * (4). 

The three coefficients c, c, c" cannot all vanish, otherwise we 
should have a system of three equations in two variables, x and 
y, a case already considered in § 8. 

Let us suppose that c =t= 0, then the following system 

ax +by + cz + d =0 (5), 

c' (ax + by + cz + d) - c(a'x + b'y + c'z + d') = (6), 

c"(ax + by + cz + d)- c(a"x + b"y + c"z + d") = (7), 

is obviously equivalent to (1), (2), and (3). Matters are so 
arranged that z disappears from (6) and (7) ; and if, for short- 
ness, we put 

A - ac' - a'c, B = be' - b'c, C = dc' - d'e, 
A' - ac" - a"c, B' = be" - b"c, C = dc" - d"c, 
we may write the system (5), (6), (7) as follows : — 

ax + by + cz + d = (5'), 

Ax + By + C - (6'), 

Ax + B'y + 0' = (7'). 



368 



GENERAL SOLUTION 



CHAP. 



(8), 

(9). 
(10). 

(11). 



Now, provided AB' - A'B 4= 

(6') and (7') have the unique solution 

, _ BC - B'C 
X ~ AB' - A'B 

_ CA - CA 

y ~ AB' - A'B 

These values of x and y enable us to derive from (5') 

g(BC - B'C) + b(CA' - CA) + d(AB' - A'B) 
Z ~ f (AB' - A'B) 

(9), (10), and (11) being equivalent to (5'), (6'), (7'), that is, 
to (1), (2), (3), constitute a unique solution of the three given 
equations. 

It only remains to show that the condition (8) is equivalent 
to (4). 

We have 
AB' - A'B = {ad - a'c) (be" - b"c) - (ac" - a"c) (be' - b'c), 

= c(ab'c" - ab"c' + be' a" - be" a' + ca'b" - ca"b') (12). 
Hence, since c 4= 0, (8) is equivalent to (4). 

Although, in practice, the general formulae are very rarely used, } r et it may 
interest the student to see the values of x, y, z given hy (9), (10), (11) ex- 
panded in terms of the coefficients. We have 

- (BC - B'C) = {dc' - d'e) (Id' - b"c) - (dc" - d"c) (be' - b'c). 

Comparing with (12), we see that -(BC'-B'C) differs from AB'-A'B 
merely in having d written everywhere in place of a (the dashes being 
imagined to stand unaltered). Hence 

- (BC - B'C)=c(db'c" - db"c' + be'd" - be'd' + cd'b" - cd"b'). 
So that we may write 



d(b'c" - b"c') + d'(b"c - be") + d"(bc' - b'c) 

X= 7- 



(13). 



a(b'c" - b"c') + a'(b"c - be") + a"(bc' - b'c) 

We obtain the values of y and 2 by interchanging a and b and a and c 
respectively, namely, 

d(a'c" - a"c') + d'(a"c - ac") + d"(ac' - a'c) 



?/= - 



b(a'c" - a"c') + b'(a"c - ac") + b"(ac' - a'c) 
d(b'a" - b"a') + d'(b"a - ba") + d"(ba'^ b'a) 
c(b'a" - b"a') + c'(b"a-ba") + c"(ba' - b'a) 



(14), 
(15). 



XVI 



HOMOGENEOUS SYSTEM OF THREE EQUATIONS 



3G9 



Written in determinant notation these would become 



V = 



d 


b 


c 




a 


b 


c 




d' 


V 


c 


-^ 


a' 


V 


e 


(13'), 


d" 


b" 


c 




a" 


b" 


c 




a 


d 


c 




a 


b 


c 




a' 


d' 


c' 


4- 


a' 


V 


c' 


an 


a" 


d" 


c" 


a" 


b" 


It 

c 




a 


h 


d 




a 


b 


c 




a' 


V 


d' 


4- 


a' 


b' 


c' 


(15'). 


a" 


b" 


d" 




a" 


b" 


c" 





§ 12.] In the special case where d — 0, d' = 0, d" = 0, the 
equations (1), (2), (3) of last paragraph become 

ax +by + cz =0 (1), 

a'x + b'y + c'z = (2), 

a"x + b"y + c'z - (3), 

which are homogeneous in x, y, z. 

If the determinant of the system, namely, a" (be - b'c) + b"(ca' - c'a) 
+ c"(ab' - a'b), do not vanish, tee see from § 11 (9), (10), (11) (or 
more easily from (13), (14), and (15) of the same section) that 

x = 0, y - 0, 3 = 0. 
If the determinant does vanish, this conclusion does not necessarily 
follow. 

In fact, if we write (1), (2), (3) in the form 



a - + b - + c 
z z 







«'- + ^ + c' = 

z z 

a' ,X - + b" V - + c" = 
z z 



0'), 

(2'), 
(3'), 



and regard xjz and yjz as variables, these equations are con- 
sistent, since 

a" {be' - b'c) + b"(ca' - c'a) + c"(ab' - a'b) = (4), 

and any two of them determine the ratios xjz, yjz ; so that we 
have 

x:y:z-bc' -b'c : at' - c'a : ab' - a'b, 
= be" - b"c : ca" - c"a : ab" - ab, 
= b : c" - b"d : c'a" - c"a' : a'b" - a"b'. 

VOL. I 2 B 



370 EXAMPLES 



CHAP. 



These different values of the ratios are in agreement, by virtue 
of (4), as the student should verify by actual calculation. 

Hence, if the determinant of a system of three homogeneous equa- 
tions of the 1st degree in x, y, z vanish, the values of x, y, z are inde- 
terminate {there being a one-fold infinity of solutions), but their ratios 
are determinate. 

§ 13.] Knowing, as we now do, that a system of three equa- 
tions of the 1st degree in x, y, z has in general one definite 
solution and no more, we may take any logically admissible 
method of obtaining the solution that happens to be convenient. 
(1) We may guess the solution, or, as it is put, solve by inspec- 
tion, verifying if necessary. (2) We may carry out, in the 
special case, the process of § 11; this is perhaps the most gene- 
rally useful plan. (3) We may solve by substitution. (4) We 
may use Bezout's method. (5) We may derive from the given 
system another which happens to be simpler, and then solve 
the derived system. The following examples illustrate these 
different methods : — 

Example 1. 

x + y + z = a + b + c, (b-c)x + {c-a)y + {a-b)z = 0, - + \ + -—3. 

' v '" ' a b c 

A glance shows ns that this system is satisfied by x = a, y = b, z — c; and, 
since the system has only one solution, nothing more is required. 

Example 2. 

Bx + 5y- 7;- 2 = (a), 

4af+8y-14z+3=0 (jS), 

Sx + 6y- 8~-3 = (7). 

Multiplying (a) by 4 and (/3) by 3, and subtracting, we obtain 

4^-143+17 = (3). 

From (a) and (7), by subtraction, 

2/-~-l = (e). 

Multiplying (e) by 4, and subtracting (5), we have, finally, 

l(b-21 = 0; 
whence z=2'l. 

Using this value of z in (e), we find 

V = 3-1; 

and, putting 3/ =8 1, z = 2'\ in (a), we find 

x =-4. 
The solution of the system (a), (/3), (7) is therefore 

x=-i, t/ = 3-1, ss=21. 



XVI 



EXAMPLES 371 



Example 3. 

Taking the equations (a), (/3), (7) of last example, we might proceed by 
substitution, as follows : — 
From (a) 

5 7 2 
X =-3 y + 3 Z + 3- 



20 28 8 



This value of x reduces (/3) to 

which is equivalent to 

4y-14z + 17 = (5')- 

Substituting the same value of as as before in (7), we deduce 

2/-~-l=0 (e'). 

Now (e') gives 

y=z+l, 
and this value of y reduces (5') to 

-102 + 21 = 0, 
hence ~ = 21. 

The values of y and x can now be obtained by using first (e') and then (a). 

Example 4. 

Taking once more the equations (a), (^), (7) of Example 2, Ave might pro- 
ceed by Bezout's method. 

If X and fi be two arbitrary multipliers, we derive from (a), (/3), (7), 
(3x + 5y-7z-2) + \(ix + 8y-liz + 3) + /jL(3z + 6y-8z-3) = (5'). 
Suppose that we wish to find the value of x. We determine X and /j. so that 
(5') shall contain neither y nor z. We thus have 

8X + 6/*+5 = (e'), 

-14X-8/x-7 = (D, 

(3 + 4X + 3/4£-2 + 3X-3m=0 (if). 

If we solve (e') and (f), we obtain 

X=--l, ^=-7. 
The last equation (rf) thus becomes 

(3--4-2-1).b-2- -3 + 2-1 = 0, 
that is, 'ox- # 2 = ; 

whence x= -2/*5 = -4. 

The values of y and z may be obtained by a similar process. 

Example 5. 

ax + by + cz = (a), 

(b + c)x + {c + a)y + (a + b)z=0 (|3), 

a 2 x + b*y + ch = a 2 {b - c) + b-{c - a) + c-(a - b) (7). 
From (a) and (/3) we derive, by addition, 

(a + b + c)(x + y + z) = 0, 
which, provided « + i + c + 0, is equivalent to 

x + y + z = (5). 

We can now, if we please, replace (a) and (/3) by the equivalent simpler pair 
(a) and (5). 



372 EXAMPLES chap. 

Now (see § 10), by virtue of (a) and (5), we have 

x y z . . 

b-c c-a a-b 
If none of the three, b-c, c-a, a-b, vanish, we may write (7) in the form 

a 2 (b - c),— + b-(c - a)^— + c-(a - b) -^ = a 2 (b -c) + b 2 (c -a) + c"(a - b). 
b — c c — a a — b 

Using (e) we can replace y/{c-a) and zfta-b) by x/(b-c), and the last equa- 
tion becomes 

{a°-(b-c) + b-(c-a) + <?(a-b)},— = a"(b-c) + b*-(c-a) + c 2 (a-b); 

c 

and, since a"(b-c) + b-{c-a)+c-(a -b) = -(b-c) {c-a) (a- b), which does not 

vanish, if our previous assumptions be granted, it follows that 

b-c 
Hence x = b-c, and, by symmetry, y = c-a, z = a-b. 

This solution might of course have been obtained at once by inspection. 

Example 6. 

x + ay + a 2 z + a 3 =0\ 

x + by + b 2 z + ¥ = ol (a). 

x + cy + c-z+c 3 -0J 
From the identity 

Z 3 +p£ z +qZ+r={£-a)$-b){Z-c), 

(see chap, iv., § 9), where 

2)= -a -b-c, q — bc + ca + ab, r=-abc, 
we have 

r + aq + a 2 p + a? = 0\ 

r + bq+b 2 j,+ b 3 =0l (£). 

r + cq+c-p+ (^-0) 

It appears, therefore, from (,3) that 

x-r, y = q, z=p 
is a solution of (a). Hence, since (a) has only one solution, that solution is 
x=-abc, y=bc + ca + ab, z='—a — b — c 
This result may be generalised and extended in various obvious ways. 

§ 14.] A system of more than three equations of the 1st degree 
in three variables will in general be inconsistent. To secure consistency 
one condition must in general be satisfied for every equation beyond 
three. This may be seen by reflecting that the first three equa- 
tions will in general uniquely determine the variables, and that 
the values thus found must satisfy each of the remaining equa- 
tions. Thus, in the case of four equations, there will be one 
condition for consistency. The equation expressing this condition 
could easily be found in its most general form ; but its expression 
would be cumbrous and practically useless without the use of 



xvi GENERAL THEORY FOR A LINEAR SYSTEM 373 

determinantal or other abbreviative notation. There is, how- 
ever, no difficulty in working out the required result directly in 
any special case. 

Example. 

Determine the numerical constant p, so that the four equations 
2x~Zy+5z=18, 3x-y + lz = 2Q, 4x + 2y-z=5, 

{p+l)x+{p+2)y+(p + S)z=76 

shall he consistent. 

If we take the first three equations, they determine the values of x, y, z, 
namely, x=l, y = B, 2 = 5. 

These values must satisfy the last equation ; hence we must have 
(p + l) + (p + 2)3 + (p + 3)5 = 76, 
which is equivalent to 

9p = 5i. 
Hence p = 6. 

§ 15.] If the reader will now reconsider the course of reason- 
ing through which we have led him in the cases of equations of 
the 1st degree in one, two, and three variables respectively, he 
will see that the spirit of that reasoning is general ; and that, 
by pursuing the same course step by step, we should arrive at 
the following general conclusions : — 

I. A system of n - r equations of the 1st degree in n variables 
has in general a solution involving r arbitrary constants; in other 
words, has an r-fold infinity of different solutions. 

II. A system of n equations of the 1st degree in n variables has 
a unique determinate solution, provided a certain function of the co- 
efficients of the system, which we may call the determinant of the 
system, does not vanish. 

III. A system of n + r equations of the 1st degree in n variables 
will in general be inconsistent. To secure consistency r different con- 
ditions must in general be satisfied. 

There would he no great difficulty in laying down a rule for calculating 
step hy step the function spoken of above as the determinant of a system of 
n equations of the 1st degree in n variables ; hut the final form in which it 
would thus be obtained would be neither elegant nor luminous. Experience 
has shown that it is better to establish independently the theory of a certain 
class of functions called determinants, and then to apply the properties of 
these functions to the general theory of equations of the 1st degree. A 
brief sketch of this way of proceeding is given in the next paragraph, and 
will be quite intelligible to those acquainted with the elements of the theory 
of determinants. 



374 



GENERAL SOLUTION OF LINEAR SYSTEM 



CHAP. 



GENERAL SOLUTION OF A SYSTEM OF LINEAR EQUATIONS 
BY MEANS OF DETERMINANTS. 



§16.] Consider the system 








"D - ! =a n x x + a 12 .r 2 + . . 


. + a m x n + c, 


= 


(1), 


KJ 2 == ^'21 1 ^22* 2 ' * * 


• + CWEji + C-2 


= 


(2), 



U„ * ffljuJB, + rt, !2 a- 2 + . . . + fl^Bn + C n = (»), 

where there are n variables, 2,, x 2 , . . ., #«, and ra equations to 
determine them. 



Let 



A = 



a n a ]2 



"«2 



and let A M A 2 , . . ., A n denote the determinants obtained from A 
by replacing the constituents of the 1st, 2nd . . . nth columns 
respectively, by the set r,, c 2 , . . ., c n . 

Also let the co-factors of a,„ a I2 , . . ., a m , a 21 , a 22 , . . ., a m , 
&c., A be denoted by A u , A I2) . . ., A m , A 21 , A 22 , . . ., A 2n , &c, 
as usual. 

Then, by the theory of determinants, we have 

OuA,, + OnAa + . . . + a m A m = A \ 
a IS A„ + rt 23 A 21 + . . . + a n2 A m = 

(a), 

a, n A u + a. m A 2i + . . . + a nn A m = 

c iAi + £3-^1 + . . . + <" n A, u = A, J 
and so on. 

If the determinant A, which we call the determinant of the 
system of equations, does not vanish, then A u , A 21 , . . ., A, n 
cannot all vanish. Let us suppose that A„ 4= 0. Then, by 
chap, xiv., § 10, the system 

A U U, + A 21 U 2 + . . . + A /U U„ = 0, 

U 2 =0, u, = o, . . ., u n =o, 

is equivalent to the system (1), (2) . . . (»). If we collect the 



XVI 



BY MEANS OF DETERMINANTS 



375 



coefficients of the variables ar, , x 2 , . . ., x n in the first of these 
equations, and attend to the relations (a), that equation reduces to 

A», + A t = 0. 
Since A + 0, this is equivalent to 

*,= -A/A. 

By exactly similar reasoning we coidd show that x 2 = - A.,/ A, . . . 
x n = - A n /A. Hence the solution, and the only solution, of (1), 
(2), . . ., (n) is 

k, = - A,/A, x 2 = - A 2 /A, . . ., x n = - A,,./ A (ft). 

Although, from the way we have conducted the demonstration, 
it is not necessary to verify that (ft) does in fact satisfy (1), 
(2), . . ., (a), yet the reader should satisfy himself by substitu- 
tion that this is really the case. 

We have thus shoivn that a system of n equations of the 1st degree 
in n variables has a unique determinate solution, provided its determin- 
ant does not vanish. 

Next, let us suppose that, in addition to the equations (1), 
(2), . . ., (n) above, we had another, namely, 

0»+i,i x i + a »+i, 2 x 2 + • • • + a w +i,« %n + Cn+i = Oj (a + 1 ), 
the system of n + 1 equations thus obtained will in general be 
inconsistent. 

The necessary and sufficient condition for consistency is that the 
solution of the first n shall satisfy the n+lth, namely, ^/A=t=0, 



— #71+1,1 A, 



tf»+i >2 A 2 - 



a 



u 

a 2l 



a™, 



a n+i,n Aji + C,!-)-! 

... i 



A = 0, that is, 



(1-2,1 






II 



rt+i,i #n+2,s 



" nil ' II 

a n+2,n ("n+i 



= 



(y)- 



Lastly, let us consider the particular case of n homogeneous 
equations of the 1st degree in n variables. In other words, let 
us suppose that, in equations (1), (2), . . ., (/<) above, we have 
c, = 0, c 2 = 0, . 



., Cn=0. 



376 CONDITION OF CONSISTENCY HOMOGENEOUS SYSTEM chap. 



1st. Suppose A 4= 0, then, since now A l = 0, A., - 0, 
A n = 0, (/3) gives Xy = 0, x 2 = 0, . . ., x n = 0. 
2nd. Suppose A = 0. 
We may write the equations in the form 



x x 



x, 



On— + «m— + • • • + a m = 0, 



■' a 



X~ 



« 2 i— + «»— + . . .+a m = 0, 



"21 T •*«! 



X\ x% 

Q"M ^ ^W2 I" 



+ a nn = 0. 



These may be regarded as a system of n equations of the 1st 
degree in the n — 1 variables xfx m x.,/x n , . . ., x n ^\x n \ and, since 
A = 0, they are a consistent system. Using only the last n-\ 
of them, we find 



x 1 

Xn 



"'in ( *22 • • • ®2,n - 1 
^371 ^32 • • • Q'3,n - 1 



™"im ™na 



l n,n-\ 



Clo, do 



(I., 



J 2\ "'22 • • • "B,n - 1 

^'31 ^32 • " * "3,71—1 



0"K\ Q-1V2 . • • tt 



n,n-i 



= "("I) 2 



A A 

\2n-3 tt!L - n 

A A * 

In a similar way we prove that 

X 2 /X n = A 12 | A m . . ., X n-l jX n = Ai jn _ 1 /A ln . 

iZe/tce we /«we 

3/ t : £ 2 • • • • • -^n ~ -"-li • -"-is • • • • ■ A lrt J 

and, fo/ 'parity of reasoning, 

x l : x. 2 : . . . : x n = A rl : A r2 : . . . : A m 
where r = 2, = 3, . . ., = n, as we please. In other words, the ratios 
of the variables are determinate, but their actual values are in- 
determinate, there being a one-fold infin iiy of different solutions. 



Exercises XXV 
Solve the following systems : — 

(1.) 



x y z 



a , v 



3 + 4 + 6-^' i5 + 20 + 9 -10, 2 + 10 + 4 _4,J - 



xvr 



EXERCISES XXV 377 



(2.) 2x + %y + \z = 2§, 3x+2y+5z=S2, 4.z + 3?/ + 2i = 25. 
(3.) '8x+l-2y+6-8z=l, SSx-2-5y-Z'82z=-5, 

•Ola:- -003y- -301;= -013 ; 
calculate as, y, z to four places of decimals. 

(4.) x + y + z = 26, x-y = <±, x-z=6. 

(5.) If (x+l) a _ A | B:r+C 



(x + 2)(ar + x+l)~x + 2 a? + x+V 
determine the numerical constants A, B, C. 

(6.) Find a linear function of x and y, which shall vanish when x=x', 
y = y', and also when x = x", y = y", and which shall have the value +1 when 
x = x'", y = y'". 

(7.) An integral function of x of the 2nd degree vanishes when x = 2, 
and when x=3, and has the value - 1 when x= - 2 ; find the function. 

Solve the following systems : — 
(8.) y + z = a, z + x = b, x + y — c. 

(9.) JL+ * =2a, -^- + — = 26, - X .+- y -,=2c. 

b + c b-c c+a c-a a+b a- b 

(10.) An integral function of a; of the 2nd degree takes the values A, B, C, 
when x has the values a, b, c respectively ; find the function. 

Solve the following systems : — 

(11.) bc{b - c)x + ca(c - a)y + ab(a - b)z=0, 

(a + b-c)x + (b + c-a)y+(c + a-b)z = a 2 + b 2 + c 2 , 

b 2 c 2 x + c 2 a 2 y + a 2 b 2 z = abc(bc + ea + ab). 

(12.) If _^_ + ^_ + _f_=l > 

a + a b+a c + a 

x y z . 



« + /3 6 + (3 c + /3 

x y z . 



then 



a + y b + y c+y 
I V ■ z - Iz£. 



(a + a)(a + p) 2 "(b + a)(b + p) 2 (c + a) (c + /3) 2 (a + j8) (b + p) (c + /3) 
(13.) aa; + fa/ + c3 = « + & + c, 

a 3 x- + fe 2 2/ + c^ = (a + 6 + c)-, 
bcx + cay + abz = 0. 
(14.) ax + cy + bz = ex + by + az = bx + ay + cz = a 3 + b* + c 3 - Zabc. 

(15.) lx + my + nz = mn + nl + hn, 

x + y + z=l + m + n, 
(m-n)x + (n- l)y + (l- m)z—0. 

(16.) h- + my + nz = 0, 

(ra + n)x + (7i + ?)y + {l + m)z=l + m + n, 
Px + m 2 y + n 2 z =p 2 . 

(17.) Show that (b- c)x + by-cz = 0, (c-a)y + cz-ax=Q, (a-b)z + ax-by 
= 0, are consistent. 



378 EXERCISES XXV chap. 

(18.) Show that the system cy-bz=f, ass — cz—g, bx-ay = h has no finite 
solution unless af+bg + ch = 0, in which case it has an infinite number of 
solutions. 

Find a symmetrical form for the indeterminate solution involving one 
arbitrary constant. 

Solve the following systems : — 

(19.) 3x-2y + 3u = 0, x-y+z=0, 3y + Sz-2u = 0, x + 2y + Bz + 4u = 8. 
(20.) Uax=p-r, by—p-s, cz = r-s, d{y + z) = s-q, e(z + x) = q-r,f(X + y) 
= q-p + g, find z in terms of a, b, c, d, e,f, g. 
iz + x 3u + x 5v + x 



(21.) 2y + x = 



3 4 8 

x + y + z + u + v - 1 _ 5x + ly + Bz + 2u + v + 2 



4 9 

(22.) ax + by=l, cx + dz=l, ez+fu = l, gu + hv=l, z + y + z + u + v = 0. 

(23.) Prove that, with a certain exception, the system U = 0, V = 0, W = 0, 

and \XJ + fiY + vW = 0, \'U + /x'V + k'W = 0, X"U + /j."V + p"W = are equivalent. 

(24.) If x=by + cz + du, y = ax + cz + du, 

z = ax + by + du, u = ax + by + cz, 

„ a b c d 

then - + i — -H 7 +-5 — =r=l- 

a+l 6+1 c+1 d+l 

(25.) Show that the system ax + by + cz + d = 0, a'x + b'y + c'z + d' = 0, 
a"x + b"y + c"z + d" = will be equivalent to only two equations if the system 
ax + a'y + a"-Q, bx + b'y + b" = Q, cx + c'y + e" = Q, dx + d'y + d" — be con- 
sistent, that is, if 

b'c" - b"c' _ b"c - be" _ be' - b'c 
a'd" - a"d' ~ a"d - ad" ~~ad'- a'd' 

Show that in the case of the system 

7 x y z . x y z . 

x + y + z = a + b + c, - + f + -=l, -v. + pj + -^ = 0, 
a b c a 6 b- f c 8 

the above two conditions reduce to one only, namely, 

bc + ca + ab = Q. 
(26.) Show that the three equations 

x = A + A'u + A"v, y = B + B'u + B"v, z = C + C'tc + C"v, 
where u and v are variable, are equivalent to a single linear equation con- 
necting x, y, z; and find that equation. 
(27.) If ax + by + cz + d—0, show that 



y= (£ +q y e -a)-£ 



where p and q are arbitrary constants. 



xvi EQUATIONS REDUCIBLE TO LINEAR SYSTEMS 379 

(28. ) If ax + by + cz + d = 0, a'x + b'y + c'z + d' = 0, show that 
x=p(bc! - b'c) + \{b' - c')d -(b- c)d'}/{a(b' - c') + b(c'-a') +c(a' -b')}, 
y=p(ca' ~ c'«) +{(e'- a')d - (c - a)d'\/{a(b' - c') + b(c' - a') + c(a' -&')}, 
z=p(ab' - a'b) + { (a' - b')d -(a- b)d'}/{a{b' - c') + b(c' - a') + c(a' -b')}, 
where p is an arbitrary constant. 

EXAMPLES OF EQUATIONS WHOSE SOLUTION IS EFFECTED BY 
MEANS OF LINEAR EQUATIONS. 

§ 17.] We have seen in chap. xiv. that every system of 
algebraical equations can be reduced to a system of rational 
integral equations such that every solution of the given system 
will be a solution of the derived system, although the derived 
system may admit of solutions, called " extraneous," which do not 
satisfy the original system. It may happen that the derived 
system is linear, or that it can, by the process of factorisation, 
be replaced by equivalent alternative linear systems. In such 
cases all we have to do is to solve these linear systems, and then 
satisfy ourselves, either by substitution or by examining the 
reversibility of the steps of the process, which, if any, of the 
solutions obtained are extraneous. The student should now 
re-examine the examples worked out in chap, xiv., find, wher- 
ever he can, all the solutions of the derived equations, and 
examine their admissibility as solutions of the original system. 
"We give two more instances here. 

Example 1. 

• + ./;.„ .L 2 1U =V{2(^ + D} («)■ 



(Positive values to be taken for all tbe square roots.) 

If we rationalise the two denominators on the left, we deduce from (a) the 
equivalent equation, 

V{^-V0« 2 -i)}+V{* + V(^-i)}=V{2(^+i)} (/s). 

From (|3) we derive, by squaring both sides, 

2x + 2\f{x 2 -(x°-~l)} =2(a* + l), 
that is, 2.e + 2 = 2x 3 + 2 (y). 

Now (7) is equivalent to x 3 -x = 0, 

that is, to x(x - 1) (se+ 1) =0 (8). 

Again (5) is equivalent to the alternatives 
that is to say, its solutions are # = 0, x=l, x = 




380 



EXAMPLES 



niAP. 



Since, however, the step from (fi) to (7) is irreversible, it is necessary to 
examine which of these solutions actually satisfy (a). 

Now x = Ogives \f - i+ \J -\-i=\j2, 

that is (see chap, xii., § 17, Example 3), 

which is correct. 

Also, £=1 obviously satisfies (a). 

But x= - 1 gives 2i = 0, which is not true, hence x= - 1 is not a solution 
of (a). 

Remark that x= -\ is a solution of the slightly different equation, 

V{z+V(z 2 -D} " Vte-VC* 8 -!)} = ^'^ + 1) ' 

Example 2. 

x 2 ~y 2 = x-y, 2x+3y-l = (a). 

Since the first of these equations is equivalent to (x~y)(x + y-l) = 0, the 
system (a) is equivalent to 

/ x-y = 0, and 2x + 3i/-l=0 N 

\x + y-l = 0, an.l 2z + 32/-l = 0, 

now the solution of x-y = 0, 2x+By-l~0 is a: =1/5, 2/ = 1/5 ; and the solu- 
tion of a: + 2/-l = 0, 2»+3y-l = is a;=2, y— -1. Hence the solutions of 
(a) are 



a; 


y 


1/5 
2 


1/5 
-1 



§ 18.] The solution of linear systems is sometimes facilitated 
by the introduction of Auxiliary Variables, or, as it is sometimes 
put, by changing the variables. This artifice sometimes enables us 
to abridge the labour of solving linear systems, and occasionally 
to use methods appropriate to linear systems in solving systems 
which are not themselves linear. The following are examples : — 

Example 1. 

(a). 



{x-af _x-2a-b 



{x + b'f x + a + 2b 

Let x + b = z, so that x = z-b ; and, for shortness, let c = a + b. 
Then (a) may be written 

(z-c) 3 _z-2c 
2 s ~ z + c 
From (/3) we derive 

{z-c)\z + c) = z 3 ^z-2c\ 



(£)■ 



xvi CHANGE OF VARIABLES 381 

that is, 

z* - 2z»c+ 2z<? - c 4 = 2 4 - 2rV, 
which is equivalent to 

2c s 3-c 4 =0 (7). 

Now (7) has the unique solution z=e/2, which evidently satisfies (a) 
Hence x = c/2 - b, that is, x — (a-b)j2, is the only finite solution of (a). 

Example 2. 

a(x + y) + b(x-y)+c = 0, a'(x + y) + b'(x-y) + c' = (a). 

Let £ = x + ?/, -q = x-y, then the system (a) may be written 

a^ + bij + c-0, a's + b'T] + e' = Q (a'). 

Now (a') is a linear system in £ and 7;, and we have, by § 4, 

be' -b'c _ca' - c'a 

*~ab'-a'b' V ~ab'-a'b &>' 

Replacing f and 7; by their values, we have 

be' - b'c ca' -c'a , . 

X + y = aV^b' X ~ y= W^Tb iy) - 

From (7), by first adding and then subtracting, we obtain 

_ be' - b'c + ca' - c'a _ be' - b'c - ca' + c'a 
X ~ 2{ab'-a'b) ' V ~ 2(ab' - a'b) ' 
Example 3. 

cy + bz — az + cx — bx + ay — abc. 

Dividing by be, by ca, and by ab, we may write the given system in the 
following equivalent form 

!+-:- » 

x y , . 

a + l= C W 

Now, if we add the equations (/3) and (7), and subtract (a), we have 

(MM;+s)-(K)-»+-^ 

x 

that is, 2- — b + c-a\ 

a 

, a(b + c-a) 
wlience x= — — . 

By symmetry, we have 

y = b{c + a-b)/2, z = c{a + b-c)/2. 

Here we virtually regard x/a, yjb, z/c as the variables, although we have 
not taken the trouble to replace them by new letters. 



382 



EXAMPLES 



CHAP. 



Example 4. 



x—ay—b z- c 
r 



(a), 

03). 

have 

(7). 



(*)• 



P 1 

Ix + my + nz = d 

Represent each of the three equal functions in (a) by p. Then we 
(x-a)/p = p, (y-b)/q = p, (z-c)/r = p, 
which are equivalent to 

x = a+pp, y = b + qp, z-c + rp 

Using (7), we reduce (j8) to 

l(a +pp) + m[b + qp) + n(c + rp) = d, 
for which we obtain, for the value of the auxiliary p, 

_d-la- mb - nc 

lp + mq + nr 

From (7) and (5) we have, finally, 

d—la — mb - nc 

x = a + v -7 , 

lp + mq + nr 

_ m{aq - bp) + n(ar - cp) -\-})d 
lp + mq + nr 

The values of y and z can be similarly found, or they can be written 
down at once by considering the symmetry of the original system. 

Example 5. 

x-2y + Zz = (a), 

2a;-3y + 4t = Q3), 

ia? + By 3 + z 3 -xyz = 216 (7). 

From (a) and (/3) we have (see § 10 above) 

x/l=y/2=z/l=p, say. 
Hence & = P, y — %P> ~ = / } (5). 

By means of (5) we deduce from (7) 

27 p 3 = 216, 
which is equivalent to p* = 8. (e). 

Now the three cube roots of 8 are (see chap, xii., § 20, Example 1) 

2, 2( - 1 + V3i). 2( - 1 - \/30- 
Hence the solutions of (e) are 

P = 2, p=2(-l + V3i). p=2(-l-V8i)- 

Hence, by (5), we obtain the three following solutions of (a), (/3), (7) ; — 



X 


V 


z 


2 
- 1 + \J%i 

- 1 - V3i 


4 
2( - 1 + V30 

2( - 1 - V30 


2 

- 1 + V3* 

- 1 - V3i 



xvi EXERCISES XXVI 383 

Since, by chap. xiv. , § 6, the system in question has only three solutions, 
we have obtained the complete solution. 

N.B. — In general, if Ui, w 2 , • • •, « n -i be homogeneous functions of the 
1st degree in n variables, and v a homogeneous function of the nth degree in 
the same variables, the solution of the system 

!/l = 0, W2 = 0, . . ., « n -i = 0, v = 

may be effected by solving a system of n - 1 linear equations in n - 1 variables, 
and then extracting an »th root. See in this connection § 16 above. 

Example 6. 

ax- + by 2 + c = 0, a'x? + b'y- + c'=0. 

If we regard x? and y' 2 as the variables, we have to do with a linear system, 
and we obtain as heretofore, 

x 2 ={bc' - b'c)/(ab' - a'b), y"- = (ca' - c'a)/(ab' - a'b). 
Hence 

x=±\/{M -b'c)/(ab' -a'b), y=±\/(ca' -c'a)/{ab' -a'b). 

Since either of the one pair of double signs may go with either of the other 
pair, we thus obtain the full number of 2 x 2 = 4 solutions. 

Example 7. 

ay + bx + cxy = 0, a'y + b'x + c'xy=0. 

These two equations evidently have the solution x=0, y = 0. 
Setting these values aside, we may divide each of the two equations by 
xy. We thus deduce the system 

a-+b- + c = 0, a'- + b'- + c' = 0, 
x y x y 

which is linear, if we regard 1/x and 1/y as the variables. Solving from this 
point of view, we obtain 

1 _ be' - b'c 1 _ ca' - c'a _ 

x~ab'-a'b' y~ab'-a'b ' 
for which we have 

x = {aV - a'b)/(bc' - b'c), y = {aV - a'b)/(ca' - c'a\ 

"We have thus found two out of the four solutions of the given system. There 
are no more finite solutions. 



Exercises XXVI. 
Solve the following by means of linear systems :- 



,, > \Jox +\/b _ \/a + \/b 

\/ax - yjb ~ \/b 

,„ , Va;+4?>i _ \/x + 27H 

\/x + 3)i ~ \Jx + n ' 
(3.) ^/( x + 22)-s/(x+ 11) = 1. 

(4.) V*+V(*+3) = 12/V(* + 3). 



384 EXERCISES XXVI 

(5.) V(^ + 2 ) + V(«-2) = 5/V(a- + 2). 

(6.) V aj +V( a + / 3 )-V(a-J 8 )=V(*+2/3). 

(7. ) \J{x + q- r) + V(a; + r-p) + s/(x +p -q) = 0. 

*j{x-p)-y/ P + V(»-p)+Vp = ^ (x ~ p) ~ 

(9.) z=v> 2 -aV(& 3 + a; 2 --a 2 )}+a. 

(10. ) V( V-' c + V«) + V(V* - V«) = V(2 V® + 2 \ /b )- 

(11.) \?x+^/{B-^(2x + x 2 )} = ^3. 

( 12 -) V^'V^)^' V(*-2)=|v(*-3). 

(13.) \/{a-x)-^J{y-x) = sJy, \J{b-x) + *J(y-x)~s/y. 

1 17 

\H.) Va ; -\ / 2/=4. ^-y^^g- 

(15.) (x-a) 2 -(y-6) 2 = 0, (a; -6) (y -«) = «( 26 -a). 

(16.) a-2/ = 3, z 2 -7/ 2 = 45. 

(17.) xfy = a/b, x^-if — d. 

(18.) a; + ay + a 2 2 + a 3 ii + a 4 =0, 

x + by + bh + b s u + b* = 0, 
x + cy + <?z + <?u + c 4 = 0, 
re + tf y + cPz + d?u + d 4 = 0. 

(19.) x+y+z=0, ax + by + cz-Q, 

bcx + cay + abz = (b- - c 2 ) (c 2 - a 2 ) (a 2 - b"). 

mx -ly _ ny -mz _ Iz-nx 1 
^ "' lm{a-b) mn(b — c) nl(c-a) hnn' 



CHAP. 



(21.) 



mnx + nly + hnz — a + b + c. 
x + 7/ — z y + z-x z + x-y , 

b+c c+a a+b 



(22.) Z(&-c)a;=0; 2a(6 2 -c 2 )x=0, 2a(b-c)x=JI(b-c). 

(23.) 2sc=l, 2te/(6-c)=0 J 2.<:/(i 2 -c 2 ) = 0. 

(24.) ax + k(y + z + ic)-0, by + k(z + u + x) = 0, 

cz + k(u + x + y)-0, du + k(x + y + z) = 0. 
(25.) x + y-\-z—a, y + z + u — b, z + u + x = c, u + x + y=d. 
(26.) 3/»-2/y=i, 4»+7y=l|ay. 

1 1 _12 ay + 2s + 3y + 2 

1 ' sb+5 y + 7 35' rry+1 

< 28 -> w+J- 11 - L + h 16 - 



X L M 2 \2 + M 2 X 2_ M 2 

y- x- + y 2 a: 2 --y- ar + y- 



(29) -^- +■ M -1 -^-- "^ =1 



XVI 



EXERCISES XXVI 



385 



(30.) 

(31.) 
(32.) 

(33.) 



a(b -y) + b(a - x) = c(a -x)(b- y), 
a\b -y) + b\a -x) = c 2 {a -x)(b- y). 

b « 2 ft 2 

+ . , . =1, — + 



c+a+x c + b + y 

b 2 c 2 , 6 

— + ~i — 1, 

I/ 2 z 2 



c+a+x c+b+y 



-a + b. 



+ * 2=] ' 



a 2 b 2 , 
— +— =1, 

yz zx 



z 2 

-+— =1, 

2a; xy 



a 2 6 2 , 

-3 + - = l. 
x 2 y- 

^2/ 2/2 



(34. ) ayz - bzx - cxy = - ayz + bzx - cry — - ayz - bzx + cxy = xyz. 

(35.) Show that (l + lx) (l + ay) = l + lz. {l+mx)[\ + by) = l+mz, 
(1 + nx) (1 +cy) = l \-nz are not consistent unless 

(6 - c) ajl + {c- a)b/m + (a-b) cfn = 0. 

If this condition be satisfied, then x = (c/n-b/m)/(b - c) ; and particular 
solutions for y and z are y— - \ja, z= - l/l. 



GRAPHICAL DISCUSSION OF LINEAR FUNCTIONS OF ONE AND 
OF TWO VARIABLES. 

§ 19.] The graph of a linear function of one variable is a 
straight line. 

Consider the function 
y = ax+ b. To find the 
point where its graph cuts 



Y 

B 




p 

N 








sk 


o i\ 


ft X 



Fio. 1. 

OY, that is, to find the 
point for which X = 0, 

we have to measure 

OB = b upwards op 

downwards, according 

as b is positive or 

negative (Figs. 1 and 2). Through B draw a line parallel to the 

x-axis. 

Let OM represent any positive value of x, and MP the cor- 
responding value of if. 

vol. I 2 c 




Fio. 2. 



386 



THE GRAPH OF CLV + b 



CHAP. 



By the equation to the graph, we have (y - b)]x = a. Now, since 
b = + OB = + MN in Fig. 1, - - OB = - MN in Fig. 2, we have 

y-b = PM - MN - PN in Fig. 1, 
y - b = PM + MN = PN in Fig. 2. 
Hence we have in both cases 

PN_PN_y-6 

BN ~ OM ~ " 



= a. 



In other words, the ratio of PN to BN is constant ; hence, 
by elementary geometry, the locus of P is a straight line. If a 
be positive, then PN and BN must have the same sign, and the 
line will slope upwards, from left to right, as in Figs. 1 and 2 ; 
if a be negative, the line will slope downwards, from left to 
right, as in Figs. 3 and 4. The student will easily complete the 
discussion by considering negative values of x. 
Y Y 



M X 




Fig. 3. 



Fio. 4. 



§ 20.] So long as the graphic line is not parallel to the axis 
of x, that is, so long as a =# 0, it will meet the axis in one point, 
A, and in one only. In analytical language, the equation 
ax + b = has one root, and one only. 

Also, since a straight line has no turning points, a linear 
function can have no turning values. In other words, if we 
increase x continuously from - oo to + oo , ax + b either increases 
continuously from - oo to + oo , or decreases continuously from 
+ oo to - oo ; the former happens when a is positive, the latter 
when a is negative. 

Since ax + b passes only once through every value between 



XVI 



THE GRAPH OF OX + b 387 



+ oo and - oo , it can pass only once through the value 0. We have 
thus another proof that the equation ax + b = has only one root. 

A purely analytical proof that ax + b has no turning values 
may be given as follows : — Let the increment of x be h, then the 
increment of ax + b is 

{a(x + h) + b) - {ax + b) = ah. 
Now ah is independent of x, and, if h be positive, is always posi- 
tive or always negative, according as a is positive or negative. 
Hence, if a be positive, ax + b always increases as x is increased ; 
and if a be negative, ax + b always decreases as x is increased. 

§ 21.] We may investigate graphically the condition that the 
two functions ax + b, a'x + b' shall have the same root ; in other 
words, that the equations, ax + b = 0, a'x + b' = 0, shall be con- 
sistent. Denote ax + b and a'x + V by y and y' respectively, so 
that the equations of the two graphs 
are y = ax + b, y' = a'x + b'. If both 
functions have the same root, the 
graphs must meet OX in the same 
point A. Now, if P'M PM be ordi- 
nates of the two graphs corresponding 
to the same abscissa OM, and if the 
graphs meet OX in the same point A, 
it is obvious that the ratio P'M/PM 
is constant. Conversely, if P'M/PM FlQ - 5 - 

is constant, then P'M must vanish when PM vanishes ; that is, 
the graphs must meet OX in the same point. Hence the neces- 
sary and sufficient analytical condition is that (a'x + b')j(ax + b) 
shall be constant, = k say. In other words, we must have 

a'x + V = k(ax + b). 
From this it follows that 

a' = ka, V = kb, 
and ab' - a'b = 0. 

These agree with the results obtained above in § 2. 

§ 22.] By means of the graph we can illustrate various limiting 
cases, some of which have hitherto been excluded from con- 
sideration. 




388 



GRAPHICAL DISCUSSION OF LIMITING CASES 



CHAP. 



I. Let b = 0, a 4= 0. In this case OB = 0, and B coincides 
with ; that is to say, the graph passes through (see Figs. 

Y 




Fig. 6. Fig. 7. 

6 and 7). Here the graph meets OX at 0, and the root of 
ax = is x = 0, as it should be. 

II. Let b =¥ and a = 0. In this case the equation to the 
graph is y = b, which represents a line parallel to the a-axis (see 
Figs. 8 and 9). In this case the point of intersection of the 



Y 


B 






O 


X 



o 



Fig. 8. 



B 



Fig. 9. 



graph with OX is at an infinite distance, and OA = oo . If we 
agree that the solution of the equation ax + b = shall in all 
cases be x — - b/a, then, when b 4= 0, a = 0, this will give x = oo , 

in agreement with the conclusion just 
derived by considering the graph. 
This case will be best understood 
-^ by approaching it, both geometric- 
— r ally and analytically, as a limit. 
Let us suppose that 6= — 1, and 
that a is very small, =1/100000, 
say. Then the graph correspond- 
ing to y = x/l 00000 - 1 is something like Fig. 10, where 



B 



Fio. 10. 



:cvi INFINITE ROOT 389 

the intersection of BL with the axis of x is very far to the right 
of ; that is to say, BL is nearly parallel to OX. 
On the other hand, the equation 

1=0 



100000 

gives x = 100000, a very large value of x. The smaller we 
make a the more nearly will BL become parallel to OX, and the 
greater will be the root of the equation ax + b = 0. 

If, therefore, in any case where an equation of the 1st degree in 
x was to be expected, we obtain the paradoxical equation 

6 = 0, 
where b is a constant, this indicates that the root of the equation has 
become infinite. 

III. If a = 0, b = 0, the equation to the graph becomes y - 0, 
which represents the axis of x itself. The graph in this case 
coincides with OX, and its point of intersection with OX becomes 
indeterminate. If we take the analytical solution of ax + b = 
to be x = - bja in all cases, it gives us, in the present instance, 
x = 0/0, an indeterminate form, as it ought to do, in accordance 
with the graphical result. 

§ 23.] Tho graphic surface of a linear function of two inde- 
pendent variables x and y, say z = ax + by + c, is a plane. It 
would not be difficult to prove this, but, for our present pur- 
poses, it is unnecessary to do so. We shall confine ourselves to 
a discussion of the contour lines of the function. 

The contour lines of the function z = ax + by + c are a series of 
parallel straight lines. 

For, if k be any constant value of z, the corresponding con- 
tour line has for its equation (see chap, xv., § 1 6) 

ax + by + c-k (1). 

Now (1) is equivalent to 

/ a\ k- c 

r-(-j>+T < 2) - 

But (2), as we have seen in § 19 above, represents a straight 
line, which meets the axes of x and y in A and B, so that 



390 



CONTOUR LINES OF ClX + by + C 



CHAP. 



b b I b a 




Fig. 11. 



Let h' be any other value of z, then the equation to the corre- 
sponding contour line is 

ax + by + c = k' (3), 



el- 



s' 



a\ k' - c 

-y x+ -b- 



(4). 



Hence, if this second contour line meet the axes in A' and B' 
respectively, we have 



OB' = 



Hence 



Jc'-c 

IT' 
OA = 
OB~ 



OA' = 
OA' 



k'-c 



OB' 



which proves that AB is parallel to A'B'. 

The zero contour line of z = ax + by + c is given by the equa- 
tion 

ax + by + c = (5). 

This straight line divides the plane XOY into two regions, such 
that the values of x and y corresponding to any point in one of 
them render ax + by + c positive, and the values of x and y cor- 
responding to any point in the other render ax + by + c negative. 
§ 24.] Let us consider the zero contour lines, L and I/, of 
two linear functions, z = ax + by + c and z' = a'x + b'y + c'. Since 
the co-ordinates of every point on L satisfy the equation 

ax + by + c = (1), 



XVI GRAPHIC ILLUSTRATION OF INFINITE SOLUTION 391 

and the co-ordinates of every point on L' satisfy the equation 

a'x + Vy + ti = (2), 

it follows that the co-ordinates of the point of intersection of L 
and L' will satisfy both (1) and (2); in other words, the co- 
ordinates of the intersection will be a solution of the system 

(1), (2). 

Now, any two straight lines L and L' in the same plane have 

one and only one finite point of intersection, provided L and L' 

be neither parallel nor coincident. Hence we infer that the 

linear system (1), (2) has in general one and only one solution. 

It remains to examine the two exceptional cases. 

I. Let L and L' (Fig. 1 1 ) be parallel, and let them meet the 
axes of X and Y in A, B and in A', B' respectively. In this case 
the point of intersection passes to an infinite distance, and both 
its co-ordinates become infinite. 

The necessary and sufficient condition that L and L' be 
parallel is OA/OB = OA'/OB'. Now, OA = - c/a, OB = - c/b ; 
and OA' = - c'fa', OB' = - c'/b'. Hence the necessary and suffi- 
cient condition for parallelism is bja - b'/a', that is, ab' - a'b - 0. 

We have thus fallen upon the excepted case of §§ 4 and 5. 
If we assume that the results of the general formulae obtained 
for the case ab' - a'b # 0, namely, 

be' - b'c ca' - c'a 



x =zr> — I7i> y 



n i 



ab' - a'b' J ab' - a'b 

hold also when ab' - a'b = 0, we see that in the present case 

neither of the numerators be' - b'c, ca' - c'a, can vanish. For if, 

say, be' - b'c = 0, then - c/b = - c'/b', that is, OB = OB' ; and the 

two lines AB, A'B', already parallel, would coincide, which is not 

supposed. 

It follows, then, that 

be' - b'c ca' - c'a 

Z = — ^- = 00, y = _ -— =co: 

and the analytical result agrees with the graphical. 

II. Let L and L' be coincident, then the intersection becomes 
indeterminate. The conditions for coincidence are 



392 INDETERMINATE SOLUTION CHAP. 





OA = OA', OB = OB', 


whence 


-c/a = -c'/a', -c/b= -c'/b'. 


These give 


a b c 
a' = b' = c" 



which again give 

be' - b'c = 0, ca' - c'a = 0, ab' - a'b = 0. 

We thus have once more the excepted case of §§ 4 and 5, but 
this time with the additional peculiarit\ r that be' - b'c = and 
ca' - c'a ~ 0. 

If we assert the truth of the general analytical solution in 

this case also, we have 



* = o' ?/ = 0' 
that is, the values of x and y are indeterminate, as they ought to 
be, in accordance with the graphical result. 

§ 25.] Since three straight lines taken at random in a plane 
have not in general a common point of intersection, it follows 
that the three equations, 

ax + by + c = 0, a'x + b'y + c' = 0, a"x + b"y + c" = (1), 
have not in general a common solution. When these have a 
common solution their three graphic lines, L, L', L", will have a 
common intersection. We found the analytical condition for 
this to be 

ab'c" - ab"c + be a" - be" a + ca'b" - ca'b' = (2). 

In our investigation of this condition we left out of account the 
cases where any one of the three functions, ab' — a'b, a"b - ab", 
a'b" - a'b', vanishes. 

We propose now to examine graphically the excepted cases. 

First, we remark that if two of the functions vanish, the 
third will also vanish ; so that we need only consider (I.) the 
case where two vanish, (II.) the case where only one vanishes. 

I. ab' - a'b = 0, a'b - ab" = 0. 

This involves that L and L' are parallel, and that L and L" are 
parallel ; so that all three, L, L', L", are parallel ; and we have, 
in addition to the two given conditions, also a'b" - a'b' = 0. 



xvi EXCEPTIONAL SYSTEMS OF THREE EQUATIONS 393 

Hence, since the condition (2) may be written 

c(a'b" - a"b') + c'(a"b - ab") + c"(ab' - a'b) = 0, 

it appears that the general analytical condition for a common 
solution is satisfied. 

This agrees with the graphical result, for three parallel 
straight lines may be regarded as having a common intersection 
at infinity. 

In the present case is of course included the two cases where 
two of the lines coincide, or all three coincide. The corre- 
sponding analytical peculiarities in the equations will be obvious 
to the reader. 

II. ab' - a'b = 0. 

Here two of the graphic lines, L and L', are parallel, and the 
third, L", is supposed to be neither coincident with nor parallel 
to either. 

Looking at the matter graphically, we see that in this case 
the three lines cannot have a common intersection unless L and 
L' coincide, that is, unless 

a' = ka, b' = kb, c' = kc, 

where k is some constant. 

Let us see whether the condition (2) also brings out this 
result, as it ought to do. 

Since ab' - a'b = 0, 

we have - = _ = /• say. 

a b J 

Hence a' = ka, b' = kb. 

Now, by virtue of these results, (2) reduces to 

a"(be' - b'c) + b"(ca - c'a) = 0, 

that is, to 

a"(bc' - kbc) + b"(cka - c'a) = 0, 
that is, to 

(a"b - ab") (c' - kc) = 0, 

which gives, since a"b - ab" 4= 0, 

c - kc = 0, 
that is, c' = kc. 



394 EXERCISES XXVII chap. 

Hence the agreement between the analysis and the geometry is 
complete.* 

§ 26.] It would lead us too far if we were to attempt here 
to take up the graphical discussion of linear functions of three 
variables. We should have, in fact, to go into a discussion of 
the disposition of planes and lines in space of three dimensions. 

We consider the subject, so far as we have pursued it, an 
essential part of the algebraic training of the student. It will 
help to give him clear ideas regarding the generality and 
coherency of analytical expression, and will enable him at the 
same time to grasp the fundamental principles of the application 
of algebra to geometry. The two sciences mutually illuminate 
each other, just as two men each with a lantern have more light 
when they walk together than when each goes a separate way. 

Exercises XXVII. 

Draw to scale the graphs of the following linear functions of x : — 
(1.) y=x + l. (4.) y = 2x + 3. 

(2.) y=-x+\. (5.) y=-\x-\. 

(3.) y=-2E-l. (6.) y=-Z[x-\). 

(7.) Draw the graphs of the two functions, 3a; -5 and 5a; +7 ; and by 
n>eans of them solve the equation Sx- 5 = 5»+7. 

(8.) Draw to scale the contour lines of z = 2x-3y + l, corresponding to 

g=-2, z=-l, z = 0, z=+l, z=+2. 
(9. ) Draw the zero contour lines of z = 5x + 6y - 3 and z' = 8x - 9y + 1 ; and 
by means of them solve the system 

5x + 6y-3 = 0, 8a:-9y+l=0. 

* It may be well to warn the reader explicitly that he must be careful to 
use the limiting cases which we have now introduced into the theory of 
equations with a proper regard to accompanying circumstances. Take, for 
instance, the case of the paradoxical equation b = 0, out of which we manu- 
factured a linear equation by writing it in the form 0x + & = 0; and to which, 
accordingly, we assigned one infinite root. Nothing in the equation itself 
prevents us from converting it in the same way into a quadratic equation, 
for we might write it Ox 2 + Ox + b = 0, and say (see chap, xviii., § 5) that it 
has two infinite roots. Before we make any such assertion we must be sure 
beforehand whether a linear, or a quadratic or other equation was, generally 
speaking, to be expected. This must, of course, be decided by the circum- 
stances of each particular case. 



XVI 



EXERCISES XXVII 395 



Also show that the two contour lines divide the plane into four regions, 
such that in two of them (5x + 6y-Z) (8x-9y + l) is always positive, and in 
the other two the same function is always negative. 

(10.) Is the system 

3x-4y + 2 = 0, 6x-8y + 3 = 0, x-$y+l=0 
consistent or inconsistent ? 

(11.) Determine the value of c in order that the system 

2se+y-l=0, 4.r + 2y + 3 = 0, {c + 1)x + (c + 2)y + 5 = Q 
may be consistent. 

(12.) Prove graphically that, if ab' -a'b — 0, then the infinite values of x 
and y, which constitute the solution of 

ax + by + c-0, a'x + b'y + c' = 0, 
have a finite ratio, namely, 

x/y=(bc' - b'c)j(ca' - c'a). 

(13. ) If (ax + by + c)/{a'x + b'y + c') be independent of x and y, show that 
ab'-a'b-O, ca'-c'a = 0, bc'-b'c = 0; 
and that two of these conditions are sufficient. 

(14.) Illustrate graphically the reasoning in the latter part of § 5 of the 
preceding chapter. 

(15.) Explain graphically the leading proposition in § 6. 



CHAPTER XVII. 
Equations of the Second Degree. 

EQUATIONS OF THE SECOND DEGREE IN ONE VARIABLE. 

§ 1.] Every equation of the 2nd degree (Quadratic Equation) 
in one variable, can be reduced to an equivalent equation of 
the form 

ax 2 + bx + c = (1). 

Either or both of the coefficients b and c may vanish ; but 
we cannot (except as a limiting case, which we shall consider 
presently) suppose a = without reducing the degree of the 
equation. 

By the general proposition of chap, xii., § 23, when a, b, c 
are given, two values of x and no more can be found which 
shall make the function ax 2 + bx + c vanish ; that is, the equation 
(1) has always two roots and no more. The roots may be equal or 
unequal, real or imaginary, according to circumstances. 

The general theory of the solution of quadratic equations is 
thus to a large extent already in our hands. It happens, 
however, that the formal solution of a quadratic equation is 
always obtainable ; so that we can verify the general proposition 
by actually finding the roots as closed functions of the coefficients 
a, b, c. 

§ 2.] We consider first the following particular cases : — 
I. c = 0. 

The equation (1) reduces to 

ax 2 + bx = 0, 



chap, xvn BOOTS EQUAL AND OF OPPOSITE SIGN 397 

that is, since a 4= 0, 

az(z + ]|) = 0, 

which is equivalent to 

a: = 

x + - b = 
a 

Hence the roots are x = 0, x = - bfa. 

II. & = 0, e = 0. 
The equation (1) now reduces to 

ax x x = 0, 
which, since a 4 0, is equivalent to 

{:::}■ 

Hence the roots are x = 0, x = 0. This might also be deduced 
from I. 

Here the roots are equal. "We might of course say that there 
is only one root, but it is more convenient, in order to maintain 
the generality of the proposition regarding the number of the 
roots of an integral equation of the nth degree in one variable, 
to say that there are two equal roots. 

III. b = 0. 

The equation (1) reduces to 

ax 2 + c = 0, 
that is, since a =t= 0, to 

W-IX*-v/-;H 

which is equivalent to 

V a 



i 



-- = o 



a J 

Hence the roots are x= - s/(-c/a), x= + ^(-cja); that is, 
the roots are equal, but of opposite sign. If cja be negative, 



398 



GENERAL CASE 



CHAP. 



both roots will be real ; if c/a be positive, both roots will be 
imaginary, and we may write them in the more appropriate form 
x = - i \/(cJa), x = + i \/(cJa). 

§ 3.] The general case, where all the three coefficients are 
different from zero, may be treated in various ways ; but a little 
examination will show the student that all the methods amount 
to reducing the equation 



ax 2 + bx + c = 



(1) 



to an equivalent form, a(x + A) 2 + jm = 0, which is treated like the 
particular case III. of last paragraph. 

1st Method. — The most direct method is to take advantage 
of the identity of chap, vii., § 5. We have 

f -b + J(b 2 -Aac)) ( -b- J(l/-iac)) 

hence the equation (1) is equivalent to 

b+ J( b 2 -4:ac) ) ( - b - y/(6 2 - iac) } 

-6+ ^(b'-Aac) 



a-lx- 
that is, to 



x- 



\ 



X- 



-b 



2a 
sl(b 2 - Aac) 



2a 



= 



= 



>■ 



The roots of (1) are therefore {-b+ s /(b 2 - iac)}/2a, and 
{-h- J(b 2 -iac)}/2a. 

2nd Method. — We may also adopt the ordinary process of 
" completing the square." We may write (1) in the equivalent form 



x + ~x = — 
a a 



(2), 



and render the left-hand side of (2) a complete square by adding 
(b/2a) 2 to both sides. We thus deduce the equivalent equation 



\ X + 2a) ~ia* a' 



b 2 - 4«c 
4a* 



(3). 



xvn VARIOUS METHODS OF SOLUTION 399 

The equation (3) is obviously equivalent to 

/ fb 2 -Aac\ 



b 

2a 



i 



b I (b 2 - iac 



x + — = - 



y. 



2a V \ 4 a* 

from which we deduce 

x = { - b + */(!? - iac) }/2a, x={-b- K /(b 2 - iac) } /2a, 

as before. 

3rd Method. — By changing the variable, we can always make 

(1) depend on an equation of the form az 2 + d = 0. Let us 

assume that x = z + h, where h is entirely at our disposal, and z 

is to be determined by means of the derived equation. Then, 

by (1), we have 

a(z + Kf + b(z + h) + c = (4). 

It is obvious that this equation is equivalent to (1), provided x 
be determined in terms of z by the equation x = z + h. 
Now (4) may be written 

az 2 + (2ah + b)z + (ah 2 + bh + c) = (5). 

Since h is at our disposal, we may so determine it that 2ah 
+ b-0 ; that is, we may put h = - bj2a. The equation (5) 
then becomes 

'♦■(-sM-a)--* 

that is, az 2 — = (6). 

From (6) we deduce z = + J(b 2 - iac)/2a, z = - ^(b 2 - 4ac)/2a. 
Hence, since x = z + h = - b/2a + z, we have 

x = {-b+ s/(b 2 -iac)}j2a, x={-b- >J(b 2 - iac)} /2a, 

as before. 

In solving any particular equation the student may either 
quote the forms { - b ± *J(b 2 - iac)} /2a, which give the roots in 
all cases, and substitute the values which a, b, c happen to have 
in the particular case, or he may work through the process of 



az 



400 DISCRIMINATION OF THE ROOTS chap. 

the 2nd method in the particular case. The latter alternative 
will often be found the more conducive to accuracy. 

§ 4.] In distinguishing the various cases that may arise when 
the coefficients a, b, c are real rational numbers, we have merely 
to repeat the discussion of chap, vii., § 7, on the nature of the 
factors of an integral quadratic function. 

We thus see that the roots of 

ax* + bx + c = 0, 

(1) Will be real and unequal if b 2 - lac be positive. 

(2) Will be real and equal if b 2 - lac = 0. 

(3) Will be two conjugate complex numbers if b 2 - lac 
be negative. The appropriate expressions in this case are 
{-b + i s /(lac - b 2 )}/2a, {-b-i ^(lac - b 2 )}j2a. 

(4) The roots will be rational if b 2 - lac be positive and the 
square of a rational number. 

(5) The roots Avill be conjugate surds of the form A ± JB 
in the case where b 1 - lac is positive, but not the square of a 
rational number. 

(6) If the coefficients a, b, c be rational functions of any 
given quantities p, q, r, s, . . . then the roots will or will not 
be rational functions of p, q, r, s, . . . according as b 2 - lac is or 
is not the square of a rational function of p, q, r, .?,... 

It should be noticed that the conditions given as characterising 
the above cases are not only sufficient but also necessary. 

The cases where a, b, c are either irrational real numbers, or 
complex numbers of the general form a + a'i, are not of sufficient 
importance to require discussion here. 

Example 1. 

2jt 2 -3.7- = 0. 

By inspection we see that the roots are x = 0, x=3j2. 
Example 2. 

This equation is equivalent to # 2 + 4 = 0, whose roots are x = 2i, x= -2i. 

Example 3. 

35x 2 -2u;-l = 0. 
The equation is equivalent to 

2 _ 1 
X '-35 X ~T$' 



XVII 



EXAMPLES, EXERCISES XXVIII 



401 



that is, to 

Henca 

Hence 



(-& 



L\ 2 _1_ 1__36_ 
: 35 2 + 35~35 2 ' 



1 -+1 

* 35 35" 

1±6 
W 



x=- 



The roots are, therefore, + 1/5 and - 1/7. 

Example 4. 

a? ! -2a!-2=0. 

The roots are 1 + v'3 and 1 - \/3. 

Example 5. 

3a; 2 + 2ix + 48 = 0. 

The given equation is equivalent to 

a? + 8^ + 16 = 0, 
that is, to (a; + 4) 2 = 0. 

Hence x= - 4±0 ; that is to say, the two roots are each equal to - 4, 

Example 6. 

a; 2 -4a;+7 = 0. 
This is equivalent to 

a; 2 -4a; + 4=-3 ) 

that is, to (as-2) a =3**. 

Hence the roots are 2 + \jZi, 2 - \JZi. 

Example 7. 

a; 2 - 2{p + qfx + 2p* + 1 Iff + 2q i = 0. 

This equation is equivalent to 

{x - (p + q)*\ 9 = (p +q)* - 2p* - 12pY - 2q*, 
= -(p-q)\ 
= {p-q)H\ 

Hence the roots are (p + qf + {p-qfi, (p + q^-ip-qfi- 



Exercises XXVIII. 



(1.) a: 2 + a; = 0. 


(2.) 


(3.) (a:+l)(as-l)+l=0. 


(4.) 


(5.) (a;-l) 2 + (a--2) 2 = 0. 


(6.) 


(7.) p(x + a) 2 -q(x + p)* = 0. 


(8.) 


(9.) 2a,- 2 + 3a:+5 = 3a; 2 + 4a: + l. 


(10.) 


(11.) 255a; 2 - 431a: + 182 = 0. 


(12.) 


(13.) x*- 22a; +170 = 0. 


(14.) 


(15.) a? + 102a; + 2597 = 0. 


(16.) 


(17.) a? + 6\/7* + 55 = 0. 


(18.) 


(19. ) a; 2 + (23 + 12i)x + 97 + 137*' = 0. 


(20.) 


VOL. I 





(2a:-l)(3aj-2) = 0. 

(a:-l) 2 + 3(a;-l) = 0. 
.3(x-l) 2 -2(a:-2) 2 =0. 
{px + q) 2 +{qx+p) 2 = 0. 
ar> + 8a: 2 + 16a:-l = (a: + 3) 3 . 
4a,- 2 -40a; + 107 = 0. 
a; 2 -201a; + 200 = 0. 
a; 2 -4a; -2597 = 0. 
a.- 2 -2(l + V2)a; + 2V2 = 0. 
a.- 2 -(8-2i)a:=38i-31. 
2 D 



402 EQUATIONS REDUCIBLE TO QUADRATICS chap. 

(21.) («-l)(aj-2) + (aj-l)(a!-3) + (a!-2)(aJ-3)=0. 

(22.) (x-l) 3 + {x-l) 2 (x-2)-2{x + l) 3 = 0. 

(23.) (x-i)(x-i) + (x-i)(x-l) = 0. 

(24.) {x-a) 2 + (x-b) 2 = a 2 + b 2 . (25.) a 2 + 4aa;=(&-c) 2 + 4(&c-a 2 ). 

(26.) x 2 +(b-c)x=a? + bc + ca + ab. 



w^i-K^ + y|> 



(28.) (a + b)(abx 2 -2) = (a 2 + b 2 )x. 

(29.) (a-b)x 2 -(a 2 + ab + b 2 )x + ab(2a+b) = 0. 

(30.) (c + «-2Z>);z 2 +(a + &-2c),?+(& + c-2«) = 0. 

(31. ) {a 2 - ax + c 2 ) (a 2 + ax + c 2 ) = a 4 + aV + c 4 . 

(32. ) x- - 2(a 2 + b 2 + c 2 )x + a i + b i + c 4 + b 2 c 2 + <?a* + a 2 b 2 = 2abc(a + b + c). 

(33.) (b-c){x-a) 3 +{c-a)(x-b) 3 + (a-b){x-c) 3 = 0. 

(34.) Evaluate V(7 + V(7 + V(7 + V(7... ad oo... )))). 



EQUATIONS "WHOSE SOLUTION CAN BE EFFECTED BY MEANS OF 

QUADRATIC EQUATIONS. 

§ 5.] Reduction by Factorisation. — If we know one root of an 
integral equation 

/(a) = (1), 

say x = a, then, by the remainder theorem, we know that f(x) = 
(x - a)cj>(x), where <j>(x) is lower in degree by one than f(x). 
Hence (1) is equivalent to 

(2). 






tip) 

The solution of (1) now depends on the solution of <f>(x) = 0. It 
may happen that <f>(x) = is a quadratic equation, in which case 
it may be solved as usual ; or, if not, Ave may be able to reduce 
the equation <f>(x) — by guessing another root ; and so on. 

Example 1. 

To find the cube roots of - 1. 

Let x be any cube root of - 1, then, by the definition of a cube root, we 
must have x 3 — - 1. We have therefore to solve the equation 

3^ + 1 = 0. 

We know one root of this equation, namely, x= -1 ; the equation, in fact, is 

equivalent to 

(x + l)(x 2 -x+l) = 0, 

that is, to { . X + } = °A. 

{ x 2 - x + 1 = J 

The quadratic x 2 -x + l = 0, solved as usual, gives a;=(l±t\/3)/2. 



xvii INTEGKALISATION AND RATIONALISATION 403 

Hence the three cube roots of - 1 are - 1, (1 + i\/S)/2, (1 - i\jZ)j2, which 
agrees with the result already obtained in chap. xii. by means of Demoivre's 
Theorem. 

Example 2. 

This equation is obviously satisfied by x=l. Hence it is equivalent to 

{7x 2 -6x-3)(x-l) = 0. 
The roots of the quadratic 7x 2 - 6x- 3 = are (3±V30)/7. Hence the three 
roots of the original cubic are 1, (3 + V30)/7, (3 - V3lr)/7. 

It may happen that we are able by some artifice to throw an 

integral equation into the form 

PQR . . . = 0, 

where P, Q, R, . . . are all integral functions of x of the 2nd 

degree. The roots of the equation in question are then found 

by solving the quadratics 

P = 0, Q=0, R = 0, ... 

Example 3. 

p(ax 2 + bx + r) 2 - q(dx 2 + ex +/ ) 2 = 0. 

This equation is obviously equivalent to 

{^p(ax 2 + bx + c) + \Jq{dx 2 + ex+f)} {^p(ax*+bx + c) - *Jq(dx 2 + ex+f)} = 0. 

Hence its roots are the four roots of the two quadratics 

(a sjp + dsjq)x 2 + (b\Jp + c\Jq) x + {csjp +f\Jq) = 0, 
(a \/p - d\/q)x 2 + {b\Jp - esjq) x + (c\/p -f\/q) = 0, 
which can be solved in the usual way. 

§ 6.] Integralisation and Rationalisation. — We have seen in 
chap. xiv. that every algebraical equation can be reduced to an 
integral equation, which will be satisfied by all the finite roots of 
the given equation, but some of whose roots may happen to be 
extraneous to the given equation. The student should recur to 
the principles of chap, xiv., and work out the full solutions of 
as many of the exercises of that chapter as he can. In the exer- 
cises that follow in the present chapter particular attention 
should be paid to the distinction between solutions which are 
and solutions which are not extraneous to the given equation. 

The following additional examples will serve to illustrate the 
point just alluded to, and to exemplify some of the artifices that 
are used in the reduction of equations having special peculiarities. 



404 EXAMPLES chap. 

Example 1. 

1 =0. 



a- b 

If we combine the first and last terms, and also the two middle terms, we 
derive the equivalent equation 

2x 2x _ 

>+ .» ,_ ra =0- 



x 2 -[a + bf x'-{a-bf 
If we now multiply by {x~ - (a + b) 2 } {x 2 -(a- b) 2 } we deduce the equation 

2x{2x 2 -2(a 2 + b 2 )}=0; 
and it may be that we introduce extraneous solutions, since the multiplier 
used is a function of x. 

The equation last derived is equivalent to 

\x 2 -{a 2 + b-) = 0j- 
Hence the roots of the last derived equation are 0, + \/{a 2 + b 2 ), - \/{a 2 + b 2 ). 
Now, the roots, if any, introduced by the factor \x 2 -(a + b) 2 ) {x 2 -(a- b) 2 } 
must be ±(a + b) or ±(a-b). Hence none of the three roots obtained from 
the last derived equation are, in the present case, extraneous. 

Example 2. 

a-x a + x . i \ 

\/a+\J{a-x) \/a+^/{a + x) 
If we rationalise the denominators on the left, we have 

(g-ar){Va-V(a-g)} + { a + x){^a- s/ja + x)} _ , 

x -x ' '' 

From (^), after multiplying both sides by x, and transposing all the terms 
that are rational in x, we obtain 

[a + as)t -(«-«)?= 3a; \/a (7). 

From (7), by squaring and transposing, we deduce 

2a 3 -3ax 2 = 2(a 2 -x 2 )i (5). 

From (5), by squaring and transposing, we have finally the integral equation 

(4^-3« 2 )a^=0 (e). 

The roots of (e) are (repeated four times, but that does not concern us so far 
as the original irrational equation * (a) is concerned) and ±a\/3/2. 

It is at once obvious that x = is a root of (a). 

If we observe that s/(l±s/Z/2) = (s/3±l)/2, we see that ±«V 3 / 2 are roots 

of (a), provided 

2:pV3 2±V3 

xl + \/3 2±l + \/3 ' 



that is, provided 



2^1 + V^ 2±l + \/3 
2- V3 , 2 + V3 = 

l + Vs^s + vs ' 



which is not true. 

Hence the only root of (a) is x = 0. 

* For we have established no theory regarding the number of the roots of 
an irrational equation as such. 



xvil EXAMPLES 405 

Example 3. 

V(ffl + a:) \/(a-x) 

V« + V( re + x ) V a _ V(« ~ a:) 

By a process almost identical with that followed in last example, we deduce 

from (a) the equation 

4z*-3ah?=0 (/3). 

The roots of (/3) are 0, and ±a\/3/2 ; but it will be found that none of these 
satisfy the original equation (a). 

Example 4. 

V(2a; 2 - 4aT+ 1)+ V(*' 2 -5x+2)= V(2a? - 2a; + 3) + V(« 2 - 3a: + 4) (a). 
The given equation is equivalent to 

V(2a? - 4«+ 1) - V(>» 2 ~ 3a: + 4) = s/(2x 2 - 2x + 3) - V(a* - Bas + 2). 
From this last, by squaring, we deduce 



3a; 2 - 7x + 5 - 2V(2a: 2 - 4a; + 1) (a; 2 - 3a; + 4) 

= 3a; 2 - 7x + 5 - 2V(2ar ! - 2a; + 3) (a- 2 - 5a; + 2), 
which is equivalent to 

V(2a, 4 - 10a? + 21a? - 19a; + 4) = yj(2x* - 12a? + 17a? - 19a- + 6) (j8). 
From (/3), by squaring and transposing and rejecting the factor 2, we deduce 

a?+2a?-l=Q (7). 

One root of (7) is x — - 1, and (7) is equivalent to 

(«+l)(aj 2 +a!-l)=0. 
Hence the roots of (7) are - 1 and ( - 1 ±\/5)/2. 

Now x= -1 obviously satisfies (a). We can show that the other two 
roots of (7) are extraneous to (a) ; for, if x have either of the values 
(-l±\/5)/2, then x* + x-l = 0, therefore a; 2 = -a; + l. Using this value of 
a; 2 , we reduce (a) to \/{ - 6x + 3)=\J{ - 4a; + 5). This last equation involves 
the truth of the equation - 6a; + 3= - 4ar+ 5, which is satisfied by x= - 1, and 
not by either of the values x— ( - 1 ± \J5)/2. 

N.B. — An interesting point in this example is the way the terms of (a) 
are disposed before we square for the first time. 



Example 5. 



1 - \/(l - a; 2 ) __ 27 V(l+aQ + V(l-g ) (a)< 



1 + V(l - * 2 ) V(l +*) - V(i - x) 
Multiply the numerator and denominator on the left by 1 - \/(l -a: 2 ), and the 
numerator and denominator on the right by y/(l+x) - \/(l -x), and we ob- 
tain the equivalent equation 

(i_vr^ ) 2 _ x 

a? - 1-V(I-^ 2 )' 

Multiply both sides of the last equation by a-^l - Vl - x% and we deduce 

{l-sj(l-x*)} 3 = 27x 3 08). 



406 



EXERCISES XXIX, XXX 



CHAP. 



If 1, w, u 2 (see chap, xii., § 20) be the three cube roots of 4-1, then (/3) is 
equivalent to 

1- \/(l-x 2 ) = 3x 

l-V(l-ar»)=3«a! ! 

l-^/(l-x 2 ) = Z<J i x j 
By rationalisation we deduce from (7) the three integral equations 

10x 2 -6.r = (f 
{l + 9w 2 )a?-6wx = - (5). 

^l+QoO^-ear.x^O, 
The roots of these equations (5) are 0, 3/5 ; 0, 6w/(l +9w 2 ) ; 0, 6w 2 /(l +9w). 

The student will have no difficulty in settling which of them satisfy the 
original equation (a). 

Exercises XXIX. 

(1.) 1 X' 6 -1 = (X 2 + |) 2 (X 2 -1). 

(2.) x 3 -(a + b + c)x 2 -{a~ + b- + c--bc-ca-ab)x + a 3 + P+c $ -Babc=0. 

(3.) x 4 - 40a; + 39 = 0. 

(4.) x A + 2(a-2)x s +(a-2) n -x 2 + 2a 2 (a-2)x + a i =0. 

(5.) 2a 3 -a 2 -2a; -8 = 0. (6.) ax 3 + x + a + l = 0. 

(7.) ar*-3a^ + 4a; 2 -3a; + l = 0. 



x 2 . x . 1 P p 2 



p p 2 p 3 X 2 X 
(9.) a; 4 -6a,- 3 + 10a^-8x + 16 = 0. (10.) x i -6 = 5x(x 2 -x-l). 

(11.) (x 2 + 6x + 9) (x 2 + 8x + 1 6) = (a; 2 + 4a; + 4) (x 2 - 12a; + 36). 



(1.) ax = 2(l + l 



Exercises XXX. 



1 !\ 
x + -- T . 
a 0/ 



(2.) 



b+x a+x 



, n ,2x 2 -x-l 2a?-3a;-8 8a; 2 -8 
(3.) -ST-+- 



a;-2 



a;-3 



2a; -3 



9a; + 5 4a;-2 _ 12a; + 3 4a; + 3 11 
( ' 12 7x-l _ 16 7z + 9 48' 



(5.) a;-3 = 



a? -27 

a^ + 8 ' 



,. . ax + b ax + b 2ax + d b 

(6 ° -r + <^+b = ~2r- + -c 

ax + b bx + a_(a + b)(x + 2) 
cx + b cx + a 



(8 .) |5±5 + |"±»-5±? 

x - a x-b x-c 



cx + a + b 

n r 



.,+ 



bx 



lex 



, n . x-a x-b b 

9.) -=— + = - 

b a x-a x-b 



a?-a 2 ^x 2 -b 2 x*-c 2 
a 



XVII 



EXERCISES XXX, XXXI 407 



a-c b-c a + b-lc 



^ l0 '' 2b + x + 2a + x a + b + :>:' 

x + a x-a _ x 2 + a? 3 2 -« 2 

(1L) x~^a + x + a~x li -a- + x 2 + a^ 
(x - a) (x-b) _ (x -c)(x- d) 



(12.) 



x-a—b x- c—d 

a + 2x 



/ gt + ax + x- 
^ \a 2 -ax + x' 



a-2x 

( ' 2z 2 + 2a: + 3 x + 1 

x 74 



( 15 ') a 2 -2a -15 cc 2 + 2x-35 a; 2 + 10a; + 2l' 

2x+3a 2x-Sa_ a + b a-b 
^ ' 2x-3a + 2x+Sa~a-b a + b' 

K ''' (x-b){x-c) (x-c){x-a) {x-a){x-b) 



(1.) 



Exercises XXXI. 

x+\/x_ x(x-l) 
x- \/x 4 



^ v^T2) = ^ +2) + 2 ^- 

(4.) (a 2 + bx)s/{a? + c-) = (a 2 + be) V(a 2 + * ! ). 

(5.) 6xtx-l)-2 s /{Z{x-2)(x + l)-2(x-5)}=±{x+Z). 

(6. ) Viz + Vz) + V(* - V*) = aV*/V(* + Va). 

(7.) (l+x)V(l-a; 2 ) + (a : - 1 ) = - 

(8.) (aj-8)M^-6* + 8 6 ) = ( aj - 4 )M^-8» + 64). 

(9. ) (2a - a)/ V(«* ~ «* + « 2 ) = ( 2a; " & V V(-* 2 " hx + ^ 

wV(.4.H-y^)-y(£iO}- 

(ii.) vV 2 + 6a + 1) - V(3' 2 + to + 4 ) + V(* 2 + 6a " - s)=o. 

(12.) V(ff 2 + &x) + V(& 2 + a*) = 3(« + &). 

(13.) V {a(te- a 2 )/6} + V{ ft («* - b ')l a ) = a ~ h - 

(14.) ^/(a + x) + s/{b + x) = 2s/(a + b + x). 

Consider more especially the case where a=b. 
(15.) \/(x + i)-^(x-i) = ^(x-l). 
(16.) 2x s /{x i + a' 2 ) + 2x'sJ{x i + V i ) = a?-b-. 
(17. ) V(* 2 + 4x + 3) - V(* 2 + 3 * + 2) = 2(x+ 1). 

Two solutions, x- - 1 and another. 

(18. ) a 2 + a 2 + V(-<-' 4 + « 4 ) - 2 * V {■<■" + V(*- 4 + « 4 ) } • 



408 CHANGE OF VARIABLE chaP. 

(19.) x=\J{ax + x 2 -a\/(ax + x 2 )}. 

(oqn 7 12 1 6 

y ~ '' V(a:-6) + 4 V(«-6) + 9 V(a'-6)-4 + V( a; - 6 )-9 _ 
(21.) V(« 2 + a-' 2 ) + V(2a») = V(« 2 + 3oaj) + VC* 2 + So*). 
(22.j_._l 1 



\/(a + x)- \Ja \Ja + \/(a + x) \/(a + x) - \/{a - x) 

(23. ) V« + V(« + «) - V(« - *) = ^/(« 2 - * 2 )- 

(24. ) BiV(a+a!) + » V(« -«)= V(™ 2 + « 2 ) #(«* - * 2 )- 

(25. ) Rationalise and solve 2,\J{x -b-c) = \Jx. 

(26.) V{(^ 3 + a 2 )(^ + & 2 )}+x{V(a; 2 + a 2 )-\/( a;2 + &2 )}= : ^ 2 + a;2 - 
(27.) a + (a; + J)V{^ 2 + « 2 )/(a; 2 + i 2 )}=6 + (a: + a)v'{(a; 2 + i 2 )/(a; 2 + a 2 )}. 

§ 7.] Reduction of Equations by change of Variable. If we have 
an equation which is reducible to the form 

{_W +*{/(*)} +2«0 (a), 

then, if we put £ =/(%), we have the quadratic equation 

to determine £. Solving (/3), we obtain for £ the values 
{ -p ± J(p 2 - 4q)}/2. Hence (a) is equivalent to 



< 



-p-J(p 2 -4q) 



m 



1 



(?)• 



If the function f{x) be of the 1st or 2nd degree in x, the 
equations (y) can be solved at once ; and all the roots obtained 
will be roots of (a). 

Even when the equations (y) are not, as they stand, linear 
or quadratic equations, it may happen that they are reducible to 
such, or that solutions can in some way be obtained, and thus 
one or more solutions will be found for the original equation (a). 

In practice it is unnecessary to actually introduce the 
auxiliary variable £. We should simply speak of (a) as a 
quadratic in f(x), and proceed to solve for f(x) accordingly. 

Example 1. 

<_*>/- + _»*'- -12=0. 

We may write this equation in the form 

(a5*- , .)_+4(_:_'/e)_12__0. 



XVII 



EXAMPLES 409 



It may therefore be regarded as a quadratic equation in o'J'i. Solving, we find 

xp ;, j=+2, xi>''J=-6. 
From the first of these we have 

.rP = 2'l. 

Hence, if 1, w, or, . . ., w^- 1 be the pth roots of +1, we find the following p 

values for x : — 

2t'i\ oj2">'p, w 2 -2i'p, . . ., wP- 1 2i'p. 

In like manner, from xp : '< = - 6, we obtain, if q be even, the p values 

&P, w6«* w 2 6i'p, . . ., w*- 1 6*'8; 

and, if q be odd, the p values 

w'6«*. w' 3 6** s w' 5 6^ ( . . ., uPP-i&l*, 

where &>', w' 3 , . . . , w' 2 ^ -1 are the ^?th roots of - 1 . 

Example 2. 

x 2 + 3 = 2 V(^' 2 ~ 2a + 2) + 2z. 

This equation may be written 

a?-2sB+2-2VO^-2a!+2) + l=0 ; 

that is, 

{ VC-e 2 - 2a + 2) } 2 - 2 { VC* 3 - 2x + 2)} + 1 =0, 

which is a quadratic in \f(3? -2x + 2). 
Solving this quadratic we have 

x /(x 2 -2x + 2) = l. 
Whence ^-2* + 2 = 1, 

that is, (x--l) 2 = 0. 

The roots of this last equation are 1, 1, and x=l satisfies the original equation. 

Example 3. 

22* _ 3-2*4-2 + 32 = 0. 

We may write this equation as follows, 

(2*)2_ 12(2*) + 32 = 0; 
that is, (2*-4)(2*-8)=0. 

Hence the given equation is equivalent to 

\2*=8J 

The first of these has for one real solution x = 2; the second has the real 
solution x=3. 

Example 4. 

(x + a) (x + a + b) (x + a + 2b) {x + a + 3b) = c 4 . 

Associating the two extreme and the two intermediate factors on the left, 
we may write this equation as follows, 

{a; 2 + (2« + 3b)x + a(a + 36) } {sc* + {2a + %b)x + {a + b) (a + 26)} = c\ 



410 RECIPROCAL BIQUADRATIC chap. 

If ^ = x i + (2a + 3b)x + {w i + dab), the last equation may be written 

^+26 a )=c*; 

that is, ? + 2b- 2 Z + b i = b i + c i . 

Hence £= -&±s/(&+ct). 

The original equation is therefore equivalent to the two quadratics 

x- + (2a + 3b)x + a 2 + 3ab + b i =±\J{b i + c i ). 

§ 8.] Reciprocal Equations. — A very important class of equa- 
tions of the 4th degree (biquadratics) can be reduced to 
quadratics by the method we are now illustrating. 

Consider the equations 

ax* + bx 3 + ex 2 + bx + a = (1), 

ax 4 + bx 3 + ex 2 - bx + a = (I.), 

where the coefficients equidistant from the ends are either equal, 
or, in the case of the second and fourth coefficients, equal or 
numerically equal with opposite signs. Such equations are 
called reciprocal* 

If we divide by x 2 , we reduce (1) and (I.) to the forms 



B (* + ?) 


+ b(x + -j + c = 


(2), 


a ( x2+ ?J 


+ b(x--J +c = 


(II.) 


r alent to 






a(x + l) + b{ 


x + -J +c-2a = 


(3), 


«(«-!) + b[ 


x — ) + c+ 2a = 
\ xJ 


(III.) 



3 and III. are quadratics in x + 1 jx and x - l/x respectively. If 
their roots be a, /?, and y, 8 respectively, then (3) is equivalent to 




* If in equation (1) we write l/£ for x, we get an equation which is equiva- 
lent to a? + b? + c? + bS + a = Q. Hence, if f be any root of (1), l/£ is also a 
root. In other words, two of the four roots of (1) are the reciprocals of the 
remaining two. In like manner it may be shown that two of the roots of (I.) 
are the reciprocals of the remaining two with the sign changed. 



XViI GENERALISED RECIPROCAL BIQUADRATIC 411 

that is, to 



j X - aX + 1 = (^ ,.. 

|/-/fc+l=0j W " 

Similarly, III. is equivalent to 

The four roots of the two quadratics (4) or (IV.) are the roots 
of the biquadratic (1) or (I.) 

Generalisation of the Reciprocal Equation. — If we treat the 
general biquadratic 

ax* + bx 3 + ex 2 + dx + e = 

in the same way as we treated equations (1) and (I.), we reduce 
it to the form 

a { x 2 + — 5 ) -r b(x + j-) + c = 0. 
\ ax"/ \ ox/ 

Now, if e/a = cPjb 2 , this last equation may be written 

( d\ 2 J d\ n ad n 

a(x + E ) + b(x + -) + c-2 T = 0, 

which is a quadratic in x + djbx. 

Cor. It should be noticed that the following reciprocal equations 
of the 5th degree can be reduced to reciprocal biquadratics, and can 
therefore be solved by means of quadratics, namely, 

ax* + bx* + ex 3 ± ex 2 ± bx ± a = 0, 

where, in the ambiguities, the upper signs go together and the 
lower signs together. 

For the above may be written 

a(x> ± 1) + bx(x 3 ± 1) + cx\x ± 1) =0, 

from which it appears that either x + 1 orz-1 is a factor on 
the left-hand side. After this factor is removed, the equation 
becomes a reciprocal biquadratic, which may be solved in the 
manner already explained. The roots of the quintic are either 
+ 1 or - 1, and the four roots of this biquadratic. 



412 EXAMPLES CHaP. 

Ill an appendix to this volume is given a discussion of the 
general solution of the cubic and biquadratic, and of the cases 
where they can be solved by means of quadratics. 

Example 1. 

To find the fifth roots of +1. Let x be any fifth root of + 1 ; then x 5 = l. 
Hence we have to solve the equation 

^-1 = 0. 
This is equivalent to 

Xx^ + a^ + x^ + x + l-OJ' 
The latter equation is a reciprocal biquadratic, and may be written 

H) 2+ H)- 1=0 - 

After solving this equation for x + \jx, we, find 

1 1 + V5 1 1 - \/5 

x 2 x 2 

These give the two quadratics 

a?+ 1 J^6 a . +1=0j ^ + 11^ + 1 = 0. 

These again give the following four values for x : — 

- (1 + V5)/4±»V(10 - 2V5)/4, - (1 - V5)/4±*V(10 + 2V5)/4, 
these, together with 1, are the five fifth roots of + 1. This will be found to 
agree with the result obtained by using chap, xii., § 19. 

Example 2. 

(cc + «.) 4 + (a: + Z>) 4 =17(a-Z>) 4 . 

This equation may be written 

(z + «) 4 + (a- + &) 4 =17{(a; + a)-(.r + &)} 4 , 
from which, by dividing by (x + b)\ we deduce 

or £ 4 + l = 17(S-l) 4 , 

where £ = (» + a)/(x + b). 

This equation in £ is reciprocal, and may be written thus — 

Hence * + F = 2' 

, 1 7 • 
From this lust pair we deduce 



{=2, or £ ; and £ = - 



±iV(15) 



8 



xvii RATIONALISATION BY MEANS OF AUXILIARY VARIABLES 413 

Hence we have the four equations 

x + a_ x + a_, x + a_ 7±i\/(15) 
x~+b~' J ' aT+l> _ *' x~+~b~ < 8 ' 

From these, four values of x can at once be deduced. The real values are 
x = a-2b and x = b- 2a. 

§ 9.] By introducing auxiliary variables, we can always make 
any irrational equation in one variable depend on a system of 
rational equations in one or more variables. For example, if we 

have 

s'(x + a)+ J(x + b)+ s/(x + c) = d, 

and we put u = \/(x + a), v= sj(x + b), w= J(x + c), then we 
deduce the rational system 

u + v + w = d, n 2 = x + a, v 2 = x + b, w 2 = x + c. 
Whether such a transformation will facilitate the solution de- 
pends on the special circumstances of any particular case. The 
following is an example of the success of the artifice in question. 
Example. 

(a +aj)* + («-»)=&. 
We may write the given equation thus — 

(re + «)* + (re - xy j {(re +z) + (a- x)} *. 

Hence we deduce 



\a-xj /t) a \t la — x ) 



(2a) 
Let now y={(a + x)l(a-x)}* t 

we then have y + 1 = -Ay* + 1 )*. 

(2a)* 
From the last equation we deduce 

2a(y+l) i =V(y*+l), 

which is a reciprocal biquadratic, and can therefore be solved by means of 
quadratics. Having thus determined y, we deduce the value of x by means 
of the equation (re + x)/(a -x) = y i . 



Exercises XXXII. 

(1. ) a; 2 " 1 - x'"(b m + c m ) + b m c m = 0. 

(2.) e?*' 2 + qe~ 3xl ' 2 =p ; show that the sum of the two real values of x is 

° 7 2 

(3.) 2xH^)^= C ^f~(x 1 'i> + x 1 ^). 



4H EXERCISES XXXII chap. 

(4.) (9*)* - 2(3*)*3*+ 1 = B 2 *-^. 

(5.) **-£l+*+ '£*=*• 

(6.) (x + \) + l/(x + \) = f i. 

(7.) (l-x + x ! )l(l + a 2 -x) = (l + a 2 + x)/(l+x + x i ). 

(8.) 6x i + 5x 3 -Z8x 2 + 5x + 6 = 0. (9.) 6a^-31x 3 + 51a; 2 -31x + 6 = 0. 

(10.) 2x i -7x 3 + 7x 2 -7x + 2 = Q. (11.) 8x 4 - 42ar* + 29a: 2 + 42a; + 8 = 0. 

(12.) ax 3 + bx 2 + bx + a-0. (13.) rta^ + i>x 2 -6x-a=0. 

(14.) aa^ + 6ar + c = 0. (15.) ax* + bx 3 - bx - a = 0. 

(16.) a 2 x i + 2abx 3 + b 2 x 2 -c 2 = 0. (17.) a: 5 + l = 0. 

(18.) x 5 + 7x i + 9x 3 -9x 2 -7x-l = 0. 
(19.) 12a 5 + x 4 + 13a- 3 - 13a; 2 -x- 12 = 0. 

(20.) Show that the biquadratic ax i + bx 3 + cx 2 + dx + e = can be solved 
by means of quadratics, provided b/2a = iad/(iac- b 2 ). 
(21.) a; 1 + 10ar 3 + 22a; 2 -15a;+2 = 0. 

(22. ) x i + 2(p- q)x 3 + (p 2 + q 2 )x 2 + 2pq{p - q)x +pq(p 2 +pq + q 2 ) = 0. 
(23.) 3/(x 2 -7x + B)-2/(x 2 + 7x + 2) = 5. 

(24.) at(l+lX-{to?+x)=7Q. (25.) VU -o?) = \ + *J(l+x 2 )/x. 
(26.) N /(a?+l) + 4=5/V(« 2 +l)- (27-) (x + 5) i + (x + 5)~ h = 2. 

™ {fc:M^-{(i*-:)*'-»W- 

(29.) 2ar + 2 v / (z 2 + 4a;-5) = 4x 2 + 8a;-|-5. 
(30.) x 2 + 7x-3 = s/(2x 2 +Ux + 2). 

(31.) (,z-7) i +(z + 9)* + 2(ar + 2:c-63) i =70-2.r. 

(32. ) ^(x 2 +px + a) + \J(x 2 +px + b) + s/(x 2 +px + c) = 0. 

(33.) Show that the imaginary 7th roots of + 1 are the roots of 
x 2 - ax + 1 = 0, a; 2 - fix + 1 = 0, x 2 - yx + 1 = 0, where a, §, y are the roots 
of the cubic x 3 + x 2 - 2x - 1 = 0. 

(34.) ^ + J = a;V2\/(^ 4 -i)- (35.) 5(l + ar>)/(l -x 3 )= {(1 +x)/(l -x)} 3 . 

(36.) {a-x) 5 + (x-b) 5 = (a-b) 5 . (37.) v'a; + v^a; - 1 ) = \/(x + 1 ). 

(38.) (as + 3)(»+8)(a! + 13)(aj + 18) = 51. 



SYSTEMS WITH MORE THAN ONE VARIABLE WHICH CAN BE 
SOLVED BY MEANS OF QUADRATICS. 

§ 10.] According to the rule stated without proof in chap, 
xiv., § 6, if we have a system of two equations of the Zth and 
with degrees respectively in two variables, x and y, that system 
has in general Im solutions. Hence, if we eliminate y and 
deduce from the given system an equation in x alone, that equa- 
tion will in general be of the Imth degree, since there must in 



XVII MOST GENERAL SYSTEM HAYING TWO SOLUTIONS 415 

general be as many different values of x as there are solutions of 
the original systems. We shall speak of this equation as the 
Resultant Equation in x. 

In like manner, if we have a system of three equations of 
the /th, wth, and nth degrees respectively, in three variables 
x, y, z, the sj r stem has in general Imn solutions ; and the re- 
sultant equation in x obtained by eliminating y and z will be of 
the Imnth degree ; and so on. 

From this it appears that the only perfectly general case in which 
the solution of a system of equations will depend on a quadratic equation 
is that in which all the equations hit one are of the 1st degree, and 
that one is of the 2nd. 

It is quite easy to obtain the solution in this case, and thus 
verify in a particular instance the general rule from which we 
have been arguing. All we have to do is to solve the n-\ 
linear equations, and thereby determine n - 1 of the variables as 
linear functions of the nth. variable. On substituting these 
values in the nth. equation, which we suppose of the 2nd 
degree in all the n variables, it becomes an equation of the 
2nd degree in the nth variable. We thus obtain two values 
of the nth variable, and hence two corresponding values for each 
of the other n - 1 variables ; that is to say, we obtain two solu- 
tions of the system. 

Example 1. 

lx + my + n = (I), 

ax 2 + 2hxy + by- + 2gx + 2fy + c=0 (2). 

(1) is equivalent to 

Ix+n 
y =-- m - (3); 

and this value of y reduces (2) to 

am 2 x* - 2hmx{lx + n) + b(lx + rif + 2gm-x - 2fm(lx + n) + cm- = 0, 
that is, 

{am 2 - 2hlm + bP)x 2 + 2{gm' 1 - hmn + bnl -flm)x + {bn~ - 2fmn + c??i 2 ) = (4). 
The original system (1), (2) is therefore equivalent to (3), (4). Now (4) gives 
two values for x, and for each of these (3) gives a corresponding value of y. 
For example, the two equations 

3.c + 2y + l=0, x 2 + 2xy + y--x + y + Z = 0, 
will be found to be equivalent to 

y=-?x-\, ar-8;c+ll-0. 



416 SYSTEM IN TWO VARIABLES chap. 

Hence the two solutions of the system are 

x= 4 + V5, 4- s/5 ; 

2/=-¥-iV5, -V+fs/6- 

Example 2. 

3a;+2y-i8=l > x + y-3z = 2, x 2 + y 2 + z 2 =l. 
The system is equivalent to 

x=-5z-3, y = 8z + 5, 90z 2 + 1102 + 33 = 0. 
The solutions are 

the upper signs going together and the lower together. 

§ 11.] For the sake of contrast with the case last considered, 
and as an illustration of an important method in elimination, let 
us consider the most general system of two equations of the 
2nd degree in two variables, namely — 

ax* + 2hxy + by 2 + 2gx + 2fy + c = (1), 

ax 2 + 2h'xy + b'tf + 2g'x + 2fy + c' = (2). 

We may write these equations in the forms — 

bif + 2(hx +f)y + (ax 2 + 2gx + c) = 0, 
b'y 2 + 2(h'x +f')y + (a'x 2 + 2g'x + c') = 0, 
say by 2 +py + q = (1'), 

Vtf+ft + j-Q (2'), 

Avhere p = 2(hx +/), q = ax 2 + 2gx + c, &c. 

If we multiply (1') and (2') by b' and by b respectively, and 
subtract, and also multiply them by q' and by q respectively, and 

subtract, we deduce 

(pb'-p'b)y + (b'q-bq') = (3), 

(b'q-bq')y 2 +(p'q-pq')y = (4); 

and provided bq' - b'q^ 0, (3) and (4) will be equivalent to (1') 
and (2'). In general, the values of x which make bq'-b'q=0 
will not belong to the solutions of (3) and (4), nor will the value 
y - belong to those solutions. Hence we may say that, in 
general, the system 

{j>h'-p'b)y + (b'q-bq) = (3'), 

(b'q-bq')y + (p'q-pq') = (4'), 

is equivalent to (1') and (2'). 

Again, if we multiply (3') and (4') by b'q - bq and by pb' - p'b 
respectively, and subtract, we deduce 

(b'q-b q y-( P b'-p'b)(p'q~pq') = (5), 



XVII 



OF THE ORDER (2, 2) 417 



and, provided bq - b'q 4= 0, (4') and (5) will be equivalent to (3') 
and (4'). 

Hence we finally arrive at the conclusion that, in general, the 
system 
{b'(ax 2 + 2gx + c) - b(a'a? + 2g'x + c')} 2 - 4={b'(hx +/) 

- b(h'x +/)}{{h'x +/') (atc a + 2gx + c) - (hx +/) (a'af + 2g'x + c')} 

-0 (6), 
{b'(ax 2 + 2gx + c) - b(a'x 2 + 2g'x + c')}y 

+ 2{{h'x +/') (ax 2 + 2gx + c) - (Iix +/) (ax 2 + 2g'x + c')} 

= (7), 
is equivalent to (1) and (2). 

The first of these is a biquadratic giving four values for x, 
and, since (7) is of the 1st degree in y, for each value of x we 
obtain one and only one value of y. We have therefore four 
solutions, as the general rule requires. 

In general, the resultant biquadratic (6) will not be reducible 
to quadratics. It may, however, happen to be so reducible in 
particular cases. The following are a few of the more im- 
portant : — 

I. If, for example, b'/b =/'//= c'/c, then (6) reduces to 
x 2 [{b'(ax + 2g) - b(a'x + 2g')} 2 - i(b'h - bh'){(h'x + f) (ax + 2g) 

- (hx+f) (a'x + 2g') + (h'c - he')}] = 0, 
two of whose roots are zero, the other two being determinable 
by means of a quadratic equation. 

II. Again, if a' /a = b'/b = h'/h, it will be found that the two 
highest terms disappear from (6). Hence in this case two of its 
roots become infinite (see chap, xviii., § 6), and the remaining 
two can be found by means of a quadratic equation. 

III. If/=0, g = 0,f = 0, g' = 0, it will be found that only 
even powers of x occur in (6). The resultant then becomes a 
quadratic in x 2 . 

IV. The resultant biquadratic may come under the reciprocal 
class discussed in § 8 above. 

Most of these exceptional cases are of interest in the theory 
of conies, because they relate to cases where the intersection of two 
conies can be constructed by means of the ruler and compasses 
alone. The general theory is given in the Appendix to this vol. 

VOL. I 2 E 



418 EXAMPLES chap. 

Example 1. 
The system 3x 2 + 2xy + y 2 =l7, x 2 -2xy + 5y 2 = 5, 

is equivalent to 12.vy + 14a; 2 - 80 = 0, 73a; 4 -692a; 2 + 1600 = 0. 
The solutions of which are 

x=+2, -2, +20/V(73), -20/V(73); 

y=+l, -1, +1/V(73), -1/V(73). 
Example 2. 

n{x 2 + 2y) = 1(1 + 2xy), n(ij 2 + 2x) = m(l + 2xy). 
Here the elimination is easy, because the first equation is of the 1st degree 
in y. We deduce from it 

?ia; 2 - 1 
y ~2{lx-n)' 
This reduces the second equation to 

n(nx 2 - l) 2 + 8nx(lx-n) 2 = irn(lx-n) 2 +imx(lx-n) (nx 2 - I), 
which is equivalent to 

(n 2 - 4.lm)x i + 4(2P + mn)x* - 18nlx 2 + i(2n 2 + lm)x + (I 2 - 4»m) = 0. 
If n = l, this biquadratic is reciprocal, and its solution depends upon 

(Z-4m)f 2 + 4(2Z + m)£+ (8m -200=0, 
where £ = a; + l/a;. 

In general, if we have an equation of the 1st degree in x 
and y together with an equation of the nth. degree in x and y, 
the resultant equation in x will be of the wth degree. In par- 
ticular cases, owing to the existence of zero or infinite roots, 
or for other special reasons, this equation may be reducible to 
quadratics. 

Example. 

x + y = 18, a; 3 + y 3 =4914, 
is equivalent to 

y=lS-x, ar ! + (18-a;) 3 = 4914. 
The second of these two last equations reduces, as it happens, to 

a; 2 - 18a; +17 = 0. 
Hence the finite solutions of the given system are 

35=17,1; 
y=l, 17. 

§ 12.] A very important class of equations are the so-called 
Homogeneous Systems. The kind that most commonly occurs is 
that in which each equation consists of a homogeneous function 
of the variables equated to a constant. The artifice usually em- 
ployed for solving such equations is to introduce as auxiliary 
variables the ratios of all but one of the variables to that one. 
Thus, for example, if the variables were x and y, we should put 
y = vx, and then treat v and x as the new variables. 






xvii HOMOGENEOUS SYSTEMS EXAMPLES 419 

Example 1. 

x 2 + xy = 12, xy -2y 2 = l. 
Put y = vx, and the two equations become 

z 2 (l + fl) = 12, x 2 (v-2v 2 ) = l. 
From these two we derive 

x 2 (l + v)-\2x 2 {v-2ir) = 0, 
that is, 

a?{24t?-llt>+l}=0. 
Since x = evidently affords no solution of the given system, we see that the 
original system is equivalent to 

a?(l+v) = 12, 24b 2 -11v + 1 = 0. 
Solving the quadratic for v, we find v—l/S or 1/8. 

Corresponding to v=l/3, the first of the last pair of equations gives x 2 = 9, 
that is, x= ±3. 

Corresponding to v—1/8, we find in like manner x= ±4n/(2/3). 
Hence, bearing in mind that y is derived from the corresponding value of 
x by using the corresponding value of v in the equation y — vx, we have, for 
the complete set of solutious, 

*=+3, -3, +4V(2/3), -4V(2/3); 
y=+l, -1, +1/V6, -VV6. 

Example 2. 

x 2 + 2yz=l, y 2 + 2zx = m, z 2 + 2xy = n. 

Let x=uz, y — vz, then the equations become 

(u 2 + 2v)z 2 = l, {v 2 + 2u)z 2 = m, (l + 2uv)z 2 = n. 

Eliminating z, we have, since 2 = forms in general no part of any solution, 
n ( v? + 2v) = I (1 + 2uv), n {v 2 + 2u) = m (1 + 2wi>). 

"We have already seen how to treat this pair of equations (see § 11, 
Example 2). The system has in general four different solutions, which can 
be obtained by solving a biquadratic equation (reducible to quadratics when 
n = I). 

If we take any one of these solutions, the equation (1 +2icv)z 2 = n gives 
two values of z. The relations x = uz, y = vz, then give one value of a; and one 
value of y corresponding to each of the two values of z. 

We thus obtain all the eight solutions of the given system. 

There is another class of equations in the solution of which 
the artifice just exemplified is sometimes successful, namely, 
that in which each equation consists of a homogeneous function 
of the variables equated to another homogeneous function of the 
variables of the same or of different degree. 

Example 3. 
The system 

ox 2 + bxy + cy 2 = dx + ey, a'x 2 + b'xy + c'y 2 = d'x + e'y (1) 

is equivalent to 

(a + bv + cv 2 )x 2 ={d + ev)x, (a' + b'v + c'v 2 )x 2 = (d' + e'v)x (2) 

where y = vx. 



420 EXAMPLES chap. 

From this last system we derive the system 

x*{(a + bv + cv°) {d' + e'v)- (a' + b'v + c'v>)(d + ev)} =0\ ,„. 

£{(« + &u + cr 2 )a;-(d + er)}=0/ (6 >> 

which is equivalent (see chap, xiv., § 11) to (2), along with 

(a + bv + cv 2 )a? = (4), 

{d + ev)x = (5). 

If we observe that x=0, y — is a solution of the system (1), and keep 
account of it separately, and observe further that values of v which satisfy both 
(1) and (5) do not in general exist, we see that the system (1) is equivalent to 

(a + bv + cv 2 ) (d' + e'v) - {a' + b'v + c'v 2 ) (d + ev) = (6) 

along with (a + bv + ci?)x- (d + ev) = (7) 

and x = 0, y = 0. 

The solution of the given system now depends on the cubic (6). The 
three roots of this cubic substituted iu (7) give us three values of x, and y = vx 
gives three corresponding solutions of (1). Thus, counting x = 0, y = 0, we 
have obtained all the four solutions of (1). 

The cubic (6) will not be reducible to quadratics except in particular cases, 
as, for example, when ad' -a'd = Q or ce' - c'e — 0. 
For example, the system 

3a; 2 - 2xy + 3y 2 = x + 1 2y, 6x 2 + 2xy - 2y 2 -2x + 29y, 
is equivalent to x=0, y = 0, together with 

7>(llli> 2 - 86*;+ 8) = 0, (3 - 2v + 3v-)x= 1 + 12t>. 
The values of v are 2/3, 4/37, and 0. Hence the solutions of the system are 

x=0, 3, 185/227, 1/3; 
y=0, 2, 20/227, 0. 

§ 13.] Symmetrical Systems. — A system of equations is said to 

be symmetrical when the interchange of any pair of the variables 

derives from the given system an identical system. For example, 

x + y = a, x 2 + y 2 = b ; z 3 + y = a, y 3 + x -a ; 

x + y + z = a, x 2 + y 2 + z* — b, yz + zx + xy = c, 

are all symmetrical systems. 

There is a peculiarity in the solutions of such systems, which 
can be foreseen from their nature. Let us suppose in the first place 
that the system is such that it would in general have an even 
number of solutions, four say. If Ave take half the solutions, say 

X = a 1 , a 2 , 
V = Pn Am 

then, since the equations are still satisfied when the values of x 
and y are interchanged, the remaining half of the solutions are 

x = (3 n ft, 
y = a u a 2 . 



xvn SYMMETRICAL SYSTEMS 421 

If the whole number of solutions were odd, five say, then 
four of the solutions would be arranged as above, and the fifth 
(if finite, which in many cases it would not be) must be such 
that the values of x and y are equal ; otherwise the interchanges 
of the two would produce a sixth solution, which is inadmissible, 
if the system have only five solutions.* 

These considerations suggest two methods of solving such 
equations. 

1st Method. — Replace the variables by a new system of vari- 
ables, consisting of one, say x, of the former, and the ratios to it 
of the others, u, v, . . . say. Eliminate x, v, . . . and obtain 
an equation in u alone ; then this equation will be a reciprocal 
equation ; for the values of u are 

a l Pi a 2 P-2 o / 1 • 1 -i \ 

u = — , — , -r-, — , &c. (and, it may be, u = 1 ), 

Pi «i Pa a 2 

that is to say, along with each root there is another, which is its 
reciprocal. The degree of this resultant equation can therefore 
in all cases be reduced by adjoining a certain quadratic, just as 
in the case of a reciprocal biquadratic. 

2nd Method. — Replace the variables x, y, z, . . . by an equal 
number of symmetric functions of x, y, z, . . ., say by Sic, i>//, 
~2xyz, . . ., ifec, and solve for these. 

The nature of the method, its details, and the reason of its 
success, will be best understood by taking the case of two 
variables, x and y. 

Let us put u = x + y, v = xy. After separating the solutions, 
if any, for which x = y, Ave may replace the given system by a 
system each equation of which is symmetrical. We know, by 
the general theory of symmetric functions (see chap, xviii., § 4), 
that every integral symmetric function can be expressed as an 

* We have supposed that for all the solutions (except one in the case of 
an odd system) x + y. It may, however, happen that x = y for one or more 
solutions. Such solutions cannot be paired with others, since an interchange 
of values does not produce a new solution. This peculiarity must always 
arise in systems which are symmetrical as a whole, but not symmetrical in the 
individual equations. As an example, we may take the symmetrical system 
x 3 + y=a, y 3 + x-a, three of whose solutions are such that x = y. 



422 SYMMETRICAL SYSTEMS chap. 

integral function of u and v. Hence it will always be possible to 
transform the given system into an equivalent system in u and v. 

We observe further that, in general, u and v will each have 
as many values as there are solutions of the given system, and 
no more ; but that the values of u and v corresponding to two 
solutions, such as x = a u y = ($„ and x = /3 l} y = a l} are equal. 
Hence in the case of symmetrical equations the number of solu- 
tions of the system in u and v must in general be less than usual. 

Corresponding to any particular values of u and v, say u = a, 
v = (3, we have the quadratic system x + y = a, xy = fi, which gives 
the two solutions 

X = {a ± v/(a 2 - 4j8)}/2, y={a T */(a S - 4/?)}/2. 

If we had a system in three variables, x, y, z, then we should 
assume u = x + y + z, v = yz + zx + xy, w = xyz, and attempt to solve 
the system in u, v, w. Let u = a, v - j3, iv - y, be any solution of 
this system ; then, since 

(f-*)tf-y)tf-*)^-«f + «f-«i 

we see that the three roots of 

e-af' + /3$-y = 

constitute a solution of the original system, and, since the 
equations are symmetrical, any one of the six permutations of 
these roots is also a solution. In this case, therefore, the number 
of solutions of the system in u, v, to would, in general, be less 
than the corresponding number for the system in x, y, z. 

The student should study the following examples in the light 
of these general remarks : — 

Example 1. A (x- + y 2 ) + Bxy +C (x f y) + T> =0 \ 

A'(x o - + y"-) + B'xy + C'{z + y) + D' = 0) ( )- 

If we put y = vx, and then eliminate x by the method employed in §11, 
the resultant equation in v is 

{ (D'A) + (D'B)» + (D'AH- 2 } 2 = (D'C) (1 + vf { (C'A) + (C'B)« + (C'A)« 2 } (2), 

where (D'A) stands for D'A - DA', (D'B) for D'B - DB', and so on. 

The biquadratic (2) is obviously reciprocal, and can therefore be solved by 
means of quadratics. 

The solution can then be completed by means of the equation 

{(D'A) + (D'B)v + (D'A)^}x + (D'C)(l + v) = (3). 



i 



XVII 



EXAMPLES 423 



As an instance of this method the student should work out in full the 

solution of the system 

2(x 2 + y-) - 3xy + 2(x + y) - 39 = 0, 
3(x 2 + y°-)-ixy + (x + i/) - 50 = 0. 

"We may treat the above example by the second method of the present 
paragraph as follows. The system (1) may be written 
A(x + y) 2 +{B -2A)xy + C(x + y) + T> = 0, 
A'(x + y) 2 + (B' - 2A!)xy + C'(x +y) + D'=0 ; 

Am 2 +(B -2A)i- + Cm+D=0\ ... 

tnatis > AV+(B'-2A> + C'«+D'=0j v h 

Eliminating first u 2 and then v, we deduce the equivalent system 

(A'B)i?+(A'C)M + (A'D)=0\ 
(A'B)u 2 + { (C'B) - 2(C'A) J u + { (D'B) - 2(D'A) } =0 J 
where (A'B), &c, have the same meaning as above. 
The system (5) has two solutions, 

u = a, a', 

say, corresponding to which we find for the original system 



x 



— I 



;a±vV-4/3)}/2, {a'±V("' 2 -4/3')}/2, 



in all four solutions. 

This method should be tested on the numerical example given above. 

Example 2. x i + y i = 82, x + y=4. 

We have x i + y i =(x + ijf - 4xy(x 2 + y 2 ) - 6x 2 y 2 , 

= (x + yf - ixy { (x + yf - 2xy } - 6x 2 if, 
= u i -4u 2 v + 2v 2 . 
Hence the given system is equivalent to 

u 4 -4u*v + 2v 2 = 82, u=i. 
Using the value of u given by the second equation, we reduce the first to 

f 2 -32w + 87 = 0. 
The roots of this quadratic are 3 and 29. Hence the solution of the u, v 
system is u = 4, 4, 

v=S, 29. 
From x+y=i, xy = 29, we derive {'--y)-= -100, that is, x-y-±10i; 
combining this with x + y=4, we have se=2±5i, y = 2:f 5i. 
From x + y=i, xy = 2, we find x = S, y = l ; x=l, ?/ = 3. 
All the four solutions have thus been found. 

Example 3. x i =mx+ny, y 4 = nx + my (I}. 

Let us put y = vx; then, removing the factor x in both equations, and 
noting the corresponding solution, x = 0, y = 0, we have 

x 3 = m + nv, v i a? = n + mv. 
These are equivalent to 

x 3 =m + nv, v*(m + nr) = mv + n (2). 



424 EXAMPLES chap. 

The second of these may be written 

n(v°-l) + mv{v?-l) = (3), 

and is therefore equivalent to 

:" +i V + (5° i > + (5 +i > +i =°}' 

The second of these is a reciprocal biquadratic. Hence all the five roots of (3) 
can be found without solving any equation of higher degree than the 2nd. 

To the root v—1 correspond the three solutions, 

x = y = (m + m)"' 3 . u{ni + n) llS , to"(m + n) 1!3 , 
of the original system, where (ra + n) 1 ' 3 is the real value of the cube root, and 
w, or are the imaginary cube roots of unity. 

In like manner three solutions of (1) are obtained for each of the remain- 
ing four roots of (3). Hence, counting x = 0, y = 0, we obtain all the sixteen 
solutions of (1). 

The reader should work out the details of the numerical case 

x i = 2x + By, y i -3x + 2y, 
and calculate all the real roots, and all the coefficients in the complex roots, 
to one or two places of decimals. 

Example 4. yz + zx + xy = 26, 

yz(y + z) + zx(z + x) + xy(x + y) = 1 62, 
yz{if + z 2 ) + zx{z- + u?) + xy{x- + y 2 ) = 538. 
If we put u = x + y + z, v = yz + zx + xy, w = xyz, the above system reduces to 

i>=26, M0-3w=162, (u 2 -2v)v-uw = 538. 
Hence 26t*-3w=162, 26ic--mc = 1890. 

Hence 26m 2 +81m- 2835 = 0. 

The roots of this quadratic are it = 9 and u= — 315/26. 

We thus obtain for the values of u, v, w, 9, 26, 24, and - 315/26, 26, 
- 159. Hence we have the two cubics 

£ 3 -9£ 2 + 26£-24 = 0, 
£ 3 + 3^£ 2 + 26f + 159 = 0. 

Twelve of the roots of the original system consist of the six permutations 
of the three roots of the first cubic, together with the six permutations of the 
roots of the second cubic. 

The first cubic evidently has the root £ = 2 ; and the other two are easily 
found to be 3 and 4. Hence we have the following six solutions : — 

x=2, 2, 3, 3, 4, 4; 
y = 3, 4, 4, 2, 2, 3; 
a = 4, 3, 2, 4, ?, 2. 
Other six are to be found by solving the second cubic. 

§ 14.] We conclude this chapter with a few miscellaneous 
examples of artifices that are suggested merely by the peculi- 



xvii MISCELLANEOUS EXAMPLES 425 

arities of the particular case. Some of them have a somewhat 
more general character, as the student will find in working the 
exercises in set xxxiv. A moderate amount of practice in solv- 
ing puzzles of this description is useful as a means of cultivating 
manipulative skill ; hut he should beware of wasting his time 
over what is after all merely a chapter of accidents. 

Example 1. 

ax by (a + b)c 



a + x b + y a + b + c' 
Let a + x = (a 4- b + c)£, b + y = {a + b + c)-q ; 

the system then reduces to 

a 2 ^ + b"lr, = (a + b) 2 , £+ij=l. 
This again is equivalent to 

{(a + b)^a} 2 = 0, f+ij=l. 
Hence we have the solution £ = a/(a + b), r) = b/(a + b) twice over. 

The solutions of the original system are therefore x = ac/(a + b), y=bc/(a + b) 
twice over. 

Example 2. 

ax 2 + bxy + cy 2 = bx 2 + cxy + ay 2 =d (1). 

This system is equivalent to 

(a - b)x 2 + (b - c)xy + (c-a)y- = (2), 

ax 2 + bxy + cy 2 =d (3). 

The equation (2) (see chap, xvi., § 9) is equivalent to 

x 2 = (c<x + l) P , xy = (aa + l)p, y 2 =(ba + l) P (4), 

where p and <r are undetermined. 
Since x 2 y 2 = (xy) 2 , we must have 

(C(r + l)(6(j + l) = (ffcr + l) 2 . 

J) A- n 0/t 

Hence we deduce c = 0, a— — r 1 — r~ (5). 

' a- -be 

The first of these, taken in conjunction with (4), gives x = y ; and hence 

4- / *~ . 

x = y = ± A / i — > 

J V a+b+c 

that is to say, two solutions of (1). If we take the second value of a we find 
, 2 _pic-af p{c-a){a-i) „ p(«^) 2 (6) 

x ~ a 2 -bc' J ~ a? -be ' T ~ a? -be K >' 

where it remains to determine p/(a 2 - be). This can be done by substituting 
in (3). We thus find 

p/(a 2 - be) = d/(a 3 + ac 2 - ca 2 + ab 2 - a 2 b - abc). 

We now deduce from (6) 

±(c-a)<P- 



x = : 



y=±, &c. 



(a 3 + ac 2 - ca- + ab 2 - a 2 b - abc) 1 ■-' 
two more solutions of the original system. 

The system (2), (3) could also be solved very simply by putting y = vx, 
as in § 12. 



426 MISCELLANEOUS EXAMPLES chap. 



Exam 
These 


pie 3. 
equations 


give 


yz = 


-a 2 , zx = 
zxxxy 


■b 2 , xy- 

b 2 xc 2 

~ a 2 ' 

b 2 c 2 
~ a 2 ' 


--c 2 . 


t is, 


yz 





Hence x=±bc/a ; the two last equations of the original system then give 
y=±ca/b, z=±ab/c. The upper signs go together and the lower together ; 
so that we have only obtained two out of the possible eight solutions. 

Example 4. 

x{y + z) = a 2 , y(z + x) = b 2 , z(x + y)=--c 2 . 

This can be reduced to last by solving for yz, zx, xy. 

Example 5. 

x{x + y + z) = a 2 , y(x + y + z) = b 2 , z(x + y + z) = c 2 . 

Let x + y + z — p. Then, if we add the three equations, we have 

p 2 = a 2 + b 2 + c 2 . 
Hence p = ± \/(« 2 + b 2 + c 2 ) ; and we have 

±a 2 ±b 2 ±c 2 



X — ///v - 2 i 12 , »2\l V~ 



«J(a 2 + b 2 + c 2 )' J ^{a 2 + b 2 + c 2 Y V^' + ^ + c 2 )' 

Example 6. 
To find the real solutions of 

a* + tf+p =t p (1), ${y+z)+tf=bc (4), 

tf' + ?- + e = b 2 (2), v{z + x) + t t =ca (5)) 

z 2 + e + T = c 2 (3), fte+y)+|ij=a5 (6). 

From (2), (3), and (4) we deduce 

U(y+s)+i?f} 2 - w+f+ewz'+e+v^^o ; 

that is, (l 2 -S») 8 +(^-f2) a +(i?-w) 3 =0 (7). 

Every solution of the given system must satisfy (7). Now, since (£ 2 -yz) 2 , 
(£77 -zf) 2 , (ft-yy) 2 are all positive, provided x, y, z, £, 77, f be all real, it 
follows that for all real solutions we must have ^ 2 — yz, §J7=fo ff = 777/. 

Hence, from the symmetry of the system, we must have 

£ 2 = yz, ■ n 2 = zx, £ 2 =xy, (8), 

— £ '-?• 8 =T (9) - 

By means of (8) we reduce (1), (2), (3) to 

x(x + y + z) = a 2 , y(x + y + z) = b 2 , z(x + y + z) = c 2 . 
Hence, by Example 5, we have 

±a 2 _ ±b 2 _ _±f 

X ~^(a 2 + b 2 + c 2 y y ~^(a 2 + b 2 + c 2 )' Z ~^/(a 2 + b 2 + c 2 Y 
From (8) we now derive 

zkbc zkca ,__ ±«£> 

* "~ . //_2 1 M 1 -2\' V ~~ . J7Z.2 , la . .-2\' i — 



V(a 2 + * 2 + c 2 )' ''-^/(a? +&+<?)' s V(a 2 + ^ + 0' 



xvil MISCELLANEOUS EXAMPLES 427 

If we take account of (4), (5), (6) we see that the upper signs must go to- 
gether throughout, and the lower together throughout ; so that we find only 
two real solutions. 

Example 7. 

x(x-a)=yz, y(y-b) = zx, z(z-c) = xy (1). 

From the first two equations we derive (x -y) (x + y + z) = ax - by, which, 
if we put p = x + y + z, may he written (p-a)x=(p-b)y. Hence, bearing in 
mind the symmetry of the system, we have 

*= — i y= — i> *= — ( 2 )> 

p-a " p-b p-c 

where p and a have to he determined. 
From the first equation of (1) we have 

<x f cr 



p-a\p-a ) {p~b)(p-c) 
Removing the factor <r, to which will correspond the solution x — y = z=0, 
we find 

o- { (2a - b - c)p + (be - a") }=a(p~ a)(p -b)(p- c) (3). 

Similarly we find 

<r{(2b-c-a)p+(ca-b"-)}=b(p-a)(p-b)(p-c) (4). 

From (3) and (4) we now eliminate <x, observing that in the process we 
reject the factors <r, p - a, p-b, p-c, which correspond to three solutions, 
namely, 

x = a, 0, ; 
y=0, b, 0; 
2 = 0, 0, c. 

We thus deduce p = . , 

r a+b+c 

which gives one more solution. We have in fact p-a = (bc-a")jZa, 
p-b = {ca- b*)/2a, p-c = (ab- c 2 )IZa. 
Hence (2) gives 

<x _(ca-b-)(ab-c-) 
p-a Babe - 2a 3 ' 

and, by symmetry, we have two corresponding values for y and z. 

This example is worthy of notice on account of the symmetrical method 
which is used for treating the given system of equations. The solution might 
be obtained fully as readily by putting x = u~, y = vz, and proceeding as in 
§ 13, Example 3. 

Exercises XXXIII. 

(1.) x + y = 30, xy = 2\Q. (2.) x-y=3, x" + if = 65. 

(3.) x-+if- = 5S, xy = 1\, (4.) x + y = 8, 3x"-2xy + y' i =5i. 

(5.) x + 2y = x\ 2x + y = y 2 . 

(6.) x- + y- + 2(x + y) = U, Zxy = 2(x + y). 

(7.) x~ + y~ — a~, x + y = b. 



428 



EXERCISES XXXIII 



CHAP. 



and 



(8. 

(9. 

10. 

11. 

the 
(12. 
(14. 
(15. 
(16. 
(17. 
(18. 
(19. 

(20. 

(21. 

(22. 
(23. 
(24. 
(25. 
(26. 
(27. 
(28. 
(29. 
(30. 
(32. 
(33. 
(35. 
(36. 
(37. 
(38. 
(39. 
(40. 
(41. 
(42. 
(43. 
(44. 
(45. 
(46. 

(47. 
(48. 

(49. 
(50. 

(51. 

(52. 



a(x- + y 2 ) =px - qy, b(x 2 + y 2 ) = qx - py. 
(x + y)/(l+xy)=a, {x-y)/(l ~xy) = b. 
ax + by = e, b/x + a/y = d. 

It ax + by = 1, ex 2 + dy 2 = 1, have only one solution, then arjc + b 2 /d = 1, 
solution in question is x = a/c, y—b/d. 

2x 2 -%xy = \, y 2 + 5xy = 34. (13.) x 2 +xy = 8i, xy + y 2 = 60. 

x 3 +4xy + y 3 = 38, x + y = 2. 
l/x 2 + l/xy = 1/a 2 , l/y 2 + l/xy = l/b 2 . 
(px + qy) (x/p + y/q) = x 2 + y 2 +p 2 + q 2 , xjp + yjq = V5. 
x 2 + a 2 = y 2 + b 2 = {x + yf + (a - bf. 
(x-yf = a 3 (x + y), {x + y) 2 = b 3 {x-y). 
(a 2 - b 2 )/(x 2 + y 2 ) + (a 2 + b^/ix 2 -y 2 ) = l, 

3 



x 2 jp 2 -y 2 /q 2 = 0. 



x+Z ,y-3_ 
x-3 + y+3~ ' 



x-Z w - 



; = 1. 



2x+3 2y + 3 
2(x-y)+xy = Bxy-(x-y) = 7. 
(x + y)/7 = 8/(x + y + l), xy=12. 
x+l/y=10/x, y + l/x = l0x. 
Z{x 2 + y 2 ) - 2xy=27, 4(x 2 + y 2 ) - 6xy = 16. 
cc 3 -?/ = 208, x-y=L 
x 2 y + xy 2 = 162, a? + y 3 = 243. 
x 2 y + xy 2 = 30, x i y 2 + x 2 y 4 =468. 
x 3 + y 3 = (a + b)(x-y), x 2 + xy + y 2 = a-b. 
x i + x 2 if + y*=741, x 2 -xy + y 2 = 19 
xy(x + ij) = 48, x 3 + y 3 = 72. 
x i + y i = Q7, x + y=5. 



(31.) x 4 + y 4 = a 4 , x + y = b. 



x i + y i = {p 2 + 2)x 2 y 2 , x + y = a. 
x 2 -y 2 = 2xy + x + y, x 3 -y 3 = 3xy(x + y). 
(x + y) {x 3 - y 3 ) = 819, (x - y) {x 3 + y 3 )= 399. 
x 2 \y + y 2 \x = 2, x + y =5. 



(34.) x i + y 5 = 33, x + y=Z. 



x 2 y 2 (x i -y i )=a' 
x i -x 2 + y i -y 2 - 



xy(x i + y 4 ) (x 2 - y 2 ) = a. 
84, x 2 + x 2 y 2 + y 2 = 49. 



y s /x=b 2 -xy. 

i/2 



x 3 /y = a 2 -xy, 

x + y+ \J(xy) = 14, x 2 + y 2 + xy = 84. 
V(l-«A') + V(l-«/2/)=\ / (l+«/^)> x + y = b. 
x+y+ VO* 2 - 2/ 2 ) = a, 2ysJ{x 2 - y 2 ) = V\ 
*J(x 2 + l2y)+sJ(y 2 + l2x) = 33, x + y = 23. 

V(*/y) + V(y/*) = 5/2, V(*W + V(s/7*) = 9 V2/2. 

\/{x + a) + V(2/ - «) = W a > s/( x ~ a ) + \/(y -a) = f \/a. 
X s + y • =a~, 



(x 2 + y 2 ) h + (2xyf = b. 



a?+a?+y*+b*=>s l /2{z( L a+y)-b{a-y)}, 

x 2 -a 2 -y 2 + b 2 ='s/2{x(a-y) + b{a + y)\. 
(x 2 + a 2 ) {y 2 + b 2 ) = m(xy + ab) 2 , {x 2 - a 2 ) (y 2 - b 2 ) = mJifix - ay) 2 . 

b b b 



x= y + ^rr z , rr 



a a a 

y+ y+ y + 

gmyn—^ym-n 

x z +y=y 4a , y 



y = x + 



x+ x+ x + 



'=uy 



*+: ' = :»•« 



xvii EXERCISES XXXIV 429 



Exercises XXXIV. 

*(1.) 2e=0, Saaj=0, Zah?=ZIL{b-c). 

(2.) (y-a){z-a) = bc, (z-b)(x-b) = ca, (x-e){y -c) = ab. 
(3.) yz+2(y+z)=U, zx+2{z+x)=8, xy+2(x+y)=16. 

(4.) ^ + ^+; = 4 > yz+xs+xy=tyxyz, 2zx + 3yz=2xy. 

(5.) z(j/ + 2) = 24, y{z + x) -18, z(x + y) = 20. 
, (6.) a<y+«)=y(«+a;)=a(a!+y)=l. 

(7.) ( 2 + a:)(a; + 2/) = « 2 , (a; + y) {y + z) = b 2 , (y + z)(z + x) = c 2 . 
(8.) ax + yz=ay + zx=az + xy=p 2 . 
(9.) a; 2 + 2?/2 = 128, y 2 + 2zx=15B, z 2 + 2xy=128. 
(10.) a 2 (j/ + 2) 2 = a 2 a; 2 + l, 6 2 (2 + cc) 2 = &V + l, c 2 (x + y) 2 = c 2 z 2 + 1 . 
(11.) a(i/ + z-a;) = (a! + i/ + z) 2 -2&2/, b(z + x-y) = (x + y + zf-2cz, 

c(x + y-z) = (x + y + z) 2 - 2ax. 
(12. ) ZOc 2 - v/2) - 2(a; 2 - yz) = a 2 , ^(x 2 - yz) - 2{y 2 -zx) = b 2 , 

2(x 2 -yz)-2(z 2 -xy) = c 2 . 
(13.) , Zbcx=0, 2aijz = Q, 2a; 2 =1. 

(14.) a(x + yz) = b(y + zx) = c(z + xy), x 2 + y 2 + z 2 + 2xyz = l. 
(15.) x(a + y + z)=y(a + z + x) = z(a + x + y) = 3a(x + y + z). 
(16.) a; 2 + ?/ 2 + 2 2 =a 2 + 2a;(?/4-2)-a; 2 , and the two equations derived from 

this one by interchanging -{ J- . 

(17.) aa?=-+- t by 2 = ~~~, cz 2 =- + -. 
v ' y z' " z x x y 

(18.) y*z*+z*x*+xhj*=i9, x 2 + if + z 2 = li, x{y + z) = 9. 

(19. ) (yz - **)/a?x=(zx - y 2 Wy= {xy - z 2 )/Sz = l/xyz. 

(20.) « 2 .c 2 (2/ + 2) 2 =(a 2 + a; 2 )2/ 2 2 2 , and the two derived therefrom by inter- 



changing-^}. 



(21.) 2x 3 = a{2x-2x) = b(2x-2y) = c(2x-2z). 

(22.) (x-l)(y+z-5)=77, (y-2)(z+x- 4)=.72, (z-3){x+y-3)=65. 

(23.) u{y-x)/(z-u)=a } z(y-x)/{z-u) = b, y(u-z)j{x- y) = c, 

x(u-z)/(x-y) = d. 
(24. ) If ar 3 + ^ + z 3 + faeyz = a, 3(y 2 z + z 2 x + x 2 y) = b, 3(yz 2 + zx 2 + xy 2 ) = c, 
show that 

x + y + z = (a + b + c) , x + wy + uPz = (a + wb + arc) , 

X + u 2 y + uz = (a + u 2 b + wc) , 

where w 2 + w + l = 0. Find all the real solutions when a— 72, b = 75, c=69. 

t C\£ \ 9 O O JO o o 

(25.) xr-yz — a-, y- -zx = b", z- -xy = c-. 

* In this set of exercises 2 and II refer to three letters only ; and 11(6 -c) 
stands for (b-c)(c-a)(a-b), and not for (b - c) (c - a) (a - b) (c - b)(a -c)(b~a), 
as, strictly speaking, it ought to do. 



430 EXERCISES XXXV CHAP, xvn 

Exercises XXXV. 
Eliminate * 

(1.) x from the system 

ax + b _ a'x + V _ a"x + b" 

cx + d~c'x + d'~<7x + d"' 
(2. ) x and y from 

Ix my a + b al bm _a-b x 2 y 2 _ 
a b ~ a - b x y a+b a 2 b 2 
(3.) x and y from 

x 2 + xy = a 2 , y 2 + xy = b 2 , x 2 + y 2 — c 2 . 
(4.) x, y, z from 

x(y + z)=a 2 , y{z + x) = b 2 , z(x + y)=c 2 , xyz = d\ 

(5.) x, y, z from 

x y z . a. . b, . c, . 
x + y + z=0, - + |+-=0, -(x-p)=-(y-l)=-{*-r). 

(6.) x, y, z from 

2Az 2 =0, 2A'x 2 = 0, 2aa:=0, 

and show that the result is 

21/ {b 2 (GA') + c 2 (AB') - « 2 (BC) } = 0, 

where (CA')-CA' - C'A, &c. 

(7.) Show that the following system of equations in x, y, z are inconsistent 

unless r 3 - p 3 = Zrgp, and that they have an infinite number of solutions if this 

condition be fulfilled. 

Skc 3 - Zxyz =p 3 , "Zyz = q 2 , 2z = r. 

Eliminate 
(8.) x and y from 

{a - x) (a - y) =p, {b-x){b-y) = q, (a-x)(b-y)/{b-x)(a-y)=c. 
(9.) x, y, z from 

x + y -z = a, x 2 + y 2 -z 2 = b 2 , x 3 + y 3 -z 3 — c 3 , xyz — a 3 . 
(10.) x, y, z from 

ax + yz = be, by + zx = ca, cz + xy — ab, xyz = abc. 

(11.) x, y, z from 

2a; 2 =p 2 , 2,3? = q 3 , 2x 4 = ?- 4 , xyz = s i . 
(12.) x, y, z from 
(x + a)(y + b){z + c) = abc, (y-c)(z-b) = a 2 , (z-a){x-c) — b 2 , {x-b){y-a) = c 2 . 

(13.) The system 

X1X2 + y 1 y 2 = h i , x& z + y 2 y 3 = kf, . . ., x„x 1 + y„y 1 = k, 2 , 
osi 2 + yi 2 = x- 2 2 + y.2 2 = . . .=x n 2 + y n 2 = a 2 , 
either has no solution, or it has an infinite number of solutions. 

* The eliminant is in all cases to be a rational integral equation. ' 



CHAPTEE XVIII. 

General Theory of Integral Functions, more 
particularly of Quadratic Functions. 

RELATIONS BETWEEN THE COEFFICIENTS OF A FUNCTION AND ITS 
ROOTS — SYMMETRICAL FUNCTIONS OF THE ROOTS. 

§ 1.] By the remainder theorem (chap, v., § 15), it follows 
that if a,, a 2 , . . ., a n be the n roots of the integral function 

p X n +p l X n ~ 1 +p^ n ~ 2 + . . . +Pn-\ X + Pn (1)> 

that is to say, the n values of x for which its value becomes 0, 
then we have the identity 



PoX n + p t x n ~ l + p.p: n " L + . . . + p n 

=jp (x - a x )(x - 03) . . . (x-a n ) (2). 

Now we have (see chap, iv., § 10) 
(X - a,)(x - og) . . . (X - a n ) = X n - P^" 1 + P 2 Z»- 2 -... + (- l)»P n> 
where P t , P 2 , . . ., P n denote the sums of the products of the n 
quantities a,, a 2 , . . ., a n , taken 1, 2, . . ., n at a time re- 
spectively. Hence, if we divide both sides of (2) by p , we have 
the identity 

x n + Pl x n-l + P* x n-2 + , _ + Pn 
Pa Pa Pa 

=x n -T l x n - 1 + T^ n - 2 -. . . + (-l)' l P n (3). 
Since (3) is an identity, we must have 

p l /p,= -P„ Wl'o = P 2 , • • -, Pn/Pa = ( ~ 1) B P« (4). 

In particular, if p = 1, so that we have the function 

x n +p l 3-7 l ~ 1 +p. 2 x n - 2 ' + . . . + p,i (5), 

then i» 1 =-P lJ #, = P 8 , . • ., i?» = (-l) n P« (6). 



432 SYMMETRIC FUNCTIONS OF TWO VARIABLES chap. 

Hence, if we consider the roots of the function 

x n + p& n ~ l + p# n " 2 + . . .+p n _ lX + p n , 
or, ivhat comes to the same thing, the roots of the equation 
x n +p 1 z n - 1 +px n - 2 + . . .+p n . 1 x+p n = 0, 
then -pi is the sum of the n roots; p a the sum of all the products of 
the roots, taken two at a time; -p 3 the sum of all the products, taken 
three at a time, and so on. 

Thus, if a and f3 be the roots of the quadratic function 
ax 2 + bx + c, that is, the values of x which satisfy the quadratic 
equation ax 2 + bx + c = 0, then 

a + (3= - b/a, a/3 = cja (7). 

Again, -if a, /3, y be the roots of the cubic function ax 3 + bx 2 
+ c x + d, then 

a + (3 + y = - b/a, {3y + ya + af3 = c/a, a/3y= - d/a (8). 

§ 2.] If s 1} s 2 , s 3 , . . ., s r stand f m- the sums of the 1st, 2nd, 
3rd, . . . , rth powers of the roots a and f3 of the quadratic equation 

x 2 +p 1 x+p 2 = (1), 

we can express s„ s,, . . ., s r as integral functions of p x and p 2 . 

In the first place, we have, by § 1 (6), 

s 1 = a + /3= - Pl (2). 

Again 

S 2 = a 2 + (3 2 = (a + (3) 2 -2a/3, 

=1K - 2p 2 (3). 

To find s 3 we may proceed as follows. Since a and (3 are 
roots of (1), we have 

a 2 +^ ia +^ = 0, {3 2 +p,{3+p 3 =0 (4). 

Multiplying these equations by a and [3 respectively, and adding, 
we obtain 

s 3 +pA+p.A = (5). 

Since s t and s 2 are integral functions of p x and p 2 , (5) determines 
5 3 as an integral function of p l and p,. We have, in fact, 
h = -lh{p;-2p 2 )+p 2 p„ 
= ~Px + 3p>P* (6). 



XVI 1 1 



SYMMETRIC FUNCTIONS OF. TWO VARIABLES 433 



Similarly, multiplying the equations (4) by a 2 and (? respectively, 

and adding, we deduce 

S4+P1S3 +#&=() (7). 

Hence s 4 may be expressed as an integral function of p x and p a , 
and so on. 

We can now express any symmetric integral function whatever of 
the roots of the quadratic (1) as an integral function of |>, and p.,. 

Since any symmetric integral function is a sum of sym- 
metrical integral homogeneous functions, it is sufficient to prove 
this proposition for a homogeneous symmetric integral function 
of the roots a and ft. Tbe most general such function of the 
? - th degree may be written 

A(a' + ft r ) + Ba/3(a'- 2 + jS*" 2 ) + Ca 2 /3V~ 4 +^" 4 ) + . . 

that is to say, 

As r - Bp 2 s,._ 2 + Cp 2 2 s,._ 4 + . . . (8), 

where A, B, C are coefficients independent of a and ft. 

Hence the proposition follows at once, for we have already 
shown that s r , s r .,, s r _ 4 , . . . can all be expressed as integral 
functions of p y and p. z . 

It is important to notice that, since a and ft may be any 
two quantities whatsoever, the result just arrived at is really a 
general proposition regarding any integral symmetric function of 
two variables, namely, that any symmetric integral function of two 
variables a, ft can be expressed as a rational integral function of the two 
elementary symmetrical functions p x = - (a + ft) and p. z = aft. 

There are two important remarks to be made regarding this 
expression. 

1st. If all the coefficients of the given integral symmetric function 
be integers, then all the coefficients in the expression for it in terms of 
l> i and p., will also be integers. 

This is at once obvious if we remark that at ever)' step in 
the successive calculation of s ly s 2 , s 3 , . . ., &c, we substitute 
directly integral values previously obtained, so that the only 
possibility of introducing fractions would be through the co- 
efficients A, B, C, ... in (8). 

VOL. I 2 F 



434 EXPRESSION IN TERMS OF p 1 AND p 2 chap. 

2nd. Since all the equations above written become identities, 
homogeneous throughout, when for p x and p g we substitute their 
values - (a + /3) and a/3 respectively; and since ^, is of the 1st 
and p 2 of the 2nd degree in a and /3, it follows that t?i ever^ 
term of any function of p x and p 2 which represents the value of 
a homogeneous symmetric function, the sum of the suffixes * of the 
p's must be equal to the degree of the symmetric function in a and ft. 
Thus, for example, in the expression (6) for s 3 the sum of the 
suffixes in the term -p*, that is, -piPiPi, is 3 ; and in the term 
3p } p 2 also 3. 

This last remark is important, because it enables us to write 
down at once all the terms that can possibly occur in the ex- 
pression for any given homogeneous symmetric function. All 
we have to do is to write down every product of p t and p s , or 
of powers of these, in which the sum of the suffixes is equal to 
the degree of the given function. 

Example 1. 

To calculate a 4 + /3 4 in terms of p\ and# 2 . 

This is a homogeneous symmetric function of the 4th degree. Hence, by 
the rule just stated, we must have 

^ + 13* = Api* + Bj?^ + Cpr, 

where A, B, C are coefficients to be determined. 

In the first place, let /3 = 0, so that ]h= - a, #2=0. We must then have 

the identity a 4 sAa'. Hence A = l. 

We now have 

a 4 + /3 4 s(a + /3) 4 + B(a + /3) 2 a/3 + Ca 2 /3 2 . 

Observing that the term a :! /3 does not occur on the left, we see that B must 
have the value - 4. 

Lastly, putting a= -/3=1, so that;>i = 0, p- 2 = -1, we see that C = 2. 

Hence 

a'+p=pS-ipi-p» + 2p.f. 

The same result might also be obtained as follows. We have 

Hence, using the values of s 2 and s 3 already calculated, we have 

Si= -]h( -pf + ZpiPt) -p-ilh 2 ~ 2?a)i 
=£i 4 -4pi s .p»+2p8*. 
Example 2. 
Calculate a 6 + ^ + o s /3 s H a'-' ( f : in terms ofjB] and jh. 

* This is called the uxiylit of the symmetric function. See Salmon's 
Higher Algebra, § 5tJ. 



xvni NEWTON'S THEOREM 435 

We have 

a 5 + /3 5 + a 3 /3 2 + a 2 /3 3 = Kp x 5 + Efefa + Cpip-j 1 . 
Putting /3 = 0, we find A= - 1 ; considering the term a 4 /i, we see that B = 5; 
and, putting a = /3 = l, we find C= - 6. Hence 

a 3 + jS 5 + a*p 2 + a 2 ^ = - p? + bpjfa - Spiff. 

Since any alternating integral function * of a, ft, say /(«, ft), 
merely changes its sign when a and ft are interchanged, it follows 
that we have /(a, ft) = -/(/?, a). Hence, if we put ft = a, we 
have/(a, a) = -/(a, a); that is, 2/(a, a) = 0. Therefore /(a, a) = 0. 
It follows from the remainder theorem that /(a, ft) is exactly 
divisible by a - ft. Let the quotient be <j(a, ft). Then g(a, ft) 
is a symmetric function of a, ft. For g(a, ft) =/(<*, /3)/(<* - /?)> an d 

,/(/?, a) =/(/?, a)/(/3-«)= -f(a,ft)l(ft-a) = /(a,0)/(a - /3); that is, 

;/((z, /?) = £(/?, a). Hence any alternating integral function of a and 

/? can be expressed as the product of a - ft and some symmetric 

function of a and ft. Hence any alternating function of a and ft can 

be expressed without difficulty as the product of ± >J(p* - 4j» s ), and 

an integral function of p. and p. 2 . 

Example 3. 

To express a 5 /3- a/3 5 in terms of pi and ^2- 

We have 

a 5 /3- a/3 5 =a/3(a 4 - /3 4 ), 

= (a-/3)a/3(a + /3)(a 2 + /3 2 ), 

= ± V( Pi 2 - 4ft) { Pi?' 2 ( Pi* - 2i? 2 )} • 

Every symmetric rational function of a and ft can be ex- 
pressed as the quotient of two integral symmetric functions of 
a and ft, and can therefore be expressed as a rational function 
of p x and p.,. 

Example 4. 

a :i + 2a 2 /3 + 2a/3 2 + ^ (a + /3) 3 - a/3(a + /3) 
a 2 /3 + a/3* a/3(a + /3) 

-7'i 3 + ;'i?'2 



_ P?-Vz 
Pi 

§ 3.] The general proposition established for symmetric 

functions of two variables can be extended without difficulty to 

symmetric functions of any number of variables. 

* See p. 77, footnote. 



436 



newton's theorem 



chap. 



We shall first prove, in its most general form, Newton's 
Theorem that the sums of the integral powers of the roots of any 
integral equation, 

x n + p^ x n - 1 + p aX n - 2 + _ _ _ + ^ _ q (\^ 

can be expressed as integral functions of p x , p s , . . ., p n , whose co- 
efficients are all integral numbers. 

Let the n roots of (1) be a,, a 2 , . . ., a n , and let the equation 
whose roots are the same as those of (1), with the exception of 
a n be 



y.n-1 



f'H ii?i« + iPi 



.11-3 



+ 



+ iPn-i = 



(2); 



also let the equation whose roots are the same as those of (1), 
with the exception of a. 2 , be 

X n ~ 1 + 2 p,X n ~ 2 + 2 p. 2 X n ' 3 + . . .+ s p n -i = Q (3), 

and so on. 
Then 

x n_1 + j^a' 1-2 + ,p 2 x n ~ 3 + . . . + x p n - x 

= (x ?l +p 1 X n ~ 1 + p 2 X n ~ 2 + . . . + p n )K% - a,), 
= X n - 1 + (a l +p ] )x^- 2 + (a i 2 +p l a 1 +p.^X n - s . . . 
+ (a 1 r +p 1 a 1 r - 1 + . . . +p r )x n - r ~ 1 + . . . 

by chap, v., § 13. 

Hence, equating coefficients, we have 

iP» = a i 2 +?i a i + P»> 



r-l 



[ p r = a l 1 +p 1 a 1 ' " + 



+ P. 



iPn-i =< 1+ .2V*i 



«-2 + 



+Pn-i 



(3'). 



Similar values can be obtained for „j\, a p s , 2 p 3 , . . ., g J?n-i 
in terms of a, and p lt p„ . . ., p»j and so on. 

Taking the (r- l)th equation in the system (3'), and multi- 
plying by a,, we have 



r-l 



Similarly 
and so on. 



i£V-i«i = <*i +j>W + 



2 p r -iOL 2 = a 2 r +p l a 2 r l + 



+jpr-i a i 



XVIII 



NEWTON S THEOREM 



43; 



Adding the 11 equations thus obtained, we have 

iPr-i*i+2Pr-i<h + - . . + „p r -ia n = S r +2> l S r - l + . . . +iV-l»j (4)- 

Now ,^ r _, is the sum of all the products r—\ at a time of the 
n— 1 quantities - a,, - a 3 , . . ., - a n . Hence i/> r -i a i is the sum, 
with the negative sign, of all those products r at a time of the 
n quantities - a 1? - a 2 , . . ., - a n which contain a,. Similarly 
the next term contains all those products rat a time in which 
a 2 occurs ; and so on. Hence on the left all the products r at 
a time of the n quantities -a,, - a 2 , . . .,-a n occur, each as 
often as there are letters in any such product, that is to say, r 
times. Hence the equation (4) becomes 

-rp r = S r +jt> 1 S r L l + . . . +p r - i S n 

Or Sr+iVr-i+ • • • + Pr~ i s i + r pr = 0. 

This will hold for any value of r from 1 to n - 1, both inclusive. 

We have therefore the system 

s i +p i =0-| 

s 2 +p v s t + 2p 2 = 

s 3 + p x s a +p a s l + 3p a =0 }■ (5). 



Sn-i +PiS n -2 + ■ • • (n- l)i>»_i = 



Again, since a, is a root of (1), we have 

a I B + ^,a, n - 1 + . . . +p n = 0. 

Similarly 



a/ 1 + p t a 2 



+ 



■ +Pn = 0; 

and so on. 

If we first add these n equations as they stand, then 
multiply them by a,, a 2 , . . ., a a and add, then multiply them 
by a, 2 , a/, . . ., a a 2 respectively, and add, and so on, we obtain 

s n +PiS n -i + • • • + np n =0- 

Sn+i+Pi*n +• ■ • +8 1 p n =0 I 
Sn+a+PJn+i +■ ■ - + S 2 p n = \ 



(6), 



and so on. 



J 



The equations (5) and (G) constitute Newton's Formula for 



438 INTEGRAL SYMMETRIC FUNCTION CHAP. 

calculating s n s 2 , s 8 , . . ., &c, in terms ofpi,p a , . . ., p n . It is 
obvious that s,, s,, s 3 , . . . are determined as rational integral 
functions of p, , p 2 , . . . , p n , in which all the coefficients are 
integral numbers. 

A little consideration of the formula? will show that in the 
expression for s r the sum of the suffixes of the p's in each term will 
be r. 

Hence to find all the terms that can possibly occur in s r we 
have simply to write down all the products of powers of p n p 2 , 
. . ., p n in which the sum of the suffixes is r. 

Example. 

To find the sum of the cubes of the roots of the equation 

a? - 2a? + 3a: + 1 = 0. 

We have 

51-2 = 0, s 2 -2s 1 + 2x3 = 0, 53- 2s 2 + 3s 1 + 3 x 1 = 0. 
Hence si = 2, s 2 =-2, s 3 =-13. 

§ 4.] We can now show that evqry integral symmetric function 
of the roots can be expressed as an integral function of p t , p 3 , . . ., p n . 
The terms of every symmetric function can be grouped into 
types, each term of a type being derivable from every other of 
that type by merely interchanging the variables a,, a,, . . ., a n 
(see chap, iv., § 22). All the terms belonging to the same type 
have the same coefficient. It is sufficient, therefore, to prove the 
above proposition for symmetric functions containing only one 
type of terms. Such symmetric functions may be classed as 
single, double, triple, &c, according as one, two, three, &c, 
of the variables a u a 2 , . . ., a n appear in each term. Thus 
^a,^, ^a,^a/, ^a^a^a/, &c, are single, double, triple, &c, 
symmetric functions. 

For the single functions, which are simply sums of powers, 
the theorem has already been established. We can make the 
double function depend on this case as follows : — 

Consider the distribution of the product 

(tt,P + a.,P + . . . + a H P) (a,? + aj + . . . + a^). 

Terms of two different types, and of two only, can occur, namely, 



XVIII 



ORDER AND WEIGHT 439 



terms derivable from a?a£, that is, af+% and terms derivable 
from a^a 2 2. We have in fact 

SpS q = SttjP+S + ^a.PaJ. 

Hence Sa^cu? = s p s q - s p+q . 

Now s p , s q , s p+q can all be expressed as integral functions of 

PitPsi • ■ •> Pn- Hence the same is true of lafa^. 

Here we have supposed p 4= q. If p = q, then the term a^a/ 
will occur twice, and we have 

s p 2 = 2 ai 2 * + S2*?aJ> ; 
but this does not affect our reasoning. 

The case of triple functions can be made to depend on that 
of double and single functions in a similar way. In the distri- 
bution of 

(ttj* + <%» + ... + a/) (a,? + a? + . . . + a,?) (a, r + a/ + . . . + a/) 

every term is of the form a M p a/a w r , where m, v, w are, 1st, all 
different ; 2nd, such that two are equal ; 3d, all equal. Any 
particular term can occur only once if p, q, r be all unequal. 
Hence we have 

SpSgSr = SoAsW + 2af+W + Sa^+W + 2<+2V + W +9+r - 
In the last equation every term, except Safafaf, can be ex- 
pressed as an integral function oip lf p 2 , . . .,p n . Hence 'Zafaja/ 
can be so expressed. 

If two or more of the numbers p, q, r be equal, tlien each 
term of lafa^a/ will occur a particular number of times ; and 
the same is true of certain of the other terms in the equa- 
tion last written. But this does not affect the conclusion in 
any way. 

We can now make the case of quadruple symmetric functions 
depend on the cases already established ; and so on. Hence the 
proposition is generally true. 

It is obvious, from the nature of each step in the above pro- 
cess, and from what has been already proved for s,, s.,, s 3 , . . ., 
that in the expression for any homogeneous symmetric function of 
degree r the sum of the suffixes of the p's will be r for each term ; 
so that we can at once write down all the terms that can possibly 



440 GENERAL STATEMENT OF THEORY CHAP. 

occur in that expression, and then determine the coefficients by 
any means that may happen to he convenient. 

It is important to remark that the degree in p lt p.,, . . . , p n 
of the expression for 'Eafa^a.f . . . in terms of p lt p s , . . .,p n 
must be equal to the highest of the indices p, q, r . . . For, let 
the term of highest degree he p^ipfo . . . p 7 ^ n , then, since 
P\ - \P\ ~ «d Pz = iPa - \P\0-ii where $ lf ,j? 2 , &c, do not contain a 1} * 
we see that p^ l p. 2 x * . . . p n Xn , when expressed in terms of 
a,, a 2 , . . ., a n , will introduce the power a,^i + X 2+' ' + *» with 
the coefficient ( — l) n iPi^p 3 3 . . . i^n-i*"- Now, since there 
are no terms of higher degree than p^ipfo . . . p,^ n , if the 
power a^i + x i + - ■ • +x » occur again, it must occur as the highest 
power, resulting from a different term of the same degree ; that 
is to say, it will occur with a different coefficient and cannot 
destroy the former term. Hence the index of the highest 
power of any letter in the symmetric function must be equal 
to the degree of the highest term in its expression in terms of 

PoPm, ■ ■ •, VnA 

Although, in establishing the leading theorem of this para- 
graph, we have used the language of the theory of equations, 
the result is really a fundamental principle in the calculus of 
algebraical identities ; and it is for this reason that we have 
introduced it here. We may state the result as follows : — 

elementary symmetric functions of the system of n variables 
x u x 2 , . . ., x n . Then we caii express any symmetric integral function 
of sc, , o: 2 , . . ., x n as an integral function of the n elementary 
symmetric functions ; and therefore any rational symmetric function 
of these variables as a rational function of the n elementary symmetric 
functions. 

On account of its great importance we give a proof of this 

* They are, in fact, the functions of a 2 , ag, . . ., a„ defined in § 3. See 
Exercises xxxvi., 51. 

t Salmon, Higher Algebra, § 58. 



XVIII 



PROOF OF GENERAL THEOREM 441 



proposition not depending on Newton's Theorem (which is itself 
merely a particular case).* 

Let n q„ n q a , . . ., n <ln denote the n elementary symmetric 
functions of the % variables x lt x 2 , . . ., x m that is to say, 
2 n K,, 2 n .r,in,, • • -, x l x a ...x n ; and let „_,&, „_,&, • • •» n-tfn-i 
denote the n~\ elementary symmetric functions of x x , x 2 , . . .,%_,, 
that is, 2a;,, ? x,r 2 , . . ., sea . . . x n ^. It is obvious that, 

?! — 1 ?(— 1 

when x n = 0, ££,, n q 2 , . . ., n <ln-i become „_,£,, „_,&, . . ., n-fln-i 
respectively. 

Let us now assume that all symmetric integral functions not 
involving more than n - 1 variables can be expressed as integral 
functions of „_,?,, n -i?8j . . ., n-i?n-i- Let/(:r.,, a- a , . . ., »„_„ « w ) 
be any symmetric integral function of the « variables x„ x„ . . ., £„, 
Then /(a;,, a:*, . . ., ar n _,, 0) is a symmetric integral function of 
x u x 2 , . . ., x n _„ and can therefore, by hypothesis, be expressed 
integrally in terms of „_,?„ »-,&, • • •, n-iin-i- Let this ex- 
pression be <K»-i?» »-i2« ■ • •» n-rfn-Oj so that <£ is a known 
function. 

Now assume 

+ xj/(x l , x 2 , . . ., %_,, .r„) (7). 

Then, since <£(»&, n </ 2 , . . ., n3W-i) is a symmetric integral 
function of a,, x 2 , . . ., x m if/(x 1} x,, . . ., x n _„ a^) is obviously 
a symmetric integral function of these variables. 

If we put x n = on both sides of the identity (7), then 

J\X X , X 2 , . . ., X n _ u 0) — <p{n-\ ( lu n-^1-21 • • '} n-iQn-i) 

+ xf (./-,, X s , . . ., Xn.\, 0) (8). 

But/(.r,, ..',, . . ., ar B _„ 0) = <f>( n _ l q 1 , w _,2„ . . ., „_,£„_,)• Hence, 
by (8), )/<(>,, x 2 , . . ., a^_ 19 0) = 0. Therefore the integral 
function \f/(x l} x 2 , . . ., x n _ u x n ) is exactly divisible by some 



* This proof is taken from a paper by Mr. R. E. Allardice, Pruc. Edinb. 
Math. Soc. for 1889. 



442 PROOF OF GENERAL THEOREM chap. 

power of x n , say x n * ; hence, on account of its symmetry, also by 
x*, x", . . ., x n _". We may therefore put 

lf,(x 1} «b, . . ., X n - 1 ,X n )= n qnfi(Vi,Xa, • ■ ■^n-i^n), 

where /, is a symmetric integral function of a,, K g , . . ., % of 
lower degree than / 

We can now deal with f x in the same way as we dealt with 
f; and so on. We shall thus resolve f(x u x 2 , . . ., x n _ u x n ) 
into a closed expression of the form 

<f> + n2»"V. + nq n ai+x % 2 + • • • + .fo" 1 W • • ' +K ' n *. (9), 
where <£,, 4>,, . . ., <f> m are, like <f>, all known functions of n q x , 
n<l>, • • ., «?«->, or else constants. 

If, therefore, the integral expression in question be possible 
for n - 1 variables, it is possible for n variables. 

Now every integral function of a single variable, x n is a 
symmetric function of that variable, and can be expressed 
integrally in terms of //,, which is simply .t,. Hence it follows 
by induction that every symmetric integral function of n variables 
can be expressed as an integral function of the n elementary 
symmetric functions. 

Cor. It follows at once, by induction, from the form of (9) that the 
coefficients of the expression for any symmetric integral function 
/(»„ x, 2 , . . ., x n ) in terms of n q u n q a , . . ., n q n are integral 
functions of the coefficients off In particular, if the coefficients of f 
be integral numbers, the coefficients of its expression in terms of 
n°i> w&j • • •> nQn w ^ a ^ so be integral numbers. 

We now give a few examples of the calculation of symmetric 
functions in terms of the elementary functions, and of the use of 
this transformation in establishing identities and in elimination. 

Example 1. 

If a, /3, 7 be the roots of the equation 

gX-pja? +p 2 x -^3 = 0, 

express /3 3 > + fiy 3 + -/a + ya? + a?p + a/3 3 in terms of ;j 1; p 2 , ]h- 

Here we have;>i = Za, p- i ='Lap, p s = af3y. Remembering that no term of 
higher degree than the 3rd can occur in the value of 2a 3 /3, we see that 



XVIII 



EXAMPLES 443 



2a 3 £ = Aprp-i + Bp a & + Cp-r ( 1 ), 

where A, B, C are numbers which we have to determine. 

Suppose 7 = ; then 2h = a + P, P-2 = *P, J>s=0 ; and (1) becomes 
a 3 (3 + ap s = A(o + /3) 2 a/3 + Ca 2 /5 2 ; 
that is to say, a 2 + /3 2 = A(a + /3) 2 + Ca/3. 

Hence A = l, C=-2. 

We now have 2a 3 /3 =^i 2 ^ 2 + Bpi p 3 - 2p 2 2 . 

Let a =/3=7 = 1, so that pi = 3, ^a = 3, p» = 1 . We then have 

6 = 27 + 3B-18. 
Hence B= - 1. 

Therefore, finally, 2a 3 /3 =£1^2 - p^ - 2i> 2 2 . 

In other words, we have the identity 

2a 3 /3=(2a) 2 2a/3 - a/3 7 2a - 2(2a/3) 2 . 
Example 2. 
To show that 
{yz - xu ) (zx - yu) (xy - zu) = (yzu + zux + uxy + xyzf - xyzu(x + y + z + u) 2 (2). 
The left-hand side of (2) is a symmetric function of a, y, z, u. Let us calcu- 
late its value in terms of/?i = 2a:, /? 2 = 2xy, p 3 -Y.xijz, p i = xyzu. 

Since the degree of U(yz - xu) in x, y, z, u is 6, and the degree in z alone 
is 3, we have 

U(yz - xu) = Api 2 pi + Bpip^&t + Cp 2 3 + BpsPi + Ep 3 2 (3). 

If we put u = 0, then pi = ~Z 3 x, p 2 = 2 3 xy, p 3 = xyz, p t =0, where the suffix 
3 under the 2 means that only three variables, x, y, z, are to be considered. If 
Pi, lh, Pz have for the moment these meanings, then (3) becomes the identity 

^ 3 2 = BpiPsPa + Cpi + Ep 3 2 . 
Hence B = 0, C=0, E=l. 

Hence Jl(yz - xu) = Aprp* + Vp-iPi + P3* ( 4 )- 

Now let x = y = l, and z = u- -\, so thatj9i = 0, #2= -2, ^3=0, p t = l. 

Then (4) becomes 

0=-2D. 

Hence D = 0. 

We now have II (yz - xu) = Apfp* +p 3 2 . 

In this put x = y = z-u = l, and we have 

= 16A + 16. 

Hence A= - 1. 

Hence, finally, H(yz - xu)=p 3 2 - p^p*, 

which establishes the identity (2). 

Example 3. 

If x + y + s = 0, show that 
g n + y ii + 8 u _^ ± ^ ± j» x * + f + z* (xs + y^ + z 3 ) 3 x 2 + y 2 + z 2 

11 ~ 3 " 2 9 2 ' ; ' 

( Wolstcnholme.) 



444 EXAMPLES chap. 

If^> 1 =Sar, p 2 = ?,xy, p 3 -xyz, s 3 = 2£ 2 , s 3 =2.r 3 , &c, then we are required to 
prove that 

su s$s a s-/s 2 ._,. 

11 _ "6" ""18 ( ,- 

We know that s n is a rational function of^i, Pi, lh- I' 1 the present case 
Pi = 0, and we need only write down those terms which do not contain p x . 
We thus have 

9n = AfrY + Bp a p s * (6), 

provided x + y + z=0. 

A may be most simply determined by putting z = -(x + y), writing out 
both sides of (6) as functions of x and y, dividing by xy, and comparing the 
coefficients of a; 9 . We thus find A = 11. 

We have therefore 

s n = llp 2 4 p 3 + B2 ] 2P3 S - 
In this last equation we may give x, y, z any values consistent with 
x + y+z=0, say x=2, y= -1, z= -1. We thus get B= - 11. Hence 

Sn = 1 lp 2 4 P3 ~ 1 l&&£ ( 7 )• 

In like manner we have 

s 8 = Ap2 i + Bp 2 ps 2 - 
Putting in this equation first x=l, y= - 1, z — 0, and then x = 2, y — - 1, 
z= -1, we find A=2, B= -8. 

Hence s 8 = Ip-t - 8p- 2 p 3 2 ( 8 ). 

We also find S3 = 3jt? 3 (9), 

s,= -2p 2 (10). 

From (8), (9), and (10) we deduce 

T - IT =^ 4 - 4 ^ 2) + 3 ^' 

"IV 
which is the required equation. 

Since we have four equations, (7), (8), (9), (10), and only two quantities, 
P2, Ps, to eliminate, we can of course obtain an infinity of different relations, 
such as (5) ; all these will, however, be equivalent to two independent equa- 
tions, say to (5), and 

725 8 = 95 2 '' + 4.V:" (11). 

Example 4. 

Eliminate x, y, z from the equations x + y + z = 0, x i + y' i + z* = a, x^ + if + z 5 
= 6, x 7 + y 7 + z 7 = c. 

Using the same notation as in last example, we can show that 

s 3 = 3;>3 , S S = - 5;j 2 ;>3 , $7 = 7prj>:i ■ 
Our elimination problem is therefore reduced to the following : — 
To eliminate p% andf7s from the equations 

3y'3 =a, - 5pa Pi = !>, "p-fl '3 = <". 
This can be done at once. The result is 

2W -25oc=0. 



xvin EXERCISES XXXVI 445 

Exercises XXXVI. 

a and /3 being the roots of the equation x 2 +px + q = 0, express the follow- 
ing in terms of p and q : — 

(1.) a» + /3» (2.)(a« + /3«)/(a-/3) 2 . (3.) a~ 5 + /3- 3 . (1 ) a" 5 - /3~ 5 . 

(5. ) (a :1 + /3 s )- 1 + (a 3 - /3»)-i. (6. ) (1 - a) 2 /3 2 + (1 - /3)V. 

(7.) If the sum of the roots of a quadratic be A, and the sum of their 
cubes B 3 , find the equation. 

(8.) If s„ denote the sum of the nth powers of the' roots of a quadratic, 
then the equation is 

(SnS»-2 - S„-i 2 )x 2 - (s„ + iS„_o - S n S n _i)x + (s n+1 S n -i - S n 2 ) = 0. 

(9.) If a and /3 be the roots of x 2 +px + q = 0, find the equation whose 
roots are (a-h) 2 , (/3-A) 2 . 

(10.) Prove that the roots of 

x 2 - {2p-q)x+p 2 -pq + q 2 = 
are p + uq, p + u-q, w and or being the imaginary cube roots of 1. 

(11.) If a, /3 be the roots of x* + z + l, prove that a" + /3" = 2, or = - 1, 
according as n is or is not a multiple of 3. 

(12.) Find the condition that the roots of ax 2 + bx + c = may be deduc- 
ible from those of a'x 2 + b'x + c' — by adding the same quantity to each root. 

(13.) If the differences between the roots of x 3 +px + q = and x 2 + qx+p 
— be the same, show that either p = q or p + q + 4 = 0. What peculiarity is 
there when p> = q1 

Calculate the following functions of a, j3, y in terms of^! = 2a, j? 2 = Sa/3, 
p 3 = aj3y:— 

(14.) a. 2 /py + p 2 /ya + y 2 /ap. (15.) a~ s + ^ 5 + y- s . 

(16.) (/ja+yxy+oW + Zn (17.) Z(a 2 + /3 7 )/(a 2 -/3 7 ). 

(18.) 2(/3- 7 )'. (19.) Z(a-0) 2 (/3- 7 )2. (20.) 2(/3 + 7 )\ 

Calculate the following functions of a, /3, 7 , 5 in terms of the elementary 
symmetric functions : — 

(21.) Za\ (22.) Sa-3. (23.) 2a 2 /3 2 . (24.) 2a 2 /3 7 . (25.) 2(a+/3)* 

(26. ) If a, /3, 7 , 3 be the roots of the biquadratic ar 1 +p 1 x :i +p 2 x 2 +psx +p 4 = 0, 
find the equation whose roots are /3 7 + a5, ya + {15, a^ + yd. 

(27.) If the roots of x 2 -2hr+p 2 — 0, x 2 -q 1 x + q 2 = Q, x 2 -rix + r 2 = 0, be /3, 
y ; y, a ; a, /3 respectively, then a, /3, 7 are the roots of 

K 3 - Ki»i + ?i + ^i)* 2 + (#! + q-i+ r 2 )x - hiPilin ~PiPa - q\q-i - nr 2 ) = 0. 
(28. ) If a, p, y be the roots of x 3 +px + q = 0, show that the equation whose 
roots are a + /3 7 , /3 + ya, y + a/3, is X s -px 2 + {p + 3q)x + q-(p + q) 2 — 0. 
(29.) If a, j8, 7 be the roots of 

p/(a + x) + q/(b + x) + r/{c + x) = 1, 
show that p = (a + a) (a + /3) (a + 7 )/(a - 6) (a - c). 

If a, /3 be the roots of x 2 +pix+p 2 = 0, and a', (3' the roots of x 2 +pi'x+p 2 
= 0, express the following in terms of p\, p 2 , p{, p 2 : — 
(30.) (a'-a)(/3'-/3)+V-/3)(/3'-«). 



446 EXERCISES XXXVI CHAP. 

(31. ) (a'-a)' + (/?' - jB)« + (a' - /3) 2 + (/3' - a)*. 
(32.) (a + a')(/3 + /3')(a + /3')(/3 + a'). 
(33.) 4(a-a')(a-/3')(/3-a')(/3-/3'). 
[The result in this case is 

icft-y*)" + ♦(pi-ftO (pip-2-piP2) = (2 P -2 + 2p 2 '- PlPl r-{pi' 2 - if*) bi' 2 - ^m 

(34.) A, A' and B, B' are four points on a straight line whose distances, 
from a fixed point on that line (right or left according as the algebraic 
values are positive or negative), are the roots of the equations 
ax 2 + bx + c = 0, a'x 2 + b'x + c' = Q. 

If AA'.BB' + AB'.BA' = 0, 

show that 1m' + 2c 'a -W = 0; 

and if AA'.BA' + AB'.BB'=0, 

that 2c«' 2 - 2cW + ab' 2 - a'W = 0. 

(35.) a, j3 are the roots of x-~2ax + b 2 = 0, and a', (3' the roots of 
x 2 - 2cx + d- = 0. If aa' + /3/3' = 4w 2 , show that 

(a 2 -b 2 )(c 2 -d 2 ) = {ae-2n-) 2 . 

(36.) a, /3, a', /3', being as in last exercise, form the equation whose roots 
are aa' + /3/3', a/3' + a'/3. 

(37.) If the roots of ax 2 + bx + c = be the squai'e roots of the roots of 
a'x 2 + b'x + c' = 0, show that a'b 2 + a-b' = 2aa'c. 

(38.) Show that when two roots of a cubic are equal, its roots can always 
be obtained by means of a quadratic equation. 

Exemplify by solving the equation 12a. 13 - 56a; 2 + 87a; - 45 = 0. 

(39.) If one of the roots of the cubic a?+pia?+p^xs+ps=0 be equal to the 
sum of the other two, solve the cubic. Show that in this case the coefficients 
must satisfy the relation 

Pi 8 -4pj#s+8p8=0. 

(40.) If the square of one of the roots of the cubic x s +piX 2 +p2X+2>3=0 be 
equal to the product of the other two, show that one of the roots is -JJg/pi ; 
and that the other two are given by the quadratic 

PiP-ix- +2h(Pi" -p-i)x +2 } i 2 P3 = °- 
As an example of this case, solve the cubic 

x 3 -9a; 2 - 63a; + 343 = 0. 
(41.) If two roots of the biquadratic x i +pix 3 +p<c 2 +p 3 x+p i = be equal, 
show that the repeated root is a common root of the two equations 
4X 3 + ZpiX 2 + 2p»z + ]>3 = 0, ./; 4 + p X 7? + p&p +2) 3 x + p 4 = 0. 
(42.) If the three variables x, y, z be connected by the relation 2x—xyz, 
show that 22a:/(l - x 2 ) = U2x/{1 - x 2 ). 

(43.) If 2z = 0, show that 22a.- 7 = 1xijzZx\ 
(44.) If Saj=0, show that Zx s =2{2yz) i - 8x 2 ifz 2 Zyz. 

(45.) If Sa=0 (three variables), then (2(6» - e?)/a 3 ) (Za^ft 8 - <?) ) =s 
36-4(2a :, )(2a- 3 ). 

(46.) If 2ar* = Q, 2.t 4 = (three variables), show that Zx 5 + xyz{{y-z)(z-x) 









xviii DISCRIMINATION OF THE ROOTS OF A QUADRATIC 447 

(47.) Ifas+y+2+tt=0, show that (2a?y i =9{2xyzf=m(yz-xu). 
(48.) Under the hypothesis of last exercise, show that 
ux{ u + xf + yziu-xf + uy(u + yf + zx(u - yf + uz(u + zf + xy(ic - zf + ixyzu = 0. 
Eliminate x, y, z between the equations 

<»•><;)='. *(:)*="■ =©'""• KS)'-* 1 . 

(50.) Sa?=a a , Zxy = V, 2xhf = c\ Zx* = d\ 

(51.) Show that Pi = B p\-a i , P2 = >P2~ >P\*s, • • ; Pr= tPr~ »Pr-\**i ■ ■ ; 
p„— - ,2hi-io. s , where 2h> P2> • • •> tPi> iPz> • • • have the same meanings as 
in § 3. 

SPECIAL PROPERTIES OF QUADRATIC FUNCTIONS. 

§ 5.] Discrimination of Boots. — We have already seen (chap, 
xvii., § 4) how, without solving a quadratic equation, to dis- 
tinguish between cases where the roots are real, equal, or imag- 
inary. There are a variety of other cases that occur in practice 
for which it is convenient to have criteria. These may be treated 
by means of the relations between the roots and the coefficients 
of the equation given in § 1 of the present chapter. If a, ft be 

the roots of 

az* + bx+c = (1), 

then a + ft - - b/a, aft = cja. 

If both a and ft be positive, then both a + ft and aft are posi- 
tive. Conversely, if aft be positive, a and ft must have like 
signs ; and if a + ft be also positive, each of the two signs must 
be positive ; but if a + ft be negative, each of the two signs 
must be negative. Hence the necessary and sufficient condition that 
both roots of (1) be positive is that b/a be negative and cja positive; 
and the necessary and sufficient condition that both roots be negative is 
tin if hja be positive and cja positive. This presupposes, of course, 
that the condition for the reality of the roots be fulfilled, namely, 
b 2 - iac > 0. 

Reality being presupposed, the necessary and sufficient condition 
that the roots have opposite signs is obviously that cja be negative. 

The necessary and sufficient condition that the two roots be numeric- 
ally equal, but of opposite sign, is a + ft = 0, that is, b/a = 0. 

If one root vanish, then aft = ; and, conversely, if aft = 0, 
then at least one of the two, a, ft, must vanish. Hence the neces- 



448 INFINITE ROOTS 



CHAP. 



sary and sufficient condition for one zero root is cja - 0, that is, c = 0, 
a being supposed finite. 

If both roots vanish, then a/3 = and a + f3 = ; and, con- 
versely, if a/3 = and a + (3 = 0, then both a = and (3 = 0; for 
the first equation requires that either a = or (3 = 0, say a = ; 
then the second gives + (3 = 0, that is, (3 = also. Hence the 
necessary and sufficient condition for two zero roots is c/a = 0, b/a = 0, 
that is, a being supposed finite, c — 0, b = 0. 

The two last conclusions have already been arrived at in 
chap, xvii., § 2. Perhaps they will be more fully understood by 
considering the case as a limit. Let us suppose that the root a 
remains finite, and that the root (3 becomes very small. Then 
a/3 becomes very small, and approaches zero as its limit, while 
a + (3 approaches a as its limit. In other words, c/a becomes 
very small, and - b/a remains finite, becoming in the limit equal 
to the finite root of the quadratic. 

If both a and (3 become infinitely small, then both a + (3 and 
a/3, that is to say, both - b/a and c/a, become infinitely small. 

Infinite Boots. — If the quadratic (1) have no zero root, it is 
equivalent to 

that is, if £= 1/x, to 

c£ 2 + b£ + a = (2). 

The roots of (2) are 1/a and \/(3; and we have 1/a+ 1/(3 
= - b/c, l/a/3 = a/c. Let us suppose that one of the two, a, f3, 
say (3, becomes infinitely great, while the other, a, remains finite j 
then \/f3 becomes infinitely small, and l/a/3, that is, a/c, becomes 
infinitely small, while l/a+l//3, that is, -b/c, approaches the 
finite value 1/a. Hence the necessary and sufficient condition 
that one root of (1) be infinite is a = 0, c being supposed finite. 

In like manner, the condition that two roots of (1) become 
infinite, that is, that two roots of (2) become zero, is a = 0, b = 0. 

If therefore in any case where a quadratic equation is in 
question we obtain an equation of the form bx + c = 0, or an equa- 
tion of the paradoxical form c = 0, ice conclude that one root of the 



XVIII 



TABLE GENERAL RESULTS CUBIC 



449 



quadratic lias become infinite in the one case, and that the two roofs 
have become infinite in the other. 

For convenience of reference we collect the criteria for dis- 
criminating the roots in the following table : — 



Roots real 


i 
b"-4ac>0. 


Both roots negative 


c/a + , b/a + . 


Roots imaginary 


b- - 4ac < 0. 


Roots of opposite 




Roots equal . 


b--4ac = 0. 


signs . 


c/a- . 


Roots equal with 




One root =0 


c = 0. 


opposite signs 


b = 0. 


Two roots = 


b = 0, c = 0. 


Both roots positive 


c/a + , bja- . 


One root = qo 


a = 0. 






Two roots = oc 


b = 0, « = 0. 



§ 6.] The reader should notice that some of the results em- 
bodied in the table of last paragraph can be easily generalised. 
Thus, for example, it can be readily shown that if in the equation 

p x n +p l x n - 1 + . . . +p n = (1) 

the last r coefficients all vanish, then the equation will have r zero 
roots ; and if the first r coefficients all vanish it will have r infinite 
roots. 

Again, if p t = 0, the algebraic sum of the roots will be zero ; 
and so on. 

It is not difficult to find the condition that two roots of any 
equation be equal. AVe have only to express, by the methods 
already explained, the sjanmetric function TI(ai - a,) 2 of the roots 
in terms of p , p u . . ., p n , and equate this to zero. For it is 
obvious that if the product of the squares of all the differences 
of the roots vanish, two roots at least must be equal, and con- 
versely. 

For example, in the case of the cubic 

a?+pix 2 +p 2 x +2^3-0 (2), 

whose roots are a, /3, 7, we find 

(/3 - 7 ) 2 (7 - a) 2 (a - /3) 2 = - Wjh +2hW + 1 Zjhlh]h ~ W - 2"i> 3 2 . 
The condition lor equal roots is therefore 

- 4]h 3 p3+Pi-pr + 18piP2i'3 ~ *Pa 8 - 27j> 3 2 = 0. 
Further, if all the roots of the cubic be real, (/3 - 7) 2 (7 - a) 2 (a - /3) 2 will be 
positive, and if two of them be imaginary, say /3=\H-^i, y = \-/xi, then 
( i 3-7)2(7- a )2 i a-^ 2 =-4 M 2 {(X-a) 2 + ^} 2 , that is, (^ - 7)^(7 - «) 2 (a - /3) 2 is 
negative. Hence the roots of (2) are real and unequal, such that two at least 
are equal, or such that two are imaginary, according as 

VOL. 1 2 G 



450 TWO QUADRATICS, CONDITION OF EQUIVALENCE CHAr. 

- ipi 3 P3 +Pi~p-? + IZpiPiPz ~ ±pi - 27p 3 2 
is positive, zero, or negative. 

The further pursuit of this matter belongs to the higher theory of equa- 
tions. 

§ 7.] If the two quadratic equations 

ax 2 + bx + c - 0, a'x 2 + b'x + c' = 
be equivalent, then b/a = b'/'a' and c/a = c'/a'. For, if the roots of 
each be a and /3, then 

bja = - (a + /5) = b'/a', cfa = afl = c'/a' ; 
and this condition is obviously sufficient. 

The above proposition leads to the following : A quadratic 
function of x is completely determined when its roots are given, and 
also the value of the function corresponding to any value of x which is 
not a root. This we may prove independently as follows. Let 
the roots of the function y be a and (3 ; then y = A(x - a) (x - /5). 
Now, if V be the value of y when x = X, say, then we must have 

V = A(A - a) (A - /8). 

This equation determines the value of A, and we have, 
finally, 

y - V (A-a)(A-/?) W 

The result thus arrived at is only a particular case of the 
following : An integral function of the nth degree is uniquely deter- 
mined when its n + 1 values, V,, V 2 , . . ., V M+1 , corresponding re- 
spectively to the Ti+1 values A n A 2 , . . ., A rt+1 of its variable x, are 
given. To prove this we may consider the case of a quadratic 
function. 

Let the required function be ax 2 + bx + c ; then, by the con- 
ditions of the problem, we have 

a A, 2 + ?>A, + c-Y u akf + bk 2 + c = V 2 , a\.f + bX x + c = V 3 . 

Tbese constitute a linear system to determine the unknown 
coefficients a, b, c. This system cannot have more than one 
definite solution. Moreover, there is in general one definite 
solution, for we can construct synthetically a function to satisfy 
the required conditions, namely, 



xvnr lagkange's interpolation formula 451 

= v C g - A *K- C ~ K) v (*- Ai)(s-Ag) v ( ■<• - a,) (x - a 2 ) 
' J \K-\,^K - A 3 ) 2 (A, - A,)(A 2 - A 3 ) 3 (A 3 - A,)(A 3 - A,) 

(2). 

The reasoning and the synthesis are obviously general. We 
obtain, as the solution of the corresponding problem for an in- 
tegral function of x of the nth degree, 

V y (■/-' - A 2 ) (■'■• - A 3 ) . . . (X - A, t+1 ) ,gX 

(A, - A.) (A! - A 3 ) . . . (A t - A )l+1 ) 
This result is called Lagrange's Interpolation Formula. 

Example 1. 

Find the quadratic equation with real coefficients one of whose roots is 
5 + 6i. 

Since the coefficients are real, the other root must be 5 - 6i. Hence the 

required equation is 

(x-5 + 6i)(x-5-6i) = 0, 
that is, (x-5f+6 2 = 0, 

that is, x*-10x + 61 = Q. 

Example 2. 

Find the quadratic equation with rational coefficients one of whose roots 
is 3 + v7. 

Since the coefficients are rational,* it follows that the other root must be 
3 - \J7. Hence the equation is 



{x-3 + ^/7)(x-3-^/7) = 0, 
that is, a; 2 - 6^ + 2 = 0. 

Example 3. 

Find the equation of lowest degree with rational coefficients one of whose 
roots is \J2 + \J3. 

By the principles of chap, x.* it follows that each of the quantities 
\/2- \J3, - \J2 + \J3, - \/2- \/3 must be a root of the required equation. 
Hence the equation is 

(x - V2 - V3) {x - V2 + V3) (* + V2 - V3) (x + s/2 + s/3) = 0, 
that is, a?-l0a?+l = 0. 

Example 4. 

Construct a quadratic function of x whose values shall be 4, 4, 6, when 
the values of x are 1, 2, 3 respectively. 



* This we have not explicitly proved ; but we can establish, by reasoning 
similar to that employed in chap, xii., § 5, Cor. 4, that, if a + b\fp be a root of 
f(x) = 0, and if a and b and also all the coefficients of f(x) be rational so far as 
\/p is concerned, then a - b\/p is also a root of/(.t') = 0. 



452 TWO QUADRATICS, CONDITION FOR COMMON ROOT chap. 

The required function is 

( 8 -2)(g-8) (■>:-!) (a- -3) (as-1) (s-2) 

4 (l-2)(l-3) + 4 (2-l)(2-3) + °(3-l)(3-2)' 

that is, a.' 2 - 3.r + 6. 

§ 8.] The condition that the two equations 

ax 2 + bx + c = 0, ft V + b'x + c' = 
have one root in common is the same as the condition that the 
two integral functions 

y = ax 2 + bx + e, y = a'x 2 + b'x + c' 

shall have a linear factor in common. Now any common factor 
of y and y' is a common factor of 

c'y - cy ', and ay' -ay, 
that is, if we denote ad - a'c by (ac'), Sec, a common factor of 
(ac')x 2 + (bc')x, and (ab')x + (ac') ; 

that is, since x is i ot a common factor of y and y' unless c = 
and c' = 0, any common factor of y and y' is a common factor of 

(uc')x + (be'), and (ab')x + (ac'). 

Now, if these two linear functions have a common factor of 
the 1st degree in x, the one must be the other multiplied by a 
constant factor. 

Hence the required condition is 

(ac') _ (be) 
(ab')~(ac'y 
or (ac' - a'c)* = (be - b'c) (ab' - a'b). 

The common root of the two equations is, of course, 

be' - b'c ac' - a'c 



x = - 



ac 



a'c ab' - a'b' 



By the process here employed we could find the r conditions 
that two integral equations should have r roots in common. 

It is important to notice that the process used in the demon- 
stration is simply that for finding the G.C.M. of two integral 
functions — a process in which no irrational operations occur. 
Hence 



xvm EXERCISES XXXVII 453 

Cor. 1. If two integral equations have r roots in common, these 
roots are the roots of an integral equation of the rth degree, whose co- 
efficients are rational functions of the coefficients of the given equations. 

In particular, if the coefficients of the two equations be real 
rational numbers, the r common roots must be the roots of an 
equation of the rth degree with rational coefficients. 

For example, two quadratics whose coefficients are all rational 
cannot have a single root in common unless it be a rational root. 

Cor 2. We may also infer that if two integral equations whose 
coefficients are rational have an odd number of roots in common, then 
one at least of these must be real. 

Exercises XXXVII. 

Discriminate the roots of the following quadratic equations without solving 
them : — 

(1.) 4a; 2 -8a; + 3 = 0. (2.) 9x- - 12*- 1 = 0. (3.) ix*-4x+6 = 0. 

(4.) 9a; 2 - 36x4-36 = 0. (5.) la? - ix - 3 = 0. (6.) 4a?+8x+8=0. 

(7.) (a5-3)(aj+4) + (a;-2)(a:+3)=0. 

(8.) Show that the roots of {b 2 - 4ac)x 2 + 4(a + c)x - 4 = are always real ; 
and find the conditions— 1° that both he positive, 2° that they have opposite 
signs, 3° that they be both negative, 4° that they be equal, 5° that they be 
equal but of opposite sign. 

(9.) Show that the roots of x 2 + 2(p + q)x + 2(jj 2 + q 2 ) = are imaginary. 

(10.) Show that the roots of 

{<?-2bc + l 2 }x 2 -2{c 2 -{a + b)c + ab}x+{2a 2 -2(b + e)a + b 2 + e 2 }=0 
are imaginary. 

(11.) Show that the roots of 

(a; - b) [x -c) + {x - e) [x - a) + (x - a) [x- b) =0 
are real ; and that they cannot be equal unless a=b = c. 

(12. ) The roots of a/(x -a) + b/(x -b) + c/(x - c) = are real ; and cannot be 
equal unless either two of the three, a, b, c, are zero, or else a = b = c. 

(13.) Find the condition that the cubic x 3 + qx + r = have equal roots. 

(14.) Show that the cubic 1 2x?- 52a; 2 + 75a; - 36 = has equal roots; and 
solve it. 

(15.) If two of the roots of a cubic be equal, and its coefficients be all 
rational, show that all its roots must be rational. 

(16.) Find the condition that two roots of the biquadratic ax i + dx + e=0 
be equal. 

(17.) If a/(x + a) + b/(x + b)-c/(x + c) + d/(x + d) have a pair of equal roots.. 
then either one of the quantities a or b is equal to c or d, or else 1/a + l/b 
= l/c + l/d. Prove also that the roots are then -a. -a, 0, - b, -b, 0, 
or 0, 0, - 2ab/(a + b). 



454 EXERCISES XXXVII chap. 

Write down and simplify the equations whose roots are as follows: — 

(18.) 1,0. (19.) h -|. (20.) 3 + V2, 3-V2. 

(21.) (a+Va 2 -l)/(«-\/« 2 -l), (a- Va 2 - 1 )/(« + Va 2 - 1 ). 

Find the equations of lowest degree with real rational coefficients which 
have respectively the following for one root : — 

(22.) a+pi. (23.) 1 + V2-V3- (21.) s/2 + i^3. 

(25.) »/2 + fyi. (Result, x 3 - 6x- 6 = 0.) 

(26. ) 8/2 + f/3. ( Result, x 9 - 1 5x 6 - 8 Tx 3 - 1 25 = 0. ) 

(27.) V(?'0 + VM + V(i>9)- 

(28.) Find the equation of the 6th degree two of whose roots are 
1 + -y/2 and 1+^-1. 

(29.) Find an equation with rational coefficients one of whose roots is 
ap 2 ' 3 + bp 1 ' 3 + c. 

Hence show how to find the greatest integer in ap 2/3 + bp 113 + c without 
extracting the cube roots. 

(30.) Form the equation whose roots are p + atf, p + a 2 q, . . ., p + a 2n q, 
where a 1; a 2 , . . ., a 2n are the imaginary (2?i + l)th roots of 1, showing that 
the coefficients are all rational, and finding the general term of the equation. 

(31.) Construct a quadratic function whose roots shall be equal with 
opposite signs, and whose values shall be 23 and 67 when x=5 and when 
x = 6 respectively. 

(32.) Construct a cubic function y corresponding to the following table ol 

values : — 

a: = 2-5, 3, 35, 4; 

y= 6, 8, 15, IS. 

(33.) If x 3 +ax + bc = 0, x 2 + bx + ca = have a common root, then their 
other roots satisfy x 2 + cx + ab = 0. 

(34.) If 2{p + q + r) = a 2 + p 2 + y 2 , and the roots of a- 2 + aa: - ;; = be/3 and y, 
and the roots of x 2 + fix - q = be y and a, then the equation whose roots are 
a and /3 is x 2 + yx -r—0. 



VARIATION OF QUADRATIC FUNCTIONS FOR REAL VALUES 
OF THE VARIABLE. 

§ 9.] The quadratic function 

y = ax 2 + bx + c 
may be put in one or other of the three forms 

y = a{(x — I) 2 - m] I., 

y = a{(;x-iy} II., 

y = a{(x - I) 3 + vi] III., 

according as its roots « and ji are real (say a = l + Jin, 



xvni VARIATION OF ca^ + bx + c 455 

(3 = I - \/m), equal (say a = l,f$ = l), or imaginary (say a = I + i sjm, 
(3 = 1 - i x /m). I and m are both essentially real quantities, and 
m is positive. 

Each of these three cases may be farther divided into two, 
according as a is positive or negative. 

In all three cases when x is very great (x - I) 3 is very great 
and positive. Hence, in all three cases, y is infinite when x is 
infinite, and it has the same sign as a. 

In all three cases the function within the crooked bracket 
diminishes in algebraical value when x diminishes, so long as 
x > I, and has an algebraically least value when x = I ; for (x - If, 
the only variable part, being essentially positive, cannot be less 
than zero. When x is diminished beyond the value x-l, (x - If 
continually increases in numerical value. 

We conclude, therefore, that in all three cases the quadratic 
function y has an algebraical minimum or maximum value when x = l, 
according as a is positive or negative; and that the function has no 
other turning value. 

In Case I., where the roots are real and unequal, y will have the 
same sign as a or not, according as the value of x does not or does lie 
between the roots. 

For y = a(x - a)(x - (3) ; and (x - a) (x - fS) will be positive if 
x be algebraically greater than both a and (3, for then x - a and 
x - (3 are both positive ; and the same will be true if x be alge- 
braically less than both a and (3, for then x - a and x- (3 are 
both negative. If x lie between a and (3, then one of the two, 
x - a, x- (3, is positive and the other negative. 

In Cases II. and III, where the roots are either equal or imaginary, 
the function y will have the same sign as a for all values of x. 

For in these cases the function within the crooked brackets 
has clearly a positive value for all real values of x. 

§ 10.] The above conclusions may be reached by a different 
but equally instructive method as follows : — 

Let us trace the graph of the function 

y = ax z + bx + c (1); 

and, for the present, suppose a to be positive. 



456 



GRAPH OF A QUADRATIC FUNCTION 



CHAP. 



To find the general character of the graph, let us inquire 
where it cuts a parallel to the axis of x, drawn at any given 
distance y from that axis. In other words, let us seek for the 
abscissa? of all points on the graph whose ordinatcs are equal to y. 

We have 

y = ax" + bx + c, 

that is, ax 2 + bx + (c - y) = (2). 

We have, therefore, a quadratic equation to determine the 
abscissa? of points on the parallel. Hence the parallel cuts the 
graph in two real distinct points, in two coincident real points, 
or in no real point, according as the roots of (2) are real and 
unequal, real and equal, or imaginary. 

Since a is positive, it follows that when x= - <x> , y = +co; 
and when x- + ao , y = + oo . Moreover, the quadratic function 
y is continuous, and can only become infinite when x becomes 
infinite. Hence there must be one minimum turning point on 
the graph. There cannot be more than one, for, if there were, 
it would be possible to draw a parallel to the &-axis to meet 
the graph in more than two points. 

The graph therefore consists of a single festoon, beginning 
and ending at an infinite distance above the axis of x. 

The main characteristic point to be determined is the 

minimum point. To obtain this we 
have only to diminish y until the 
parallel UV (Fig. 1) just ceases to 
meet the graph. At this stage it is 
obvious that the two points U and 
V run together ; that is to say, the 
two abscissa? corresponding to y 
become equal. Hence, to find //, we 
have simply to express the condition 
that the roots of (2) be equal. This 
condition is 
4a (c - y) = 0. 

y=- h lz^ (3). 

J 4a v 





V 




/ 




V^-f^M/ 




/ 

/ 
/ 
/ 
/ 
/ 
/ 
/ 

/ 


' A S 

\ 
\ 
\ 

\ 

V 

\ 
\ 

\ 



Fig. 1. 



lr 



Hence 



XVIII 



THREE FUNDAMENTAL CASES 



457 



The corresponding value of x is easily obtained, if we notice 
that the sum of the roots of (2) is in all cases - b/a, and that 
when the two are equal each must be equal to - b/2a. Hence 
the abscissa of the minimum point is given by 

b_ 

9. a. 



X = 



(0. 



There are obviously three possible cases — ■ 

I. The value of y given by (3) may be negative. Since a is 
supposed positive, this will happen when b 2 - iac is positive. 

In this case the minimum point A will lie below the axis of 
.<■, and the graph will be like the fully drawn curve in Fig. 1. 

Here the graph must cut the a'-axis, hence the function y 
must have two real and unequal roots, namely, x = OL, x = OM ; 
and it is obvious that y is positive or negative, that is, has the 
same sign as a or the opposite, 
according as x does not or does 
lie between OL and OM. 

II. The value of y given 
by (3) will be zero, provided 
b 2 - inc = 0. 





Fig. 2. 

In this case the minimum point 
A falls on the axis of x, and the 
graph will be like the fully drawn 
curve in Fig. 2. 

Here the two roots of the function 
are equal, namely, each is equal to OA. 
It is obvious that here y is always 
positive, that is, has the same sign as a. 
III. The value of y given by (3) 
will be positive, provided b 2 - iac be negative. 

In this case the graph will be like the fully drawn curve in 



Fig. 3. 



Fig. 



3. 



458 EXAMPLES 



CHAP. 



Here the graph does not cut the axis of x, so that the 
function has no real roots. Also y is always positive, that is, 
has the same sign as a. 

If we suppose a to be negative, the discussion proceeds 
exactly as before, except that for positive we must say negative, 
and for minimum maximum. The typical graphs in the three 
cases will be obtained by taking the mirror-images in the axis of 
x of those already given. These graphs are indicated by dotted 
lines in Figs. 1, 2, 3. 

For simplicity we have supposed the abscissae of the points 
L, M, N, A to be positive in all cases. It will of course happen in 
certain cases that one or more of these are negative. The cor- 
responding figures are obtained in all cases simply by shifting 
the axis of y through a proper distance to the right. 

Example 1. 

To find for what valuesof x the function y = 2x 2 - 12.T + 13 is negative, and 
to find its turning value. 

We have y = 2(.r 2 - 6ar + 9) - 5, 



= 2{(z-3) 2 - 



4i 



= 2{z-(3-Vf)}{a;-(3+Vf)}- 
Hence ?/ will be negative if x lie between 3- V(°/2) and 3 + \/(5/2), and will 
be positive for all other values of x. 

Again, it is obvious, from the second form of the function, that y is 
algebraically least when (x- 3) 2 = 0. Hence y= - 5 is a minimum value of y 
corresponding to x=Z. 

Example 2. 

To find the turning values of (a; - 8a; + 15)/a;. 

15 „ 



We have y—x + 



x 



First, suppose x to be positive, then we may write 



from which it appears that y has a minimum value, -8 + 2a/15, when 
\Jx- \/(15/.c) = 0, that is, when x=\J\5. 

Next, let x be negative, = -£ say, then we may write 



» 15 



2Vi5-(vf- V ^)* 



win MAXIMA AND MINIMA 459 

from which we see that - 8 - 2\/15 is a maximum value of y corresponding to 
£ = /^/l 5, that is, to x — - \/l 5. 

Example 3. 

If a and y be both positive, then — 

If x + y be given, the greatest and least values of xy correspond to the 
least and greatest values of (x-y) 2 ; so that the maximum value of xy is 
obtained by putting x=y, if that be possible under the circumstances of the 
problem. 

If xy be given, the greatest and least values of x + y correspond to the 
greatest and least values of (x-y)- ; so that the minimum of x + y is obtained 
by putting x = y, if that be possible under the circumstances of the problem. 

These statements follow at once from the identity 

(x + y) 2 -(x-y) 2 =4xy. 



i,,y = c--(x-y) 2 . 



For, iix + y = c, then 

And, if xy=d a , then 

(x + y) 2 =id 2 + (.r-y) 2 . 

Hence the conclusions follow immediately, provided x and y, and therefore 
xy and x+y, be both positive. 

These results might also be arrived at by eliminating the value of y by 
means of the given relation. Thus, if x + y = c, then xy = x(c-x) = cx-x 2 
= c 2 /4-(c/2-x)' 2 . Hence xy is made as large as possible by making x as 
nearly =c/2 as possible, and so on. 

Many important problems in geometry regarding maxima 
and minima may be treated by the simple method illustrated in 
Example 3. 

Example 4. 

To draw through a point A within a circle a chord such that the sum of 
the squares of its segments shall be a maximum or a minimum. 

Let r be the radius of the circle, d the distance of A from the centre, x and 
y the lengths of the segments of the chords. 

Then, by a well-known geometrical proposition, 

xy=i*-d* (1). 

Under this condition we have to make 

u = x- + y- (2) 

a maximum or minimum. 

Now, if we denote x" and y- by £ and 77, then £ and 77 are two positive 
quantities ; and, by (1), we have 

!;V=(i*-d*)* (3). 

Hence, by Example 3, { + 77 is a minimum when = 77, and is a maximum 
when (£ -tj) 2 is made as great as possible. If we diminish 77, it follows, by 
(3), that I increases. Hence (f-77) 2 will be made as great as possible by 
making £ as great as possible. 



460 GEOMETRICAL MAXIMA AND MINIMA chap. 

Hence the sum of the squares on the segments of the chord is a minimum 
when it is bisected, and a maximum when it passes through the centre of 
the circle. 

Example 5. 

A and B are two points on the diameter of a circle, FQ a chord through 
B. To find the positions of PQ for which the area APQ is a maximum or a 
minimum. 

Let be the centre of the circle. The area OPQ bears to the area APQ 
the constant ratio OB : AB. Hence we have merely to find the turning 
values of the area OPQ. 

Let OB = a, and let x denote the perpendicular from on PQ. Then, if 
u denote the area OPQ, u = x\J(r 2 -x 2 ). 

We have therefore to find the turning values of u. Since u is positive, 
this is the same thing as finding the turning values of u 2 . Now 

,.4 / r 2N 

u- = x-(r 2 - x-) = — - f x- - — 

There are two cases to consider. First, suppose a>r/\J'2. Then, since 
the least and greatest values of x allowable under the circumstances are and 
a, we have, confining ourselves to half a revolution of the chord about A, 
three turning values. If we put x=0 we give to (* 2 -r 2 /2) 2 the greatest 
value which we can give it by diminishing x below r(\j2. Hence x = gives 
a minimum value of OPQ. 

If we put x = rj\j2, we give (x 2 -r 2 /2) 2 its least possible numerical value. 
Hence, for x = r/\j2, OPQ is a maximum. 

If we put x = a, we give (ar-r 2 /2) 2 the greatest value which we can give 
it by increasing x beyond rj\j2. Hence to x = a corresponds a minimum 
value of OPQ. 

Next, suppose a<rj\j2. In this case we cannot make x= or >rj\j2. 
Hence, corresponding to a' = 0, we have, as before, OPQ a minimum. But 
now (x 2 - r 2 /\/2j 2 diminishes continually as x increases up to a. Hence, for 
x = a, OPQ is a maximum. 

Remark. — This example has been chosen to illustrate a peculiarity that 
very often arises in practical questions regarding maxima and minima, 
namely, that all the theoretically possible values of the variable may not be 
admissible under the circumstances of the problem. 

Example 6. 

Given the perimeter of a right-angled triangle, to show that the sum of 
the sides containing the right angle is greatest when the triangle is isosceles. 

Let x and y denote the two sides, p the given perimeter. Then the 
hypotenuse i&p-x-y ; and we have, by the condition of the problem, 

{p-(x+y)}*=x*+y*. 

2 

Hence xy-p{x+y) = -^-. 

This again may be written 

{p-x){p-y)j£ (1). 



XVIII 



MAXIMA AND MINIMA, GENERAL METHOD 



461 



Under the condition (1) we have to make 

u — x-\-y 
a maximum. 

If we put £ —p - x, f) —p - y, we have 

to-2 



(2) 



(3); 



and we have to make u = 2p -(£ + *?) 

a maximum ; this is, to make £ + ?? a minimum. Now, under the condition 

(3), i + v is a minimum when £ = y. Hence x + y is a maximum when v — y. 

§ 11.] The method employed in § 10 for finding the turning 
points of* a quadratic function is merely an example of the 




Fm. 4. 



general method indicated in chap, xv., § 13. Consider any 

function whatever, say 

?/=/('•) (!)■ 

Let A be a maximum turning point on its graph, whose 

abscissa and ordinate are x and y. If we draw a parallel to OX 
a little below A, it will intersect the graph in a certain number 
of points, TUVW say. Two of these will be in the neighbour- 
hood of A, left and right of AL. If we move the parallel up- 
wards until it pass through A, the two points U and V will run 
together at A, and their two abscissae will become equal. If we 
move the parallel a little farther upwards, we lose two of the 
real intersections altogether. 

Hence to find y we have simply to express the condition that the 
roots of the equation « v _ _ ( . / \ 



462 



EXAMPLES 



CHAP. 



be equal, and then examine whether, if we increase y by a small 
amount, we lose two real roots or not. If we do, then y is a maximum 
value. 

If it appears that two real roots are lost, not by increasing but by 
diminishing y, then y is a minimum value. 

Example 1. 

To find the turning values of 

y = x s -9x° + 24x + 3. 

The values of x corresponding to a given ordinate y are given by 

x 3 -9x- + 2ix + (3 - y) = 0. 

If D denote the product of the squares of the differences of the roots of this 

cubic, then all its roots will be real, two roots will be ecpial or two imaginary, 

according as D is positive, zero, or negative. 

Using the value of D calculated in § 6, and putting pi= - 9, ^ 2 =24, 

p s = 3 - y, we find 

T)=-2?(y-19)(y-2S). 

Hence y = 19, ?/ = 23 are turning values of y. If we make y a little less than 
19, D is negative, that is, two real roots of the cubic are lost. Hence 19 is a 
minimum value of y. If we make y a little greater than 23, D is again 
negative ; hence 23 is a maximum value of y. 

It is easy to obtain 
the corresponding val- 
ues of a;, if we remember 
that two of the roots of 
the cubic become equal 
when there is a turning 
value. In fact, if the 
two equal roots be a, a, 




and the third root y, we 

have, by § 1, 

2a + 7 = 9, a 2 + 2ct7 = 24. 

Hence 

a 2 -6a 4- 8 = 0, 
which gives 

a = 2, or a = 4. 



Fig. 6. 



It will be found that x = i corresponds to the minimum value y = 19 ; and 
that x=2 corresponds to the maximum y = 23. 



XVIII 



EXAMPLES 



463 



Remark. — The above method is obviously applicable to any cubic integral 
function whatsoever, and we see that such a function has in general two 
turning values, which are the roots of a certain quadratic equation easily ob- 
tainable by means of the function D. 

If the roots of this quadratic be real and unequal, there are two distinct 
turning points, one a maximum, the other a minimum. 

If the roots be equal, we have a point which may be regarded as an 
amalgamation of a maximum point with a minimum, which is sometimes 
called a maximum-minimum point. 

If the roots be imaginary, the function has no real turning point. 

If the coefficient of a? be positive, the graphs in the first two cases have 
the general characters shown in Figs. 5 and 6 respectively. 

Example 2. 

To discuss the turning values of 

ar-S.T + 15 .... 

y= — — (D- 

The equation for the values of x corresponding to any given value of y is 

ai 2 -(?/ + 8);r+15 = 0. 
Let D be the function b 2 - Aac of § 5, whose sign discriminates the roots of a 
quadratic. In the present instance we have 

D = (y + 8) 2 -60={y-(-8-VG0)}{2/-(-8 + V60)} (2). 

Hence the turning values of y are 

y = _8- V(60), and y= -8 + V(60). 

If y has any value between 
these, D is negative, and the 
roots of (1) are imaginary. 
Hence the algebraically less of 
the two, namely, - 8 - V(60), 
is a maximum ; and the 
algebraically greater, namely, 
- 8 + V(60), a minimum. 

The values of x correspond- 
iugtotheseare atonce obtained 
from the equation x = {y + 8)/2. 
They are x= - \J{15) and 
x= + \/(15) respectively. 

The reader should examine 
carefully the graph of this 
function, which lias a discon- 
tinuity when x=0 (see chnp. xv 
corresponding values : — 

»=-«, -1, -0, 




§ 5). 



Fig. 7. 
We have the following series of 



+ 0, 



y=—ao, -24, — 00, + oo , 

Hence the graph is represented by Fig. 7 



+ 3, 
0, 



>3 
<5' 



>o, 



0, + 



+ 00, 

+ 00. 



464 GENERAL DISCUSSION OF chap. 

Example 3. 

To discuss generally the turning values of the function 

_ a,r 2 + b.v + c 
V ~a'x 2 + b'x + c' (1) ' 

The equation which gives the values of x corresponding to any given value 
of y is 

(a - a'y)x 2 + (b- b'y)x + (c - c'y) = 0. 

Let ~D = {b-b'yf-4{a-a'i/){c-c'y), 

= (ft' 2 - ia'c')y 2 + 2(2a'c + 2ac' - W)y + (Ir - lac), 
= Ay 2 + By + C, say. 

Then we have x= - (b-b'y)±^D 

2{a-ay) K ' 

The turning values of y are therefore given by the equation 

A2/ 2 + By + C = (3). 

I. If B 2 -4AC>0, this equation will have real unequal roots, and there 
will be two real turning values of y. 

If A be positive, then, for real values of x, y cannot lie between the roots 
of the equation (3). Hence the less root will be a maximum and the greater 
a minimum value of y. 

If A be negative, then, for real values of x, y must lie between the roots of 
(3). Hence the less root will be a minimum and the greater a maximum 
value of y. 

II. IfB 2 -4AC<0, the equation has no real root, and D has always the 
same sign as A. In this case the sign of A must of necessity be positive ; 
for, if it were not, there would be no real value of x corresponding to any 
value of y whatever. 

Hence there is a real value of x corresponding to any given value of y 
whatever ; and y has no turning values. 

III. If B 2 - 4AC = 0, we may apply the same general reasoning as in Case 
II. The present case has, however, a special peculiarity, as we shall see im- 
mediately. 

The criteria for distinguishing these three cases may be expressed in terms 
of the roots a, /3 and a', 13' of the two functions axr + bx-rC and a'x? + b'x + c', 
and in this form they are very useful. 
We have 
B 2 - 4AC = 4(2«c' + 2a 'c - bb'f - 4{b" - lac) (6' s - Aa'c'), 



f2 i.^y-Cg-4iUS-^}, 



a' a a a J \a- aj \a* a' 
= 4aV 2 [2ct'/3' + 2a/3 - (a + /S) (a' + /3') ] 2 - (a - /3) 2 (a' - /3') 2 } , 
= 4a 2 a' 2 { 2a' 13' + 2a/3 - (a + j8) (a' + /3') - (a - /3) (a' - /3') } 

x {2a'/3' + 2ai8-(o + j8)(a' + j8') + (a-j8)(o'-j8')}, 
= 16feV 2 (a - a') (a - /3') (/3 - a') (/3 - /3'). 
Hence it appears that the sign of B 2 - 4 AC depends merely on the sign of 
E = (a-a')(a-/3')(/3-a')(/3-/3') (4) 



xvi ir (ax~ + bx + c)l(a'r + b'x + c') 465 

Since ft, b, c, a', b', c' are all real, the roots of an? + bx + c and of 
a'x 2 + b'x + c', if imaginary, must be conjugate imaginaries. Hence, by 
reasoning as in § 6, we see that, if the roots of ax 2 + bx + c, or of a'x" + b'x + c', 
or of both, be imaginary, E is positive. 

The same is true if the roots of either or of both of these functions be 
equal. 

Consider, next, the case where a, /3, a, /3' are all real and all unequal. 

Since the sign of E is not altered if we interchange both a with a' and /3 
with fi', or both a with /3 and a' with /3'. we may, without losing generality, 
suppose that a is the algebraically least of the four, a, /3, a', /?', and that a' is 
algebraically less than /3'. If we now arrange the four roots in ascending 
order of magnitude, there are just three possible cases, namely, a, p, a', /3' ; 
a, a', /3', /3 ; a, a', /3, /3'. In the first case, a -a', a-/3', /3-a', /3-/3' have 
all negative signs ; in the second, two have negative signs, and two positive ; 
in the third, three have negative signs, and one the positive sign. It is, 
therefore, in the third case alone that E has the negative sign. The peculi- 
arity of this case is that each pair of roots is separated as to magnitude by 
one of the other pair. We shall describe this by saying that the roots inter- 
lace. 

Lastly, suppose E = 0. In this case one at least of the four factors, a - a', 
/? - /3', (3 - a', /3 - j8', must vanish ; that is to say, the two functions ax 2 + bx + c 
and a'x 2 + b'x + c' must have at least one root, and therefore at least one linear 
factor in common.* 

Hence, in this case, (1) reduces to 



say. Hence we have 



ft(*-a) 
J a'(x-a') {0) > 

a:-a' + a'-a a a(a' - a) 
V = a ■-«'(*-- a')" = a' + a^^') (6) " 



From (6) it appears that y has a discontinuity when x = a', passing from 
the value + co to - co , or the reverse, as X passes through that value ; but 
that, for all other values of x, y either increases or decreases continuously as 
x increases. Hence y has no real turning values in this case, unless we choose 
to consider the value y = aja, which corresponds to x= ±°o, as a maximum- 
minimum value. 

The graph in this case, supposing a/a', a, and a' - a to be both positive, is 
like Fig. 8, where OA = a, OA' = a', OB=aJa', 

To sum up — 

Case I. occurs when the roots of either or of both of the functions 
aa? + bx + c, a'x 2 + b'x + c' are imaginary or equal, and when all the roots 
are real but not interlaced. 

Case II. occurs when the roots of both quadratic functions are real and 
interlaced. 

* In the case where they have two linear factors in common, y reduces to 
a constant, a case too simple to require any discussion. 

VOL. I. 2 H 



466 



(ax* + bx + c)/(a'a? + b'x + c') 



CHAP. 



Case III. occurs when the two quadratic functions have one or both roots 
in common. In this case y reduces to the quotient of two linear functions, 
or to a constant, and has no maximum or minimum value properly so called. 

In the above discussion we have assumed that neither a nor a' vanish ; in 
other words, that neither of the two quadratic functions has an infinite root. 
The cases where infinite roots occur are, however, really covered by the above 



Y 


B 


^ 










X 





M 


A' 



Fig. 8. 

statements, as may be seen either by considering them as limits, or by work- 
ing out the expression for B 2 - 4AC in terms of the finite roots in the particular 
instances in question. 

In stating the above conclusions so generally as this, the student must 
remember that one of the turning values may either become infinite or corre- 
spond to an infinite value of x ; otherwise he ma}' find himself at a loss in 
certain cases to account for the apparent disappearance of a turning value. 

A great variety of particular cases are included under the general case of 
this example. If we put a' = 0, c' = 0, for instance, we have the special case 
of Example 2. 

As our object here is merely to illustrate methods, it will be sufficient to 
give the results in two more particular cases. 

Example 4. 

To trace the variation of the function 

_ a 2 -7a: + 6 
V ~x--8x + lo 
The quadratic for x in terms of y is 

(l-yy--(7-8ij)x+(6-15y) = Q. 
Hence 

D = (7-8 2 /)2-4(l-2/)(6-15 2 /) = 4{ 2 /-(t-V6)}{y-a+V6)}. 
Hence 7/2 -\/6 and 7/2 + \/6 are maximum and minimum values of y re- 
spectively. The corresponding values of a; are given by 



XVIII 



GRAPHS FOR PARTICULAR CASES 



467 



X = is 



r-8y 
1-2/ 



and are 9 + 2\/6 and 9 - 2\/6 respectively. We observe farther that y is dis- 
continuous when x = 3 and when x = 5 ; that when a; = + oo or = — oo , y = 1 ; 
and that the other value of a- for which ?/ = l is a:=9. 

We have thus the following table of corresponding values : — 
x=-co, 0, +1, +3-0, +3 + 0, +4-1, 



y= 


+ 1, +"4, 0, 


- 


°°, 


+ °°, 


+ 5-9, 

mill. 


x= 


+ 5-0, +5 + 0, 


+ 6, 


+ 9, 


+ 13-9, 


+ oo 


y= 


+ CO, -CO, 


o, 


+ 1, 


+ 1-05, 


+ 1. 



The graph has the 
general form indicated in 
Fig. 9, which is not 
drawn to scale, but dis- 
torted in order to bring 
out more clearly the 
maximum point 15. 

Example 5. 
To trace the variation 
of the function 



The quadratic for x is 



Y 




y 

i 
i 

i 


max. 

B 




„ m "" ™- 






\ 


/ i x 







M 


1 " 






x?-5x+4 

' x--8x+ 15' 



Fig. 9. 



(l-y)x 2 -(5-8y)x + (i-15y) = 0. 
Here we find 

D=4{(y-i) 2 +2}. 

Hence there are no real turning values. 

The graph will be found to be as in Fig. 10. 



Y 




\ 


^__ 







1 


X 



Fig. 10. 



468 MAXIMA AND MINIMA, METHOD OF INCREMENTS chap. 

Example 6. 

To find the turning values of z = x 2 + y 2 , given that cut?+bxy+cy 2 =l. 

We have, since ax 2 + bxy + cy 2 — l, 

x 2 + y 2 £ 2 + l 

~ ~ ax' 2 + bxy + cy 2 a£- + b% + c' 
where £ = ,r/y. 

We have now to find the turning values of z considered as a function of f. 
The quadratic for £ is 

(az-l)£- + bz£ + (cz-l) = 0. 

Hence the turning values of z are given by 

&V=4(oa-l)(cs-l), 

that is, by 

(b 2 -4ac)z 2 +i(a + c)z-i = 0. 

The result thus arrived at constitutes an analytical solution of the well- 
known problem to find the greatest and least central radii (that is, the semi- 
axes) of the ellipse whose equation is ax 2 + bxy + cy 2 = 1 . 

Remark. — The artifice used in this example will obviously enable us to 
find the turning values of u=J{x, y), when <p(x, y) = c, provided /(.r, y) and 
<p(x, y) be homogeneous functions of x and y whose degree does not exceed 
the 2nd. Indeed it has a general application to all cases where f(x, y) and 
<f>(x, y) are homogeneous functions ; the only difficulty is in discriminating 
the roots of the resulting equation. 

§ 12.] Examination of the Increment. — There is yet another 
method which is very useful in discussing the variation of 
integral functions. Suppose we give x any small increment, 
h, then the corresponding increment of the function fix) is 
f(x + h) -f(x). If this is positive, the function increases when x 
increases ; if it is negative, the function decreases when x increases. 
The condition that x = a corresponds to a maximum value of f(x) 
is therefore that, as x passes through the value a, f(x + h) -f(x) 
shall cease to he positive and begin to be negative, and for a 
minimum shall cease to be negative and begin to be positive. 

The practical application of the method will be best under- 
stood by studying the following example : — 

Example. 

To find the turning values of 

y = x 3 -9x 2 + 2ix + 3. 
Let I denote the increment of y corresponding to a \ ery small increment, 
h, of x ; then 

I = (./• + hf - 9(a! + h) 2 + 24(.c + h) + 8 - a? + 9.r ; - 24a - 3, 
= (3z 2 -18a: + 24)A + (3x- 9)/r + A :! . 



XVIII 



EXERCISES XXXVIII 469 



Now, since for our present purpose it does not matter how small h may 

be, we may make it so small that (3x - 9)h" + h 3 is as small a fraction of 

(3a; 2 - 18x+24)A as we please. Hence, so far as determining the sign of I is 

concerned, we may write 

I = (3x 2 -18a; + 24)A. 

Here h is supposed positive, hence the sign of I depends merely on the sign 

of 3ar-18a; + 24. Hence I will change sign when, and only when, x passes 

through a root of the equation 

3a; 2 - 18a: + 24 = 0. 

Hence the turning values of y correspond to x = 2 and x = i. 

Moreover, we have 

l=8(x-2)(x-4)h. 

Therefore, when a; is a little less than 2, I is positive ; and when x is a 
little greater than 2, I is negative. Hence the value of y corresponding to 
a' = 2 is a maximum. 

In like manner we may show that the value of y corresponding to a;=4 is 
a minimum. 

Exercises XXXVIII. 

(1. ) Find the limits within which x must lie in order that 8(.r 2 - a 2 ) - 65xa 
may be negative. 

Trace the graphs of 

(2.) y = x 2 -5x + 6. (3.) y= -Sx- + l2x-6. 

(4.) y= - 4a; 2 + 20a; - 25. 

Find the turning values of the following ; and discriminate between 
maxima and minima : — 

(5.) ae** + be~ kx . (6.) afx+a/(a-x). 

(7.) V(l+aO + V(l-*)- ( 8 -) x-l + s/(x+l). 

Trace the graphs of the following, and mark, in particular, the points 
where the graph cuts the axes, and the points where y has a turning value : — 
(9.) y = (a: 2 + 8a; + 16)/(a; 2 -7a; + 12). 
(10.) i/ = (a; 2 -7a; + 12)/(a; 2 +8a' + 16). 
(11.) y = (a; 2 + 8a; + 16)/(a; 2 -6a;+9). 
(12.) y = (a- 2 -10aj + 27)/(a; 2 -8,r + 15). 
(13.) y = (x 2 -8x + 15)/(x 2 -\0x + 27). 
(14.) 2/ = (ar-10a: + 27)/(a; 2 -14a; + 52). 

(15.) i/=(x 2 -9a: + 14)/(a^ + 2a;-15). (16.) y = (a; 2 +a;- 6)/(se*- 1). 

(17.) y = (x 2 + 5x + Q)/{2x + 3). (18.) y = l/(x? + $x + 5). 

(19.) y = (2a^ + a;-6)/(2a; 2 +5a;-12). 

(20. ) Show that the algebraically greatest and least values of (a; 2 + 2x- 2)/ 
(x 2 + 3a; + 5) are \/( 12 Al) an< i ~ V( 12 /H) J an( l n °d the corresponding values 
of a;. 

(21.) Show that (ax-b) (dz-c)/(bx-a)(cx- d) may have all real values, 
provided (a 2 - b-) (c- - d 2 ) > 0. 

(22.) Show that (ax 2 + bx + c)/(cx 2 + bx + a) is capable of all values if 



470 EXERCISES XXXV1I1 CHAP, xvtn 

b 2 >(a + c) 2 ; that there are two values between which it cannot lie if 
(a + c) 2 >b 2 >lac; and that there are two values between which it must lie 
if b 2 < iac (Wolstenholme). 

(23. ) If ra >pb, tlien the turning value of [ax + b)/(2}x + r) 2 is ar/4p(ra ~pb). 

Find the turning values of the following ; and discriminate maxima and 
minima : — 

(24.) (x-l)(x-3)jx 2 . (25.) (x-3)/(x 2 + x-3). 

(26. ) l(ax + b) 2 + l'(a'x + b'f + l"(a"x + b") 2 . 

(27. ) ax + by, given x 2 + y 2 = c 2 . (28. ) a 2 x 2 + b"-y 2 , given x + y = a. 

(29.) xy, given a 2 /x 2 + b 2 Jy 2 =l. (30.) x 3 y + x 2 y 2 + xy 3 , given xy = a 2 . 

(31. ) ax 2 + 2hxy + by 2 , given Ax 2 + 2Hxy + By 2 = 1. 

(32.) xy/^J(x 2 + y 2 ). (33.) (2x - 1) (Bx - 4) (x - 3). 

(34.) \l\Jx + \j\Jy, given x + y = c. 

(35.) To inscribe in a given square the square of minimum area. 

(36. ) To circumscribe about a given square the square of maximum area. 

(37.) To inscribe in a triangle the rectangle of maximum area. 

(38.) P and Q are two points on two given parallel straight lines. PQ 
subtends a right angle at a fixed point 0. To find P and Q so that the area 
POQ may be a minimum. 

(39.) ABC is a right-angled triangle, P a movable point on its hypotenuse. 
To find P so that the sum of the squares of the perpendiculars from P on the 
two sides of the triangle may be a minimum. 

(40.) To circumscribe about a circle the isosceles trapezium of minimum 
area. 

(41.) Two particles start from given points on two intersecting straight 
lines, and move with uniform velocities u and v along the two straight lines. 
Show how to find the instant at which the distance between the particles is 
least. 

(42.) OX, OY are two given straight lines ; A, B fixed points on OX ; P a 
movable point on OY. To find P so that AP 2 + BP 2 shall be a minimum. 

(43. ) To find the rectangle of greatest area inscribed in a given circle. 

(44.) To draw a tangent to a given circle which shall form with two given 
perpendicular tangents the triangle of minimum area. 

(45.) Given the aperture and thickness of a biconvex lens, to find the radii 
of its two surfaces when its volume is a maximum or a minimum. 

(46. ) A box is made out of a square sheet of cardboard by cutting four 
equal squares out of the corners of the sheet, and then turning up the flaps. 
Show how to construct in this way the box of maximum capacity. 

(47.) Find the cylinder of greatest volume inscribed in a given sphere. 

(48.) Find the cylinders of greatest surface and of greatest volume in- 
scribed in a given right circular cone. 

(49.) Find the cylinder of minimum surface, the volume being given. 

(50.) Find the cylinder of maximum volume, the surface being given. 



CHAPTEE XIX. 

Solution of Arithmetical and Geometrical Problems 
by means of Equations. 

§ 1.] The solution of isolated arithmetical and geometrical 
problems by means of conditional equations is one of the most 
important parts of a mathematical training. This species of 
exercise can be taken, and ought to be taken, before the student 
commences the study of algebra in the most general sense. It 
is chiefly in the applications of algebra to the systematic investi- 
gation of the properties of space that the full power of formal 
algebra is seen. All that we need do here is to illustrate one or 
two points which the reader will readily understand after what 
has been explained in the foregoing chapters. 

§ 2.] The two special points that require consideration in 
solving problems by means of conditional equations are the 
choice of variables, and the discussion or interpretation of the solution. 

With regard to the choice of variables it should be remarked 
that, while the selection of one set of variables in preference to 
another will never alter the order of the system of equations on 
whose solution any given problem depends, yet, as we have 
already had occasion to see in foregoing chapters, a judicious 
selection may very greatly diminish the complexity of the system, 
and thus materially aid in suggesting special artifices for its 
solution. 

With regard to the interpretation of the solution, it is im- 
portant to notice that it is by no means necessarily true that 
all the solutions, or even that any of the solutions, of the system 
of equations to which any problem leads are solutions of the 



472 INTERPRETATION OF THE SOLUTION chap. 

problem. Every algebraical solution furnishes numbers which 
satisfy certain abstract requirements ; but these numbers may 
in themselves be such that they do not constitute a solution of 
the concrete problem. They may, for example, be imaginary, 
whereas real numbers are required by the conditions of the con- 
crete case ; they may be negative, whereas positive numbers are 
demanded ; or (as constantly happens in arithmetical problems 
involving discrete quantity) they may be fractional, whereas 
integral solutions alone are admissible. 

In every concrete case an examination is necessary to settle 
the admissibility or inadmissibility of the algebraical solutions. 
All that we can be sure of, a priori, is that, if the concrete 
problem have any solution, it will be found among the algebraical 
solutions ; and that, if none of these are admissible, there is no 
solution of the concrete problem at all. 

These points will be illustrated by the following examples. 
For the sake of such as may not already have had a sufficiency 
of this kind of mental gymnastic, we append to the present 
chapter a collection of exercises for the most part of no great 
difficulty. 

Example 1. 

There are three bottles, A, B, C, containing mixtures of three substances, 
P, Q, 11, in the following proportions : — ■ 

A, aP + «'Q + «"R; 

B, bY + b'Q + b"R ; 

C, cP+c'Q + c"R. 

It i3 required to find what proportions of a mixture must be taken from A, 
B, C, in order that its constitution may be dP + d'Q + d"R (Newton, Arithmdica 
Universalis). 

Let x, y, z be the proportions in question ; then the constitution of the 
mixture is 

{ax + by + cz) P + (a'x + b'y + c'z)Q + [a"x + b"y ■*- c'z)R. 
Hence we must have 

ax + by + cz-d, a'x + b'y + c'z = d', a"x + b"y + c"z = d". 

The system of equations to which we are thus led is that discussed in 
chap, xvi., § 11, with the sole difference that the signs of d, d', d" are re- 
versed. 

If, therefore, ab'c" - ab"c' + bc'a" - be" a' + ca'b" - ca"b' 4= 0, we shall obtain a 
unique finite solution. Unless, however, the values of x, y, z all come out 
positive, there will be no proper solution of the concrete problem. It is in 



XIX 



EXAMPLES 473 



fact obvious, a priori, that there are restrictions ; for it is clearly impossible, 
for instance, to obtain, by mixing from A, P., C, any mixture which shall 
contain one of the substances in a proportion greater than the greatest in which 
it occurs in A, B, or C. 

Example 2. 

A farmer bought a certain number of oxen (of equal value) for £350. He 
lost 5, and then sold the remainder at an advance of £6 a head on the original 
price. He gained £365 by the transaction ; how many oxen did he buy ? 

Let x be the number bought ; then the original price in pounds is 350/a:. 
The selling price is therefore 350/a- + 6. Since the number sold was as -5, we 
must therefore have 

(x-5)(—+6j- 350=365. 

This equation is equivalent to 

6a; 2 - 395a; -1750 = 0, 
which has the two roots a; = 70 and x = -25/6. The latter number is in- 
admissible, both because it is negative and because it is fractional ; hence the 
only solution is a; = 70. 

Example 3. 

A'OA is a limited straight line such that OA = OA' = a. P is a point in 
OA, or in OA produced, such that OP=;?. To find a point Q in A'A such 
that PQ 2 = AQ.QA'. Discuss the different positions of Q asp varies from 
to its greatest admissible value. 

Let OQ = a?, a* denoting a positive or negative quantity, according as Q is 

right or left of O. Then PQ=±(aj-j>), A'Q = a + x, AQ=a-z; and we have 

in all cases 

(x-pf = (a + x)(a-x) = a 2 -x- (1). 

Hence x 2 -px + h(p 2 -a n -) = (2). 

The roots of (2) are ^±V(i« 2 ~ lP% 

These roots will be real if pr<2a- ; that is to say, confining ourselves to 
positive values ofp, \{p<\j2a. 

From (1) we see that in all cases where x is real it must be numerically 
less than a. Hence Q always lies between A' and A. 

When p = 0, the roots of (2) are ±a\J2 ; that is to say, the two positions of 
Q are equidistant from O. 

So long as p is <a, ftp 3 - a?) will be negative, and the roots of (2) will be 
of opposite sign ; that is to say, the two positions of Q will lie on opposite 
sides of O. Since the sum of the two roots is p( = OV), if QiQ 2 be the two 
positions of Q, the relative positions of the points will be as in Fig. 1, where 
OQ 2 = PQi. 

i n m 

A' Q 2 PQiA 

Fig. 1. 

When ,p = a, Qa moves up to O, and Qi up to A. 

If p>a, then both roots are positive, and the points will be as in Fig. 2, 
where OQ2=QiP. 



474 EXAMPLES 



CHAP. 



I, I IMI II 

A' O Q,>C<M PB 

Fig. 2. 

If OB = \/2a, then B is the limiting position of P for which a solution of 
the problem is possible. When P moves up to B, Qi and Q 2 coincide at C 
(OC being £OB). 

Example 4. 

To find four real positive numbers in continued proportion such that their 
sum is a and the sum of their squares b 2 . 

Let us take for variables the first of the four numbers, say x, and the 
common value, say y, of the ratio of each number to the preceding. Then 
the four numbers are x, xy, xy 2 , xy 3 . Hence, by our data, 

x + xy + xy 2 + xy 3 = a, 

x 2 + x 2 y 2 + xY + xhf = b 2 ; 

that is to say, x{l+y){l+y 2 )-a (1), 

x 2 (l + y 2 )(l + i/) = b 2 (2). 

From (1) we derive x 2 (l +y)\l + y 2 )" = ar (3), 

and from (2) and (3), rejecting the factor y 2 +l, which is clearly irrelevant, 
we derive 

a 2 (l+7/) = b%l+y 2 )(l+y) 2 (4). 

The equation (4) is a reciprocal biquadratic in y, which can be solved by the 
methods of chap. xvii. , § 8. 

For every value of y (1) gives a corresponding value of x. 

The student will have no difficulty in showing that there will be two 
proper solutions of the problem, provided a be > b. Since, however, the two 
values of y are reciprocals, and since x(l +y) (1 + y 2 ) = xy 3 {l + l/y) (1 + l/i/ 2 ), 
these two solutions consist merely of the same set of four numbers read for- 
wards and backwards. There is, therefore, never more than one distinct 
solution. 

Newton, in his Arithmctica Universalis, solves this problem by taking as 
variables the sum of the two mean numbers, and the common value of the 
product of the two means and of the two extremes. He expresses the four 
numbers in terms of these and of a and b, then equates the product of the 
second and fourth to the square of the third, and the product of the first and 
third to the square of the second. It will be a good exercise to work out the 
problem in this way. 

Example 5. 

In a circle of given radius a to inscribe an isosceles triangle the sum of 
the squares of whose sides is 2b 2 . 

Let x be the length of one of the two equal sides of the triangle, 2y the 
length of the base. 

If ABC be the triangle, and if AD, the diameter through A, meet BC in 
E, then, since ABD is a right angle, we have AB 2 = AD . AE. Hence 

x 2 = 2a s /(. l "-y 2 ) (1). 



XIX 



EXAMPLES 475 



Again, by the conditions of the problem, we have 

2x 2 + 4y 2 =2i 2 , 

that is, x 2 + 2y 2 =b 2 (2). 

From (1) and (2) we derive 

x i -6aW + 2a 2 b' i = (3). 

The roots of (3) are 

*V{«y( l -5)}; 

and the corresponding values of y are given by (2). 

The necessary and sufficient condition that the values of x and of y be real 
is that b<3a/\/2. When this condition is satisfied, there are two real posi- 
tive values of ,r, and if b>2a there are two corresponding real positive values 
ofy. 

It follows from the above that, for the inscribed isosceles triangle the 
sum of the squares of whose sides is a maximum, b = Ba/\/2. Corresponding 
to this we have x=\J2>a, 2y=\/3a ; that is to say, the inscribed triangle, the 
sum of the squares of whose sides is a maximum, is equilateral, as is well 
known. 

Example 6. 

Find the isosceles triangle of given perimeter 2p inscribed in a circle of 
radius a ; show that, if 2p be less than 3\/3, and greater than 2a, there are 
two solutions of the problem ; and that the inscribed triangle of maximum 
perimeter is equilateral. 

Taking the variables as in last example, we find 

x* = 2a^(x'>-y") (1), 

x + y=p (2). 

Hence a- 4 - 8a 2 px + idy" = (3). 

We cannot reduce the biquadratic (3) to quadratics, as in last example ; 
but we can easily show that, provided p be less than a certain value, it has 
two real positive roots. 

Let us consider the function 

y = x*-8a' i px+iay- (4); 

and let I be the increment of y corresponding to a very small positive incre- 
ment (h) of x. Then we find, as in chap, xviii., § 12, that 

l = 4(.^-2a-p)h (5). 

Hence, so long as x i <2a"p, I is negative ; and when x 3 >2a 2 p, I is positive. 
Hence, observing that y— + <x> when x— ±oo , we see that the minimum value 
ofy corresponds to x= ^/(2arp), and that the graph of (4) consists of a single 
festoon. Hence (3) will have two real roots, provided the minimum point be 
below the ar-axis ; that is, provided y be negative when x= ^/(2a 2 p) ; that is, 
provided 

4n-])U — -«s +p§ I 



476 EXERCISES XXXIX chap. 

be negative ; that is, provided 2p<Z\J%a. It is obvious that both the roots 
are positive ; for when x = we have y = 4a 2 p 2 , which is positive ; hence the 
graph does not descend below the axis of x until it reaches the right-hand 
side of the axis of y. 

From the above reasoning it follows that the greatest admissible perimeter 
is SsjSa. When 2p has this value, the minimum point of the graph lies on the 
axis of x, and x— £/(2a 2 p>) = */(3\/3a 3 )= \JBa corresponds to two equal roots 
of (3). The corresponding value of 2y is given by 2y = 2p - 2x = B\/3a - 2\/3a 
= \jZa; in other words, the inscribed isosceles triangle of maximum peri- 
meter is equilateral. 

Another interesting way of showing that (3) has two equal roots is to dis- 
cuss the graphs (referred to one and the same pair of axes) of the functions 

y = x*, and y = 8a-px - iarp 2 . 
These can be easily constructed ; and it is obvious that the abscissae of their 
intersections are the real roots of (3). 

Exercises XXXIX. 

(1.) How long will an up and a down train take to pass each other, each 
being 44 yards long, and each travelling 30 miles an hour ? 

(2.) Diophantus passed in infancy the sixth part of his life, in adolescence 
a twelfth, then he married and in this state he passed a seventh of his life 
and five years more. Then he had a son whom he survived four years and 
who only reached the half of his father's age. How old was Diophantus when 
he died ? 

(3.) A man met several beggars and wished to give 25 pence to each ; but, 
on counting his money, he found that he had 10 pence too little for that ; and 
then made up his mind to give each 20 pence. After doing this he had 25 
pence over. What had he at first, and how many beggars were there ? 

(4.) Two bills on the same person are sent to a banker, the first for £580 
payable in 7 months, the second for £730 payable in 4 months. The banker 
gives £1300 for the two. What was the rate of discount, simple interest 
being allowed in lieu of discount ? 

(5. ) A basin containing 1 200 cubic metres of water is fed by three fountains, 
and can be emptied by a discharging pipe in 4 hours. The basin is emptied 
and the three fountains set on ; how long does it take to fill with the dis- 
charging pipe open ? — given that the three fountains each running alone 
would fill the basin in 3, 6, and 7 hours respectively. 

(6. ) If I subtract from the double of my present age the treble of my 
age 6 years ago, the result is my present age. What is my age ? 

(7.) A vessel is filled with a mixture of spirit and water, 70% of which is 
spirit. After 9 gallons is taken out and the vessel filled up with water, there 
remains 58 J°/ °' spirit : find the contents of the vessel. 

(8. ) Find the time between 8 and 9 o'clock when the hour and minute 
hands of a clock are perpendicular. 

(9. ) A and B move on two paths intersecting at O. B is 500 yards short 



XIX 



EXERCISES XXXIX 477 



of when A is at ; in two minutes they are equidistant from 0, and in eight 
minutes more they are again equidistant from 0. Find the speeds of A 
and B. 

(10.) I have a sum to buy a certain number of nuts. If I buy at the rate 
of 40 a penny, I shall spend 5d. too much, if at the rate of 50 a penny, 
lOd. too little. How much have I to spend ? 

(11.) If two numbers be increased by 1 and diminished by 1 respectively, 
their product is diminished by 4. If they be diminished by 1 and increased 
by 2 respectively, their product is increased by 16. Find the numbers. 

(12.) A is faster than B by p miles an hour. He overtakes B, who has 
a start of h miles, after a run of q miles. Required the speeds of A and B. 

(13.) To divide a given number a into two parts whose squares shall be 
in the ratio m: 1. 

(14.) Four apples are worth as much as five plums ; three pears as much as 
seven apples ; eight apricots as much as fifteen pears ; and five apples sell for 
twopence. I wish to buy an equal number of each of the four fruits, and to 
spend an exact number of pence ; find the least sum I can spend. 

(15.) A man now living said he was x years of age in the year x 2 . What 
is his age and when was he bom .' 

Remark on the nature of this and the preceding problem. 
(16.) OABCD are five points in order on a straight line. If OA = a, 
OB = 6, OC = c, OB = d, find the distance of P from O in order that 
PA : PD = FB : PC. (Assume P to lie between B and C.) 

(17.) A man can walk from P to Q and back in a certain time at the rate 
of 3£ miles an hour. If he walks 3 miles an hour to and 4 miles an hour back, 
he takes 5 minutes longer ; find the distance PQ. 

(18.) A starts to walk from P to Q half an hour alter B ; overtakes B mid- 
way between P and Q ; and arrives at Q at 2 P.M. After resting 7i minutes, 
he starts back and meets B in 10 minutes more. When did each start 
from P ? 

(19.) At two stations, A and B, on a line of railway the prices of coals are 
£p per ton and £q per ton respectively. If the distance between A and B be 
d, and the rate for the carriage of coal be £r per ton per mile, find the distance 
from A of a station on the line at which it is indifferent to a consumer whether 
he buys coals from A or from B. 

(20.) A merchant takes every year £1000 out of his income for personal 
expenses. Nevertheless his capital increases every year by a third of what 
remains ; and at the end of three years it is doubled. How much had he at 
first? 

(21.) A takes m times as long to do a piece of work as B and C together ; 
B n times as long as C and A together ; C x times as long as A and B together. 
Find a;; and show that l/(se+l) + l/(m+l)+l(»+l)=l. 

(22.) The total increase in the number of patients in a certain hospital in 
a certain year over the number in the preceding year was 2|%. In the 
number of out-patients there was an increase of 4% ; but in the number of 
in-patients a decrease of 11 %• Find the ratio of the number of out to the 
number of in-patients. 



478 EXERCISES XXXIX CHAP. 

(23.) The sum of the ages of A and B is now 60; 10 years ago their ages 
were as 5 to 3. Find their ages now. 

(24.) Divide 111 into three parts, so that one-third of the first part is 
greater by 4 than one-fourth of the second, and less by 5 than one-fifth of the 
third. 

(25.) In a hundred yards' race A can beat B by \" ; but lie is handicapped 
by 3 yards, and loses by 1 T V yards. Find the times of A and B. 

(26.) A and B run a mile, and A beats B by 100 yards. A then runs with 
C, and beats him by 200 yards. Finally, B runs with C ; by how much does 
he beat him ? 

(27. ) A person rows a miles down a river and back in t hours. He can 
row b miles with the stream in the same time as c miles against. Find the 
times of going and returning, and the velocity of the stream. 

(28. ) A mixture of black and green tea sold at a certain price brings a 
profit of 4°/ on the cost price. The teas sold separately at the same price 
would bring 5% and 3% profit respectively. In what proportion were the 
two mixed ? 

(29. ) If a rectangle were made a feet longer and b feet narrower, or a' feet 
longer and V feet narrower, its area would in each case be unaltered. Find 
its area. 

(30.) Two vessels, A and B, each contain 1 oz. of a mixture of spirit and 
water. If 1/mth oz. of spirit be added to A and l/?ith oz. of spirit to B, or if 
l/;ith oz. of water be added to A and 1/mth oz. of water to B, the percentages 
of spirit in A and B in each case become equal. What percentage of spirit 
is there in each ? 

(31.) A wine-merchant mixes wine at 10s. per gallon with spirit at 20s. per 
gallon and with water, and makes 25% profit by selling the mixture at lis. 8d. 
per gallon. If he had added twice as much spirit and twice as much water, he 
would have made the same profit by selling at lis. 3d. per gallon. How much 
spirit and how much water does he add to each 100 gallons of wine ? 

(32. ) Find the points on the dial of a watch where the two hands cross. 

(33.) Three gamesters agree that the loser shall always double the capital 
of the two others. They play three games, and each loses one. At the end 
they have each £«. What had they at first ? 

(34. ) A cistern can be filled in 6 hours by one pipe, and in 8 hours by 
another. It was filled in 5 hours by the two running partly together and 
partly separately. The time they ran together was two-thirds of the time 
they ran separately. How long did each run ? 

(35.) A horse is sold for £24, and the number expressing the profit per 
cent expresses also the cost price. Find the cost price. 

(36.) I spent £18 in cigars. If I had got one box more for the money, 
each box would have been 5s. cheaper. How many boxes did I buy ? 

(37.) A person about to invest in 3% consols observed that, if the price 
had been £5 less, he would have received ^°/ more interest on his money. 
Find the price of consols. 

(38.) Out of a cask containing 360 quarts of pure alcohol a quantity is 
drawn and replaced by water. Of the mixture a second quantity, 84 quarts 



XIX 



EXERCISES XXXIX 479 



more than the first, is drawn and replaced by water. The cask now contains 
as much alcohol as water. "What quantity was drawn out at first ? 

(39.) Find four consecutive integers such that the product of two of them 
may be a number which has the other two for digits. 

(40.) The consumption of an important commodity is found to increase as 
the square of the decrease of its price below a certain standard price (j>). If 
the customs' duty be levied at a given percentage on the value (a) of the 
commodity before the duty is paid, show that, provided the rate be below a 
certain limit, there are two other rates which will yield the same total 
revenue, and determine the rates which will yield the greatest and least 
revenues. 

(41.) A number has two digits, the sum of the squares of which is 130. 
If the order of the digits be reversed the number is increased by 18. Find 
the number. 

(42.) Three numbers are in arithmetical progression. The square of the 
first, together with the product of the second and third, is 16 ; and the square 
of the second, together with the product of the first and third, is 14. 

(43. ) To find three numbers in arithmetical progression such that their sum 
is 2a, and the sum of their squares 46 2 . 

(44.) The sides of a triangle are the roots of x 3 -ax 2 + bx-c = 0. Show 
that its area is \ \J {a( 4ab -a 3 - 8c) } . 

(45. ) The. hypotenuse of a right-angled triangle is h, and the radius of 
the inscribed circle r. Find the sides of the triangle. Find the greatest 
admissible value of r for a given value of h. 

The following are from Newton's Arithmetica Universalis, q.v., pp. 119 

ct scq. : — 

(46.) Given the sides of a triangle, to find the segments of any side made 
by the foot of the perpendicular from the opposite vertex. 

(47.) Given the perimeter and area of a right-angled triangle, to find the 
hypotenuse. 

(48.) Given the perimeter and altitude of a right-angled triangle, to find 

its sides. 

(49.) The same, given the hypotenuse and the sum of the altitude and the 

two sides. 

(50.) Find the sides of a triangle which is such that the three sides, a, b, c, 
and the perpendicular on a form an arithmetical progression. 

(51.) The same, the progression being geometric. 

(52.) To find a point in a given straight line such that the difference of 
its distances from two given points shall be a given length. 



CHAPTER XX. 

Arithmetic and Geometric Progressions and the 
Series allied to them. 

§ 1.] By a series is meant the sum of a number vf terms formed 
according to some common law. 

For example, if f(n) be any function of n whatsoever, the 
function 

/(l) +/(2) +/(3) + . . .+/(/•) + . . .+/(») (1) 

is called a series. 

/(I) is called the first term; /(2) the second term, &c; and/(r) 
is called the rth, or general, term. 

For the present we consider only series which have a finite 
number n of terms. 

As examples of this new kind of function, let f(n) — n, then we have the 
series 

1+2 + 3 + . . .+% (2) ; 

let f(n) — I /(a + bn), and we have the series 

a + b + ^+T2 + a + b3 + • • • + ^Ui (3) ; 

let f(n) — \Jnl{2- \Jn), and we have the series 



2- VI 2- V2 2-V3 ' ' 2-V» 
and so on. 

It is obvious that when the nth term of a series is given we 

can write down all the terms by simply substituting for n 1, 2, 

3, . . . successively. 

Thus, if the ?>th term be n- + 2n, the series is 

(l'- ! + 2.1) + (2 2 + 2.2) + (3- , + 2.3) + . . .+(» 8 +2tt), 

or 3 + 8 + 15 + . . .+{n 2 + 2n). 

It is not true, however, that when the first few terms are 



CHAT\ xx MEANING OF SUMMATION 481 

given we can in general find the nth term, if nothing is told us 
regarding the form of that term. This is sufficiently obvious 
from the second form in which the last series was written ; for 
in the earlier terms all trace of the law of formation is lost. 

If we have some general description of the nth term, it may 
in certain cases be possible to find it from the values of a certain 
number of particular terms. If, for example,, we were told that 
the nth term is an integral function of n of the 2nd degree, 
then, by chap, xviii., § 7, we could determine that function if the 
values of three terms of the series of known order were given. 

§ 2.] If we regard the series 

/(l)+/(2)+. . .+/(») 
as a function of n, and call it <f>(n), it has a striking peculiarity, 
shared by no function of n that we have as yet fully discussed, 
namely, that the number of terms in the function <$>(n) depends on the 
value of its variable. For example, 

</>(l)=l, 0(2) = 1 + 2, <£(3)=l + 2 + 3, 
and so on. 

It happens in certain cases that an expression can be found 

for (f>(n) which has not this peculiarity ; for example, we shall 

show presently that 

l2 2 02 a n (n + 1) (2n+ 1) 

1 +2 +3 + . . . +n " = -* '-? '. 

o 

On the left of this equation the number of terms is n ; on the 
right we have an ordinary integral function of n, the number of 
terms in which is independent of n, and which is therefore called 
a closed function of n. 

When, as in the example quoted, toe can find for- the sum of a 
series an expression involving only known functions and constructed by 
a fixed number of steps, then the series is said to admit of summation ; 
and the closed expression in question is spoken of as the sum, par 
excellence, of the series. 

The property of having a sum in the sense just explained is 

an exceptional one ; and the sum, where it exists, must always 

be found by some artifice depending on the nature of the series. 

What the student should endeavour to do is to group together, 

VOL. I 2 I 



482 ARITHMETIC SERIES chap. 

and be sure that he can recognise, all the series that can be 
summed by any given artifice. This is not so difficult as might 
be supposed ; for the number of different artifices is by no means 
very large. 

In this chapter we shall discuss two very important cases, 
leaving the consideration of several general principles and of 
several interesting particular cases to the second part of this work. 

SERIES WHOSE WTH TERM IS AN INTEGRAL FUNCTION OF n. 

§ 3.] An Arithmetic Series, or an Arithmetic Progression, as it 
is often called, is a series in which each term exceeds the pre- 
ceding by a fixed quantity, called the common difference. Let 
a be the first term, and b the common difference ; then the terms 
are a, a + b, a + lb, a + 3b, &c, the nth term being obviously 
a + (n - 1 )b. 

Here a and b may be any algebraical quantities whatsoever, 
the word " exceed " in the definition being taken in the algebraical 
sense. 

Since the nth. term may be written (a — b) + bn, where a - b 
and b are constants, we see that the nth. term of an arithmetical 
series is an integral function of n of the 1st degree. Such a 
series is therefore the simplest of the general class to be con- 
sidered in this section. 

The usual method of summing an A.P. is as follows. Let 
2 denote the sum of n terms, then 
2= a+ {a + b) + {a + 2b}+. . .+ {a + (n-l)b}. 

If we write the terms in the reverse order, we have 

2= {a+(n-l)b} + {a + {n-2)b}+ {a + (n-3)b} +. , . + a. 

If we now add, taking the pairs of terms in the same vertical 

line together, we find 

2S= {&* + (»- 1)6}+ {2a + {n-l)b} + {2a + (n-l)b}+. . . + {2a + (n-l)b}. 

Hence, since there are n terms, 

^ = ~{2a + (n-l)b} (1). 

This gives 2 in terms of n, a, b. 



xx EXAMPLES 483 

If we denote the last term of the series by I, we have 
l = a + (n-l)b. Hence 

^ a + 1 /«\ 

2 = »-y- (2). 

That is to say, the sum of n terms of an A. P. is n times the 
average of the first and last terms, a proposition which is con- 
venient in practice. 

Example 1. 

To sum the arithmetical series 5 + 3 + 1-1-3- . . . to 100 terms. Here 
a — 5 and b= - 2. Hence 

S=a^{2x5+(100-1)(-2)} > 

= 50(10-198), 

= - 9400. 
Example 2. 

To find the sum of the first n odd integers. 
The nth. odd integer is 2n - 1. Hence 

2 = 1 + 3 + 5+ . . . +(2»-l), 
l+(2n-l) 

= n 2 ' 

= n 2 . 
It appears, therefore, that the sum of any number of consecutive odd integers, 
beginning with unity, is the scpuare of their number. This proposition was 
known to the Greek geometers. 

Example 3. 

Sum the series 1-2 + 3-4 + 5 . . . to n terms. First suppose n to be 
even, —2m say. Then the series is 

2 = 1-2 + 3-4+. . . + (2m -I) -2m, 
= 1+3+ . . . +(2to-1) 
- 2 - 4 - . . . -2m. 

In each line there are m terms. The first line has for its sum m 2 , by Example 2. 
The second gives -m(2 + 2m)/2, that is, -m(m + l). Hence 

2 = wi 2 -m(m + l)= -m= — ■=. 

Next suppose n to be odd, =2»i- 1, say. 
Then we have 

2 = 1-2 + 3-4+ . . . +(2m-l). 

To find the sum in this case, all we have to do is to add 2m to our former 
result. We thus find 

2 = 2to-to = wi, 
_ n+\ 
2 • 
This result might be obtained even more simply by associating the terms of 
the given series in pairs. 



484 GENERAL INTEGRAL SERIES chap. 

§ 4.] The artifice of § 3 will not apply to the case where the 
nth term of a series is an integral function of n of higher degree 
than the 1st. We proceed, therefore, to develop a more general 
method. 

Let the nth term of the series be 

p n r +p 1 n r ~ 1 +p a n r ~ 2 + . . . +p r (1), 

where_p , p l} p. 2 , . . ., p r are independent of n. 

And let us denote the sums of the first, second, third, . . ., 
rth powers of the first n integral numbers by n s 1} n s 2 , n s a , . . ., 
n s r ; so that 



jjOj — x -r U T <J T . . . T l*j 

n * 2 =l 2 + 2 2 + 3 2 + . . . +n\ 

n . S 3=l 3 +2 3 +3 3 + . . . +n 3 , 
and so on. 

If 2 denote the sum of n terms of the series whose rath term 
is (1), we have, 

2=p l r +p l V- 1 +p 2 l r - 2 + . . . +p r , 
+ 2V2 r +p l 2 r ~ 1 +p 2 2 r - 2 + . . . +p r , 
+ p y+p l 3 r - l +p 2 3 r ~ 2 + . . . +p r , 



+ p n r +p i 7i r ~ 1 +p 2 n r ~ 2 + . . . +p r . 
Hence, adding in vertical columns, we have 

2 = 7WV + Pi n S r - i + P» n Sr-» + ■ • ■ + Wr (2). 

From this formula we see that we could sum the series whose 
general term is (1) if oidy we knew the sums of the first, second, 
third, . . ., 7th powers of the first n integers. 

These sums can be calculated successively by a uniform 
process, as we shall now show. 

§ 5.] To calculate n s,. 

If in the identity (,c + 1 )' - x' = 2x +• 1 we put successively 
z = n, x - n - 1 , . . ., x = 2, x - 1 , we have the following equations — 

(jh- l) 2 - n' = 2w +1, 

n' -(n~l) 2 = 2(n- 1)4-1, 



3 e -2 8 =2.2 -t 1, 
2 2 - 1 2 = 2 1 +1. 



XX 



SUMMATION OF l r + 2 r + . . . + n r 485 



If we add all these equations, the terms on the left mutually 
destroy each other, with the exception of two, which give 
(ft + l) 2 - 1 ; and those on the right, added in vertical columns, 
give 2 n s l + n. Hence 

(n+l) 2 -l = 2 n s 1 + ft (1). 

From this we have 

2 n s 1 = (ft+l) 2 ->+l), . 
= (ft + 1)«, 



wOi 



ft(ft + 1) 



(2); 



2 

a result which we might have obtained by the method of § 3, 
for 1 + 2 + . . . + n is an A.P. 

Cor. The sum of the first powers of the first n integers is an 
integral function of n of the 2nd degree. 
§ 6.] To calculate n s 2 . 

In the identity (x + l) 3 - z* = 3x 2 + Sx + 1 put successively 
x = n, x = n-l, . . ., x = 2, x=l, and we have 

(ft + l) 3 - ft 3 = 3ft 2 +Sn +1, 

ft 3 -(?i-l) 3 = 3(ft-l) 2 + 3(ft-l)fl, 



3 3 -2 3 = 3.2 2 +3.2 +1, 

2 3 -l 3 =3.1 2 +3.1 +1. 

Hence, adding all these equations, we have 

(n+lf-l = S n s 2 +S n s l + n (1). 

Using in (1) the value already found for „*„ we have 

3 n s 2 = (ft+l) 3 -|n(ft+l)-(ft + l), 

= ^{2(ft + l) 2 -3, i -2}, 



Hence 



= — — (2n* + n). 



_n(n + l)(2ft+ 1) 
~6~ 



(2) 



Cor. The sum of the squares of the first n integers is an integral 
function of n of the 3rd degree. 



486 SUMMATION OF V + 2 r + . . . + n f chap. 

§ 7.] To calculate n s 3 . 

In the identity (x + l) 4 - x* = 4x 3 + 6x 2 + Ax + 1 put success- 
ively x = n, x — n - 1, . . ., x— 2, x= I; add the n equations so 
obtained, and we find, as before, 

(n + l) 4 - 1 = 4„s 8 + 6„s 2 + 4 n s, + », 
or (ra + l) 4 -(%+ l) = 4 n s a + 6 n Sj + 4 n s, (1). 

Using the values of n s 2 and n Sj already found, we have 

4 w s 3 = n(n +l)(n 2 + 3n+ 3)- n(n + 1) (2w + 1) - 2»(w + 1 ), 
= »i(w + 1) (n 2 + 3n + 3 - 2rc - 1 - 2), 
= n 2 (?i + If. 
Hence 

A= |«^i)|' (3) . 

Cor. 1. w 5 3 zs aw integral function of n of the 4th degree. 
Cor. 2. T/je swm o/ <Ae cubes of the first n integers is the square 
of the sum of their first powers. 

§ 8.] Exactly as in § 7 we can show that 

(n + l) 5 - (n + 1) = 5 n s t + 10 n s 3 + I0 n s, + 5 n s, (1); 

and from this equation, knowing n Si, n s 2 , n s 3 , we can calculate 
n s 4 . The result is 

_ n(n + 1) (6n 3 + 9?i 2 + n - 1) 9 v 

wS< ~ " "30 - W 

§ 9.] This process may be continued indefinitely, and the 
functions „s 1} n s 2 , . . ., n s r _ l . . . calculated one after the other. 

Suppose, in fact, that n s u n s 2 , . . ., n s r . x had all been cal- 
culated. Then, just as in §§ 5-8, we deduce the equation 
(n + l) r+1 - (n + 1) = r+i C in s r + r+1 C 2n s r _, +. . . + r+1 C rn S, (1), 

where r +iC,, r +iC 2 , &c, are the binomial coefficients of ther + 1th 
order. 

The equation (1) enables us to calculate n s r . 

Cor. 1. n s r is an integral function of n of the r + \th degree, so 
that we may write 

n s r = q n r+1 + q^ r + qjf- 1 + . . . + &■+, 

and it is obvious from (1) that 



-i ) 



XX SUM OF ANY INTEGRAL SERIES 487 

1 1 

q ° ~ r+1 C, - r + 1 * 

Cor. 2. n s r is divisible by n(n + 1), so that we may write 

[n r - 1 1 

n s r = n(n + 1)J - — +p l n r -* + p. 2 n r ~ s + . . . +^ r _, V ; 

for this is true when r = 1, r = 2, r = 3, r = 4 ; hence it must be 
true for r = 5, for we have 

(n + 1)' - (n + 1) = 6 C in s 5 + 6 C 2n s 4 + 6 C 3n s s + 6 C 27i s a + AiAi 

and (■« + l) 6 - (/i + 1) is divisible by »(% + 1) ; and so on. 

§ 10.] We can now sum any series whose Tith term is re- 
ducible to an integral function of n. By § 4 and § 9, Cor. 1, 
we see that the sum of n terms of any series whose nth term is an 
integral function of n of the rth degree, is an integral function of n of 
the r + \th degree. We may, therefore, if we choose, in summing 
any such series, assume the sum to be A?i ,,+1 + ~Bn r + . . . + K ; 
and determine the coefficients A, B, . . ., K by giving particular 
values to ?7. If S,, S 2 , . . ., S r+2 be the sums of 1, 2, . . ., r + 2 
terms of the series, then it is obvious, by Lagrange's Theorem, 
chap, xviii., § 7, that the sum is 

r + 2 s (n - 1 ) ( n - 2) . . . (tt. - s + 1) (n - s - 1) . . . (n - r - 2) 
\ ' \s~\)(s-2) 1 (- 1) .. . (s-r- 2)' 

The following are a few examples : — 



Example 1. 

To sum the series 1, = a + (a + b) + (a + 2b) + . . . + {a + {n-l)b}. 

The ?ith term is (a-b) + nb. 

The n - 1th term is (a - b) + (n -l)b. 



The 2nd term is (a-b) + lb. 

The 1st term is (a - b) + lb. 

Hence 2 = (a- b)7i + b n si, 

= (a - V) n + b-~- — -, 



rn 



as was found in § 3. 



488 EXAMPLES 



CHAP. 



Example 2. 2 = l 2 + 3 2 + 5 2 + to n 


terms. 


The nth. term is (2m - 1 ) 2 = 4 m 2 - in + 1 . 




Hence 2 = 4?i 2 - in 


+ 1 


+ 4(m-1) 2 -4(m- 


■1) + 1 


+ 4.2 2 -4.2 


+ 1 


+ 4.1 2 -4.1 


+1. 


Hence, adding in vertical columns, we have 




2 = 4„s 2 - 4„si + n, 




?i(m + 1)(2m + 1) 
6 


-4^- 2 +1 Vn, 


_(2;i-1)m(2m+1) 





3 
Example 3. 

2 = 2.3.4 + 3.4.5 + 4.5.6+. . . to n terms. 

The Mth term is (m + 1) (?i + 2) (m + 3) = m 3 + 6?i 2 + 11m + 6. 

Hence 2 = „s 3 + 6„S2+11„Si + 6m, 

= | (n 4 + 10m 3 + 35m 2 + 50m). 

Example 4. 

A wedge-shaped pile of shot stands on a rectangular base. There are m 
and n shot respectively in the two sides of the lowest rectangular layer, m - 1 
and m - 1 in the two sides of the next rectangular layer, and so on, the upper- 
most layer being a single line of shot. Find the whole number of shot in the 
pile, m being greater than n. 

The number in the lowest layer is mil ; in the next (m - 1) (n - 1) ; in the 
next {in - 2) (m - 2), and so on ; the number in the last layer is {in-n-\) 
[n - m - 1 ), that is, (m -n + 1 ). 

Hence we have to sum the series 

S=m?i+(j»-l)(M-l) + (m-2)(»-2)+. . . +(m-n-l)(n-n- 1), 
in which there are n terms. 



The rth term of the series is (m-r-l)(n-r-l), that is, (m+l-r) 



(M + l-r), that is, (wi + 1)(m + 1)- (m + M + 2)?- + r 2 . 
Hence we may write the series as follows : — 

2= (wi + 1)(m + 1) -(?>i + m + 2)m +m 2 

+(m+l)(»+l) -(mi + m + 2)(?i-1) +(m-1) 2 



+ (wt + l)(/i + l) -(m + M + 2)2 +2 2 

+(m+l)(m+l) -(??i + m + 2)1 +1 2 , 

= n{m + l)(n + l)-(m + n + 2)„si + n « 2 > 

= {in + l)n(n+l)-§{m + n+2)n{n+l) + l / t(n+~l)(2n + l), 
= ln(n+l){Sm-n+l). 
Remark. — In working examples by this method the student must be care- 
ful to see that the scries is complete ; in other words, that there are exactly 
n terms, all formed according to the same law. If any terms are wanting, or 
if there are redundant terms, allowance must be made by adding or subtract- 
ing terms, as the case may be. 



XX 



GEOMETRIC SERIES 489 



SERIES WHOSE ftTH TERM IS THE PRODUCT OF AN INTEGRAL 
FUNCTION OF ft AND A SIMPLE EXPONENTIAL FUNCTION OF ft. 

§ 11.] The typical form of the ftth term in the class of series 
now to be considered is 

(p n g +p 1 n s ~ 1 + . • • + P& n , 
where p , p u . . ., p 8 , r are all independent of n, and s is any 
positive integer. 

The simplest case is that in which the integral function re- 
duces to a constant. The ftth term is then of the form p s r n , or 
sa,jp s r . i*" 1 , that is, at*" 1 , where a =p 8 r is a constant. 

The ratio of the ftth to the (ft - l)th term in this special case 
is ar n /ar n ~ l = r, that is to say, is constant. 

A series in which the ratio of each term to the preceding is con- 
stant is called a geometric series or geometric progression ; 
and the constant ratio in question is called the common ratio. 

If the first term be a and the common ratio r, the second 
term is ar • the third (ar)r, that is, ar 2 ; the fourth (ar*)r, that is, 
or 3 ; and so on. The ?ith term is ar n ~ l . A geometric series 
is therefore neither more nor less general than that particular 
case of the general class of series now under discussion which 
introduced it to onr notice. 

§ 12.] To sum a geometrical series. 

Let 2 = a + ar + ar 2 + . . .+ar n ~ 1 (1). 

Multiply both sides of (1) by 1 - r and we have 
(1 - r)2 = a + ar + ar 2 + . . . + ar n ~ l 

- ar - ar 2 - ... - ar n ~ 1 - ar' 1 , 
= a- ar n (2). 

1 - r n 

Hence 2 = a (3). 

1 — r 

Since the number of operations on the right-hand side of (3) 
is independent of n* we have thus obtained the sum of the 
series (1). 

Cor. If I be the last term of the series, then l = ar n - 1 and 
ar n = rl. Hence (3) may be written 

* Here we regard the raising of r to the ttth power as a single operation. 



490 



EXAMPLES 

a - rl 



CHAP. 



2 = 



Example 1. 



2=f+£+f+. 

In this case « = $, r = £. Hence 



1 -r 

. . to 10 terms. 



(*>• 



Example 2. 
Here a=l, r 






2=1-2+4-8+16 
2. Hence 

l-(- 2)"_l -(-l)"2" 
S_1 * l-(-2)~ 3 ' 

=i(l - 2 n ), if ?i. be even, 
= 1(1+2"), if ?i be odd. 
Example 3. 

2 = (a; + y) + (a; ? + xy + ?/ 2 ) + (a? + a% + xy 2 + ■>/) + . 
_x 2 -y" x?-y 3 x i -y i 
x-y x-y x-y 



to n terms. 



+ 



'WH~1 _ i/«+l 



a; -1/ 



= — (x 3 + £c 3 + . . .+a-»+ 1 )--^-(?/- + 2 / 3 + . 
x-y ' aj-y 



to ?i terms, 



+ 2/«+i), 



x-y 



{l+x- 



y 



Now 



and 
hnnce 



h. . . +ic"- 1 )-^— (l + w + . . .+7/"- 1 ). 
x-y 

l+ai+. . .+x n ~ 1 = {l-x n )/(l-x), 

1 + 1/ + . . .+2/ m_1 = (l-2/")/(l-2/). 

_ a: 2 (l - x 71 ) y-{\ - y n ) 



~{x-y){\-x) {x-y){l-y)' 

§ 13.] "We next proceed to consider the case where the integral 
function which multiplies r n is of the 1st degree. 

The general term in this case is 

(a + bn)r n (1 . 

where a and b are constants. 

It will be observed that a term of this form would result if 
we multiplied together the wth term of any arithmetic series by 
the nt\\ term of any geometric series. For this reason a series 
whose wth term has the form (1) is often called an arithmetico- 
geometric series. 

The series may be summed by an extension of the artifice 
employed to sum a G.P. 

Let 

2 = (a + b. 1>' + (a + b. 2)r 2 + (a + b. 3)/- s + . . . + (a + b. n)r n . 



XX 



ARITHMETICO-GEOMETRIC SERIES 491 



Multiply by 1 - r, and we have 

(1-rJZ 

= (a + b.iy + (a + b. 2)r 2 + (a + b.3)r 3 + . . . + (a + b n)r n 

- (a + b. l)r 2 - (a + b. 2)r 3 - . . . - (o + bn - 1 )r n 

- (a + bn)r n +\ 

= (a + b.l)r + \br + br 3 +. . . + br n \- (a + bn)r n + l (1). 

Looking merely at the terms within the two vertical lines, 
we see that these constitute a geometric series. Hence, if we 
multiply by 1 - r a second time, there will be no series left on 
the right-hand side ; and we shall in effect have found the 
required expression for 2. We have, in fact, 

= (1 - r) (a + b)r + h ? + br 3 + . . . + br n 

-br 3 -. . . - br n - br n+1 

- (1 - r) (a + bn)r n +\ 
= (1 - r) (a + b)r + br 2 - br n + l - (1 - r) (a + bn)r n +\ 
= (a + b)r - (a + b)r 2 + br 2 -{<* + (»+ l)b}r n + l + (a + bn)r n + 2 (2). 
Hence 

v (a + b)r - (a + b)r 2 + br 2 - {a + (n + I )6}r»+ 1 + (a + bn)r n + 2 
2 = ~ (l-r) s {6) - 

§ 14.] If the reader has not already perceived that the 
artifice of multiplying repeatedly by 1 - r will sum any series of 
the general form indicated in § 11, probably the following argu- 
ment will convince him that such is the case. 

Let f s (n) denote an integral function of n of the 5th degree ; 
then the degree of f g (n) -f/n - 1) is the (s - l)th, since the two 
terms in n s destroy each other. Hence we may denote /,(n) 
-fin - 1) by /,_,(»). Similarly, /,_,(«) -/,_,(« - 1) will be an 
integral function of n of the (5 - 2)th degree, and may be denoted 
by/,_ g (n), and so on. 

Consider now the series 

2=/s(l>-'+/*(2>- 2 + . • .+//»>" (I)- 

Multiply by 1 - r, and we have 



492 INTEGRO-GEOMETRIC SERIES chap. 

(1-rjS 

=/ s (iy+ //ay+ / s (3>- 3 +//»>• 

- f s (iy- /,(2)r 3 . . .-/ g (»-l)r» -fJny»+\ 
=f s (\y + I /,_ 1 (2)r 2 +f a .pY + . . . +/.. 1 (n>- | -/»"+ 1 (2). 

The series between the vertical lines in (2) is now simpler 
than that in (1) ; since the integral function which multiplies r n 
is now of the (s - l)th degree only. 

If we multiply once more by 1 - r we shall find on the right 
certain terms at the beginning and end, together with a series 
whose nth. term is now f s _ 2 (n)r n . 

Each time we multiply by 1 - r we reduce the degree of the 
multiplier of r n by unity. Hence by multiplying by (1 - r) s+1 
we shall extirpate the series on the right-hand side altogether, 
and there will remain only a fixed number of terms. 

It follows that any series whose nth term consists of an integral 
function of n of the sth degree multiplied by r n can be summed by 
simply multiplying by (1 —r) s+1 . 

This simple proposition contains the whole theory of the sum- 
mation of the class of series now under discussion. 

Example 1. 2 = IV + 2 V 2 + 3 V + . . . + n 2 r". 

Here the degree of the multiplier of r n is 2. Hence, in order to effect the 
summation, we must multiply by (1 - r) 3 . We thus find 
(1 - r) 3 2 
= 1V + 2V + 3V + 4V + . . .+ »V» 

-3.1V 2 -3.2V 3 -3. 3V 4 -. . . -3(?i-l)V*- 2,n-r n + l 

+ 3. IV + 3. 2V 4 -. . .+3(7i-2)V"-l-3(?i-l)V I + 1 + 3/iV"+ 2 
- IV-...- (»-3)V*- (?i-2)V"+ 1 -(7(.-iyV»+ 2 

- m V+ 3 , 
= r + ,-2 _ ( }l + 1 f r n+i + (2n- + 2>i- 1 >-"+ 2 - ?iV"+ 3 . 

Hence 

_r + r" - (n + 1 ) V"* 1 + (2?t 2 + 2ra - I )r n + 2 - WV+ 3 

S - (1-r) 8 

Example 2. 2=1- 2r+3r 9 - 4r 3 + . . . - 2?ir 2 "" 1 . 

Multiply by (1 + r) 2 , and we. have 
(l+r) 2 2 = l-2/- + 3r 2 - 4r»+. . . - 2»ir 2 "- 1 

+ 2;--2.2r 2 + 2.3?- 3 -. . . +2(2?i- l)?- 2 "- 1 - 2.2nr 2 " 

+ r 2_ 2r 3 + . . .- (2n,-2)r 2 »- 1 + (2w-l>r 2 ' ! -2«r 2 ' , + 1 , 
= 1 - (2n + l)r" n - 27ir"" +1 . 
Hence 

^ l-(2»,+ l)r 2 ' t -2?tr 2 "+ 1 

2 ~ (1 + r) 2 ~~ " 



xx CONVERGENCY OF GEOMETRIC SERIES 493 

If we put r=l, we deduce 

1-2 + 3-4 . . . -2«= -n, 
which agrees with § 3, Example 3, above. 

CONVERGENCY AND DIVERGENCY OF THE ABOVE SERIES. 

§ 15.] We have seen that the sum of n terms of a series 
whose nth term is an integral function of n is an integral function 
of n ; and we have seen that every integral function becomes 
infinite for an infinite value of its variable. Hence the sum of n 
terms of any series whose nth. term is an integral function of 
n may be made to exceed (numerically) any quantity, however 
great, by sufficiently increasing n. 

This is expressed by saying that every such series is 
divergent. 

§ 16.] Consider the geometric series 

2 = a + ar + ar 2 + . . . + ar n ~ \ 
If r = 1, the series becomes 

"S = a + a + a + . . . + a = na. 
Hence, by sufficiently increasing n, we may cause 2 to surpass 
any value, however great. 

If r be numerically greater than 1, the same is true, for we have 

a(r n - 1) 



2 = 



r-1 ' 

ir n a 



r-1 r-1 
Now, since r> 1, we can, by sufficiently increasing n, make r", 
and therefore ar n /(r - 1), as great as we please. Hence, by suffi- 
ciently increasing n, we can cause 2 to surpass any value, how- 
ever great (see Ex. ix. 46). 

In these two cases the geometric series is said to be divergent. 

If r be numerically less than 1, we can, by sufficiently increas- 
ing n, make r n as small as we please, and therefore ar n /(l - r) as 
small as we please. Hence, by sufficiently increasing n, we can 
cause 2 to differ from «/(l - r) as little as we please. This is 
often expressed by saying that when r is numerically less than 1, 
the sum to infinity of the series a + ar + ar 2 + ... is a! (I - r). 



494 EXAMPLE OF INFINITE GEOMETRIC SERIES chai\ 

In this case the series is said to be convergent, and to converge to 
the value a/ (I -r). 

There is yet another case worthy of notice. 
If r - - 1 , the series becomes 

~2 = a-a + a-a + . . . 

Hence the sum of an odd number of its terms is always a, and 
the sum of an even number of them always 0. The sum, there- 
fore, does not become infinite when an infinite number of terms 
are taken ; but neither does it converge to one definite value. A 
series having this property is sometimes said to oscillate. 

Example 1. 

Find the limit of the sum of an infinite number of terms of the series 



For n terms we have 



1 1 1 



s =*lzi£=i. 



2 1-4 ' 2"' 

Hence, when n is made infinitely great, 

2=1. 
This case may be illustrated geometrically as follows : — 

Let AB be a line of unit length. 

|— j j j — — | Bisect AB in P x ; bisect PiB in P 2 , 

A Pi Po P3 P4B P 2 B in P 3 ; and so on indefinitely. 

It is obvious that by a sufficient 
number of these operations we can come nearer to B than any assigned dis- 
tance, however small. In other words, if we take a sufficient number of 
terms of the series 

AP 1 + P 1 P 2 + P 2 P 3 + P3P 4 +. . ., 

we shall have a result differing from AB, that is, from unity, as little as we 
please. 

This is simply a geometrical way of saying that 

111 , 

2 + 2i + 2» + - • * adco=1 - 
Example 2. 

To evaluate the recurring decimal "34. 
Let 

^ S/ 34 34 34 

s= ' 34= Too + io^ + ioo"3 + - - ad ™' 

Then 2 is obviously a geometric series, whose common ratio, 1/100, is less 
than 1. Hence 

34 1 34 



2 = 



100 1-xfo 99 



xx PROBLEMS ON ARITHMETIC PROGRESSION 495 



PROPERTIES OF QUANTITIES WHICH ARE IN ARITHMETIC, 
GEOMETRIC, OR HARMONIC PROGRESSION. 

§ 17.] If a be the first term, b the common difference, n the 
number of terms, and 2 the sum of an arithmetic progression, 
we have 

2 = ^{2a + (n-l)b] (1). 

This equation enables us to determine any one of the four quan- 
tities, 2, a, b, n, when the other three are given. The equation 
is an integral equation of the 1st degree in all cases, except 
when n is the unknown quantity, in which case the equation is a 
quadratic. This last case presents some points of interest, which 
we may illustrate by a couple of examples. 

Example 1. 

Given 2 = 36, a = 15, &=-3, to find n. We have by the formula (1) 
above 

36=|{S0-(»-l)3}. 

Hence n 2 - lira + 24 = 0. 

The roots of this equation are n = B and n = 8. It may seem strange that 
there should be two different numbers of terms for which the sum is the same. 
The mystery is explained by the fact that the common difference is negative. 
The series is, in fact, 

15 + 12 + 9| +6 + 3 + 0-3-6] -9-. . .; 
and, inasmuch as the sum of the part between the vertical lines is zero, the 
sum of 8 terms of the series is the same as the sum of 3 terms. 

Example 2. 

2=14, a = Z, 6 = 2. 

The equation for n in this case is 

tt 2 + 2?i = 14. 

Hence ji= - 1± V(15)= +2-87 . . ., or -4-87 . . . 

The second of the roots, being negative, has no immediate reference to our 
problem. The first root is admissible so far as its sign is concerned, but it is 
open to objection because it is fractional, for, from the nature ot the case, n 
must be integral. It may be conjectured, therefore, that we have set our- 
selves an impossible problem. Analytically considered, the function n 2 + 2n 
varies continuously, and there is in the abstract no difficulty in giving to it 
any value whatsoever. The sum of an arithmetic series, on the other hand, 
varies per saltum ; and it so happens that 14 is not one of the values that 2 
can assume when a — B and b — 2. There are, however, two values which 2 



496 DETERMINATION OF ARITHMETIC SERIES BY TWO DATA chap. 

can assume between which 14 lies ; and we should expect that the integers 
next lower and next higher than 2 "87 would correspond to these values of 2. 
So, in fact, it is ; for, when w = 2 2 = 8, and when n = 3 2=15. 

§ 18.] An arithmetic progression is determined when its first 
term and common difference are given ; that is to say, when 
these are given we can write down as many terms of the pro- 
gression as we please. An arithmetic progression is therefore 
what mathematicians call a twofold manifoldness ; that is, it is 
determined by any two independent data. 

Bearing this in mind, we can write the most general arith- 
metic progressions of 3, 4, 5, &c. terms as follows : — 

a - (3, a, a. + {3, 

a - 3/3, a - (3, a + fi, a + 3/?, 

a -2(3, a -ft, a, a + (3, a + 2/3, 

&c, 

where a and (3 are any quantities whatsoever. It will be 
observed that in the cases where we have an odd number of 
terms the common difference is (3, in the cases where we have 
an even number 2(3. These formulas are sometimes useful in 
establishing equations of condition between quantities in A.P. 

Example 1. 

Given that the ^th term of an A.P. is P, and that the qth term is Q, to 
find the A.P. Let a be the first term and b the common difference ; then 
the ^th and qth terms are a + (j» - 1 )b and a + [q - l)b respectively. Hence 

a + (p-l)b = ~P, a + {q-l)b = Q. 
These are two equations of the 1st degree to determine a and b. 
We find 

& = (P-Q)/(i>-0). ct={(p-l)Q-(q-l)?}l(p-q). 

Example 2. 

If a, b, c be in A. P., show that 

a\b + c) + b-{c + a) + c\a + b) = y(a + b + cf. 

We may put a = a-p, b = a, c = a+/3. 

The equation to be established is now 

(a-/3)-(2a + /i) + a-.2a + (a + i3) 2 (2a-/3)=?(3c l ) 3 , 

— 6a s . 
Since a and /3 are independent of one another, this equation must be an 
identity. The left-hand side reduces to 



XX ARITHMETIC MEANS 497 

2a {(a - (3f- + (a + j8) 2 } + j8 {(a - /3) 2 - (a + 0)*} + 2a-\ 
= 2a{2a 2 + 2/3'} + /3{-4a/3} +2a 3 , 
= 6a :! . 
Hence the required result is established. 

§ 19.] If three quantities, a, b, c, be in A. P., we have 
b - a = c - b by definition. Hence 

b = (c + «)/2. 

In this case b is spoken of as the arithmetic mean between a 
and c. The arithmetic mean between two quantities is therefore 
merely what is popularly called their average. 

If a and c be any two quantities whatsoever, and A x , A 2 , . . ., A n 
n others, such that a, A,, A 2 , . . ., A n , c form an A. P., then A n A 2 , 
. . ., A n are said to be n arithmetic means inserted between a and c. 

There is no difficulty in finding A,, A 2 , . . ., A n when a and 
c are given. For, if b be the common difference of the A.P., 
a, A u A 2 , . . ., A, i; c, then 

A! = a + b, A 2 = a + 2b, . . ., A n = a + nb, 
and 6 = a + (n + 1 )b. 

From the last of these we deduce b — (c - a)j{n +1). Hence 
we have 

A C ~ a A O C ~ a S 

A,=a + -, A„ = a + 2 -, 6zc. 

n + 1 " n + 1 

N.B. — By the arithmetic mean or average of n quantities a,, a 2 , 
. . ., a a is meant («, + a„ + . . . + a tl )/n. 

In the particular case where two quantities only are in 

question, the arithmetic mean in this sense agrees with the 

definitions given above ; but in other cases the meanings of the 

phrases have nothing in common. 

Example 1. 

Insert 30 arithmetic means between 5 and 90 ; and find the arithmetic 

mean of these means. 

Let b be the common difference of the A. P. 5, Ai, A 2 , . . ., A M , 90. 

Then 

6=(90-5)/(80+l)=85/31. 

Hence the means are 

,85 n 85 _ 85 c 

5 + 3l' 5 + 2 -3l' 5 + 3 -3T &C ' ; 

,. t . 240 325 410 . 

that is, — , — , — , &c. 

VOL. I 2 K 



498 EXAMPLES chap. 

We have A 1 + A 2+ . . . + A„ = 1 ( r A^+A. | 

n n \ 2 J 

_A! + A» 
■~ 2~ ' 

_ 85 nn 85 \ /„ 

= (5 + 90)/2 = 95/2. 
Remark. — It is true generally that the arithmetic mean of the n arith- 
metic means between a and c is the arithmetic mean between a and c. 

Example 2. 

The arithmetic mean of the squares of n quantities in A. P. exceeds the 
square of their arithmetic mean by a quantity which depends only upon n 
and upon their common difference. 

Let the ?i quantities be 

a + b, a + 2b, . . ., a + nb. 
Then, by §§ 5 and 6, 

{a + bT + {a + 2b?+. . . +(a + nbf 
n 



If. , , , , ., , ,.,« (n + 1 2?i + l)\ 

:-■{ a-n + abnln + 1) + b 2 — ^ — - V, 

n I 6 J 



6 2 
= a 2 + ab(n + l) + -{2n 2 + 3n + l). 

. . f(a + b) + (a + 2b) + . . . +(a + nb)\ 3 / ra + l,\ a 
Again, |- — ) a= ^+__»ji 

TO 

= « 2 + «&( -,i + 1 ) + -( w 2 + 2n + 1 ). 

M 2 - 1 
Hence A. M. of squares - square of A.M. = 5 2 , 

which proves the proposition. 

§ 20.] If Z be the sum of n terms of a geometric progression 
whose first term and common ratio are a and r respectively, we 
have 

r n _ 1 

When any three of the four, 2, a, r, n, are given, this equation 
determines the fourth. "When either i: or a is the unknown 
quantity, we have to solve an equation of the 1st degree. When 
r is the unknown quantity, we have to solve an integral equation 
of the ?<th degree, which, if n exceeds 2, will in general be 
effected by graphical or other approximative methods. If n be 
the unknown quantity, we have to solve an exponential equation 
of the form r 11 = s, where r and s are known. This may be 



xx DETERMINATION OF GEOMETRIC SERIES BY TWO DATA 499 

accomplished at once by means of a table of logarithms, as we 
shall see in the next chapter. 

§ 21.] Like an A.P., a G.P. is a twofold manifoldness, and 
may be determined by means of its first term and common ratio, 
or by any other two independent data. 

In establishing any equation between quantities in G.P., it 
is usual to express all the quantities involved in terms of the 
first term and common ratio. Since these two are independent, 
the equation in question must then become an identity. 

Example 1. 

The ^th term of a G.P. is P, and the qth. term is Q ; find the first term 
and common ratio. 

Let a be the first term, r the common ratio. Then we have, by our data, 

Prom these, by division, we deduce 

?*-«=: P/Q, whence r=(?/Q) 1! (>>-<>K 

Using this value of r in the first equation, we find 

a = P/(P/Q)Ip- 1 V(p-«) = PP-d /(/>-«)Q(i-*>)/(«-*). 

Hence we have 

a = YV-iV(p-q)Qj}-p)!(<i-p) } r = pi/(i>-«)Qi/(«-j>). 

Example 2. 

If a, b, c, d be four quantities in G. P., prove that 

4(« 2 + b 2 + c 2 + d') - (a + b + c + df={a - bf + (c - df + 2(a - df. 
If the common ratio be denoted by r, we may put b — ra, c = r 2 a, d = r i a. 
The equation to be established then becomes 

4cr(l + >~ + r 4 + I s ) - a 2 {l + r + r 2 + r 3 ) 2 = a'-'(l - r) 2 + aV(l - r) 2 + 2a 2 (l - r 3 ) 2 , 
that is, 
4(1 + r 2 +r*+ r«) - (1 + It + 3r 2 + 4r= + dr 4 + 2r 5 + r 6 ) 

= l-2r + r 2 + r i -2r 5 + r K + 2- 4r 3 + 2r 6 , 
which is obviously true. 

§ 22.] When three quantities, a, b, c, are in G.P., b is called the 
geometric mean between a and c. 

We have, by definition, c/b = b/a. Hence b 2 = ac. Hence, if 
we suppose a, b, c to be all positive real quantities, b = + \/(ac). 
That is to say, the geometric mean between two real positive quantities 
is the positive value of the square root of their product. 

If a and c be two given positive quantities, and G 1; G 2 , . . ., G n 
n quantities, such that a, G lt G 2 , . , ., G n , c form a G.P., then 
G„ G 2 , . . ., G n are said to be n geometric means inserted between a 
and c. 



500 GEOMETRIC MEANS 



CHAP. 



Let r be the common ratio of the supposed progression. 
Then we have G, = ar, G 2 = ar 2 , . . ., G M = ar n , c = ar n+1 . From 
the last of these equations we deduce r = (c/a) 1/( - n+l \ the real 
positive value of the root being, of course, taken. Since r is thus 
determined, we can find the value of all the geometric means. 

The geometric mean of n positive real quantities is the positive 
value of the nth root of their product. This definition agrees with 
the former definition when there are two quantities only. 

Example. 

The geometric mean of the n geometric means between a and c is the 
geometric mean between a and c. 

Let the n geometric means in question be ar, ar 2 , . . ., ar n , so that 
c = ar n+1 . Then 

{ar. ar" . . . ar n ) Vn — (a"r l ^ 2 + " ■ • +»)i«, 

= {a 2 r"+ 1 ) 1 ' 2 , 

= {acr- } 
which proves the proposition. 

§ 23.] A series of quantities which are such that their reciprocals 
form an arithmetic progression are said to be in harmonic progression. 

From this definition we can deduce the following, which is 
sometimes given as the defining property : — 

If a, b, c be three consecutive terms of a harmonic progression, then 

ajc = {a-b)l(b-c) (1). 

For, by definition, I /a, 1/6, 1/e are in A. P., therefore 

1_1 1 _1 

b a c b 
a-b b - c 



Hence 
Hence 



ah be 
a-b ab a 
b — c be c ' 
which proves the property in question. 

§ 24.] A harmonic progression, like the arithmetic j)rogression, 
from which it may be derived, is a twofold munifoldness. The 
following is therefore a perfectly general form for a harmonic 



xx HARMONIC PROGRESSION, HARMONIC MEANS 501 

series, 1 J(a + b), l/(a + 2b), l/(a + 36), . . ., lfca + rib), . . ., 
for it contains two independent constants a and b ; and the 
reciprocals of the terms are in A. P. 

The following forms (see § 18) are perfectly general for 
harmonic progressions consisting of 3, 4, 5, . . . terms respect- 
ively : — 

l/(a-£), 1/a, l/(a + P); 

l/(a-3/3), l/(«-/3), l/(a + /3), l/(a+3/3); 
1/(0.-2(3), l/(«-/3), 1/a, l/(a + j8), l/(a+2/3); 

&c. 

The above formulae may he used like those in § 18. 

§ 25.] If a, b, c be in IIP., b is called the harmonic mean 
between a and c. We have, by definition, 1/c — 1/6 = 1/b — 1/a- 
Hence 2/b - 1/a + 1/c, and b = 2ae/(a + c). 

If a, H,, H B , . . ., H„, c form a harmonic progression, H,, H,, 
. . ., H w are said to be n harmonic means inserted between a and c. 

Since 1/a, 1/H,, 1/H 2 , . . ., 1/H n , 1/c in this case form an 
A. P., whose common difference is d, say, we have 

d = (l/c- l/a)/(n + 1) = (a - c)j(n + \)ac. 
Hence 
1/H, = 1/a + (a - c)/(n + l)ac, 1/H g = 1/a * 2(a - c)j(n + l)ac, &c; 
and H, = (n + l)ae/(a + nc), H 2 = (n + l)ac/(2a + (n - l)c), &c. 

If a quantify H be such that its reciprocal is the arithmetic mean 
of the reciprocals of n given quantities, H is said to be the harmonic 
mean of the n quantities. 

It is easy to see, from the corresponding proposition regard- 
ing arithmetic means, that the harmonic mean of the n harmonic 
means between a and c is the harmonic mean of a and c. 

§ 26.] The geometric mean between two real positive quantities a 
and c is the geometric mean between the arithmetic and the harmonic 
means between a and c ; and the arithmetic, geometric, and harmonic 
means are in descending order of magnitude. 

Let A, G, H be the arithmetic, geometric, and harmonic 
means between a and c, then 

A = (a + c)/2, G - + */(ac), H = 2ac/(a + c). 



502 ARITHMETIC, GEOMETRIC, AND HARMONIC MEANS CHAP. 

XT K-a a + c %ac 2 

Hence AH = —— x = ac = G , 

2 a + c 

which proves the first part of the proposition. 
Again, A - G = — \/(oc) = h( y/a - \U) 2 . 

G-H= sl(ac)- — =^ks/a- *Jc)\ 
a + c a+c 

Now, since a and c are both positive, Ja and N /c are both real, 
therefore ( s/a - Jcf is an essentially positive quantity ; also 
*J(ac) and a + c are both positive. Hence both A - G and G - H 
are positive. 

Therefore A >G>H. 

The proposition of this paragraph (which was known to the 
Greek geometers) is merely a particular case of a more general 
proposition, which will be proved in chap. xxiv. 

§ 27.] Notwithstanding the comparative simplicity of the 
law of its formation, the harmonic series does not belong to the cate- 
gory of series that can be summed. Various expressions can be 
found to represent the sum to n terms, but all of them partake 
of the nature of a series in this respect, that the number of steps 
in their synthesis is a function of n. 

It will be a good exercise in algebraic logic to prove that 
the sum of a harmonic series to n terms cannot be represented 
by any rational algebraical function of n. The demonstration 
will be found to require nothing beyond the elementary principles 
of algebraic form laid down in the earlier chapters of this work. 

Exercises XL. 

Sum the following arithmetical progressions : — 

(1.) 5 + 9 + 13+ ... to 15 terms. (2.) 3 + 3^ + 4 + ... to 30 terms. 

(3.) 13 + 12 + 11+. . . to 24 terms. (4.) i + l+ ■ ■ • to 16 term* 

,-k 1 , n- 1 

(o.) - + — + ... to 7t terms. 
n n 

(6.) {a~ny t + {a? + n-) + {a + n)-+ . . . to to terms. 

r) l + l, 41 7 + 

(<•) T _ r7 + -n772+ • • • to ( terms. 

(8.) The 20th term of an A. P. is 100, and the sum of 30 terms is 500 ; 
find the sum of 1000 terms of the progression. 



XX 



EXERCISES XL 503 



(9.) The first term of an A. P. is 5, the number of its terms is 15, and the 
sum is 390 ; find the common difference. 

(10.) How many of the natural numbers, beginning with unity, amount 
to 500500 ? 

(11.) Show that an infinite number of A.P.'s can be found which have 
the property that the sum of the first 2m terms is equal to the sum of the 
next m terms, m being a given integer. Find that particular A. P. having 
the above property whose first term is unity. 

(12.) An author wished to buy up the whole 1000 copies of a work which 
he had published. For the first copy he paid Is. ' But the demand raised 
the price, and for each successive copy he had to pay Id. more, until the 
whole had been bought up. What did it cost him ? 

(13.) 100 stones are placed on the ground at intervals of 5 yards apart. 
A runner has to start from a basket 5 yards from the first stone, pick up the 
stones, and bring them back to the basket one by one. How many yards 
has he to run altogether ? 

(14.) AB is a straight line 100 yards long. At A and B are erected per- 
pendiculars, AL, BM, whose lengths are 4 yards and 46 yards respectively. At 
intervals of a yard along AB perpendiculars are erected to meet the line LM. 
Find the sum of the lengths of all these perpendiculars, including AL and BM. 
(15.) Two travellers start together on the same road. One of them 
travels uniformly 10 miles a day. The other travels 8 miles the first day, 
and increases his pace by half a mile a day each succeeding day. After how 
many days will the latter overtake the former ? 

(16.) Two men set out from the two ends of a road which is I miles long. 
The first travels a miles the first day, a + b the next, a + 2b the next, and so 
on. The second travels at such a rate that the sum of the number of miles 
travelled by him and the number travelled by the first is always the same for 
any one day, namely c. After how many days will they meet ? 
(17.) Insert 15 arithmetic means between 3 and 30. 
(18.) Insert 10 arithmetic means between - 3 and +3. 
(19.) A certain even number of arithmetic means are inserted between 30 
and 40, and it is found that the ratio of the sum of the first half of these 
means to the second half is 137 : 157. Find the number of means inserted. 

(20.) Find the number of terms of the A. P. 1 +8 + 15+ . . . the sum of 
which approaches most closely to 1356. 

(21.) If the common difference of an A. P. be double the first term, the 
sum of m terms : the sum of n terms = m- : n-. 

(22.) Find four numbers in A. P. such that the sum of the squares of the 
means shall be 106, and the sum of the squares of the extremes 170. 

(23.) If four quantities be in A. P., show that the sum of the squares of 
the extremes is greater than the sum of the squares of the means, and that 
the product of the extremes is less than the product of the means. 

(24.) Find the sum of n terms of the series whose rth term is |(3r + 1). 
(25. ) Find the sum of n terms of the series obtained by taking the 1st, rth, 
2rth, 3rth, &c. terms of the A. P. whose first term and common difference are 
a and b respectively. 



504 EXERCISES XL 



CHAP. 



(26.) If the sum of n terms of a series be always n(n + 2), show that the 
series is an A. P. ; and find its first term and common difference. 

(27.) Show by general reasoning regarding the form of the sum of an 
A. P. that if the sum of p terms be P, and the sum of q terms Q, then the 
sum of n terms is Pn(?i - q)/p{p -q) + Qn(n -p>)lq{<l -p). 

(28.) Any even square, (2n) 2 , is the sum of n terms of one arithmetic 
series of integers ; and any odd square, (2ft + l) 2 , is the sum of n terms of 
another arithmetic series increased by 1. 

(29.) Find n consecutive odd numbers whose sum shall be nP. 

Show that any integral cube is the difference of two integral squares. 

(30.) Find the nth term and the sum of the series 
1-3 + 6-10 + 15-21+ .... 

(31.) Sum the series 3 + 6+ . . . +3w. 

(32.) If si, s 2 , . . ., s p be the sums of p arithmetical progressions, each 
having n terms, the fh-st terms of which are 1, 2, . . ., p, and the common 
differences 1, 3, . . ., 2p-l respectively, show that 5i + s L .+ . . . +g p is 
equal to the sum of the first np integral numbers. 

(33.) The series of integral numbers is divided into groups as follows : — 
1, | 2, 3, | 4, 5, 6, | 7, 8, 9, 10, | . . ., show that the sum of the nth group 
is |(?i 3 + ?i). 

If the series of odd integers be divided in the same way, find the sum of 
the ?ith group. 

Sum the following series : — 

(34.) 4 2 + 7 2 + . . . +(3u+l) 2 . (35.) 2„(?i 3 -l)(?i-l). 

(36.) Z n \p + q(n-l)}{p + q(n-2)}. 

(37.) l 2 -2 2 + 3 2 - . . . + (2?i-l) 2 -(2ra) 2 . 

(38.) a s + (a + bf+. . . +( a + n~^lb)*. 

(39.) (l3-i) + ( 2 3_2) + (3 3 -3)+. . . tow terms. 

(40.) 1.2 2 +2.3 2 + 3.4 2 +. . . ton terms. 

(41.) l + 2.3 2 + 3.5 2 + 4.7 2 + 5.9 2 +. . . to n terms. 

(42.) 1.3.7 + 3.5.9 + 5.7.11+ . . . to n terms. 

(43.) l 2 + (l 2 + 2 2 ) + (l 2 + 2 2 + 3 2 )+ . . . ton terms. 

(44. ) A pyramid of shot stands on an equilateral triangular base having 
30 shot in each side. How many shot are there in the pyramid ? 

(45.) A pyramid of shot stands on a square base having m shot in each 
side. How many shot in the pyramid ? 

(46.) A symmetrical wedge-shaped pile of shot ends in a line of m shot 
and consists of I layers. How many shot in the pile ? 

(47.) If B S r =l' , +2'-+ . . . +n',then n 8 r =pon<+ 1 +p 1 nr+. . . +p r+h where 
Po, Pi, • . . can be calculated by means of the equations 

r+lC 3 po + rCiJ?i = rQi, 
H-lQs p + pC2.Pl + r-lCi ^2 = rC» , 



„C r denoting, as usual, the rth binomial coefficient of the nth rank. 



XX 



EXERCISES XLI 



505 



(48.) Show that 
„S r =»(»+l)(n-l)(H-2)...(»-r)| 

r-lS r 



H 

r tn r 



r(r+l)(»-r)(r-l)! 

r -nO r 



(r-i)r(n-r+l)l\(r-2)\ ' (r-2)(r-l)(n-r + 2)2!(r-3)! 

( ~ ) r iS r 1 

1.2(7i-l)(r-l)!i ' 

where r ! stands for 1.2.3 . . . ?•. 

(49.) If ? r denote the sum of the products r at a time of 1, 2, 3, . . ., n, 
and S r denote l r + 2 r + . . . + W, show that ?-P r =SiP r _r-S 2 P,-2 + S3Pr-3~ . . . 
Hence calculate P 2 and P3. 

(50.) UJ[x) he an integral function of x of the (?•- l)th degree, show that 
f(x)- r C 1 f(x-l)+ r Q i f(x-2) . . . (-) r f{n-r)=0, r Ci, & 
coefficients. 

Exercises XLI. 
Sum the following geometric progressions : — 

(1.) 6 + 18 + 54+ . . . to 12 terms. (2.) 6-18 + 54- 

1 1 



being binomial 



(3.) -3333. . . to n terms. 
(5.) 6-4+ . . . to 10 terms 

^ V3 + 1 V3 + 2 



( 4 -) J-i + P" 



. to 12 terms, 
to n terms. 



to 20 terms. 



('■) 1+3+32+- • • 

(9.) l-x + x 2 -x 3 + 
(10.) V2 + ^+. • 

(12.) a ±?- a -^ + 



to n terms. 

. . to oo, a;<l. 

to 00. (11.) 

, . to 00 . 



(8.) 1-5+i- 



2 2' 

\/3 



V3 

V3 + 1 ' V 3 + 3 



+ 



to 00 . 



+ 



to 00. 



a-x a+x 

Sum, by means of the formula for a G.P., the following : — 
(13.) l+x-x 2 -x 3 + x 4 + x 5 -x 6 -x 7 + . . . tooo,a;<l. 

(14.) {x -y)+(l-^y(t 3 -ty. . . to * terms. 

(15.) l+(x + y) + (x 2 + o:y + y 2 ){-{x i + x 2 y + xy 2 + y 5 )+. . . to n terms. 

(16.) -33 +-333+ -3333 + . . . to n terms. 

Sum the series whose ?ith terms are as follows : — 

(17.) 2"3"+ 1 . (18.) {x n + n)(x n -n). 

(19.) Ae»-lVy»-.i). (20.) (jt»-gr)(p*+r). 

(21.) 2»(3"- 1 + 3»- 2 +. . .+1). (22.) (-l)"a 3 ''. 

(23. ) Sum to « terms the series (r n + l/>-") 2 + (r^ 1 + l/r"+ 1 ) 2 + . . . . 
(24.) Sum to ?i terms (l + l//-) 2 + (l + l/r 2 )-+. . . . 
(25.) Show that {a + b) n -b" = ab n ~ l + ab"~ 2 (a + b)+. . . +a(a + b) n -\ 
(26.) If S, denote the sum of n terms of a G.P. beginning with the tth 
term, sum the series S1 + S0+ . . . +S ( . 
(27.) Show that £/('037)=-3. 



506 EXEECISES XLI chap. 

Sum the following series to n terms, and, where admissible, to infinity : — 

(28.) l-2x + 3x 2 -lx s +. . . . (29.) 1-f + £-•£+. . . . 
9- S 2 4 2 12 2 3 3 4 

(so.) 1-I44+-. • • < 31 -> 1+ ir + 1r + ir + - ■ • ■ 

(32.) l 3 + 2 3 x + S s x 2 +. . . . 

12 3 12 3 
(33.) = + =5 + =; + =3 + ss + =a+. • . to oo, where the numerators recur 

7 7- i J i r r 

with the period 1, 2, 3. 

(34. ) - + — , + -= + -: + —. + -a + . . . to 3)i terms, where the numerators recur 

with the period a, J, c. 

(35.) A servant agrees to serve his master for twelve months, his wages to 
be one farthing for the first month, a penny for the second, fourpence for the 
third, and so on. What did he receive for the year's service ? 

(36. ) A precipitate at the bottom of a beaker of volume V always retains 
about it a volume v of liquid. It was originally precipitated in an alkaline 
solution ; find what percentage of this solution remains about it after it has 
been washed n times by filling the beaker with distilled water and emptying 
it. Neglect the volume of the precipitate itself. 

(37.) The middle points of the sides of a triangle of area Ai form the 
vertices of a second triangle of area A 2 ; from A 2 a third triangle of area A 3 is 
derived in the same way ; and so on, ad infinitum. Find the sum of the 
areas of all the triangles thus formed. 

(38.) OX, OY are two given straight lines. From a point in OX a perpend- 
icular is drawn to OY ; from the foot of that perpendicular a perpendicular 
on OX ; and so on, ad infinitum. If the lengths of the first and second per- 
pendicular be a and b respectively, find the sum of the lengths of all the per- 
pendiculars ; and also the sum of the areas of all the right-angled triangles in 
the figure whose vertices lie on OY and whose bases lie on OX. 

(39.) The population of a certain town is P at a certain epoch. Annually 
it loses d per cent by deaths, and gains b per cent by births, and annually a 
fixed number E emigrate. Find the population after the lapse of n years. 

2, a, r, n, I, having the meanings assigned to them in § 12, solve the 
following problems : — 

(40. ) Express 2 in terms of a, n, I ; and also in terms of r, n, I. 

(41.) 2= 4400, « = 11, »=4,.findr. 

(42.) 2 = 180, r = 3, n = 5, find a. 

(43.) 2 = 95, a = 20, n = S, find r. 

(44.) 2 = 155, « = 5, »=3, findr. 

(45.) 2 = 605, a=5, r = 3, find n. 

(46.) If the second term of a G.P. be 40, and the fourth term 1000, find 
the sum of 10 terms. 



XX 



EXEKCISES XLI, XLII 507 



(47.) Insert one geometric mean between \/3/\/2 and 3V3/2V2- 

(48.) Insert three geometric means between 27/8 and 2/3. 

(49.) Insert four geometric means between 2 and 64. 

(50.) Find the geometric mean of 4, 48, and 405. 

(51.) The geometric mean between two numbers is 12, and the arithmetic 
mean is 25§ : find the numbers. 

(52.) Four numbers are in G.P., the sum of the first two is 44, and of the 
last two 396 : find them. 

(53.) Find what common quantity must be added to a, b, c to bring them 
into G.P. 

(54.) To each of the first two of the four numbers 3, 35, 190, 990 is added 
SB, and to each of the last two y. The numbers then form a G.P. : find x 
and y. 

(55.) Given the sum to infinity of a convergent G.P., and also the sum to 
infinity of the squares of its terms, find the first term and the common ratio. 

(56.) IfS=Oi + a2+. . . + «„be a G.P., then S' = l/a 1 + l/a- 2 + . . . +l/a n 
is a G.P., and S/S' = «!«„. 

(57.) If four quantities be in G.P., the sum of the squares of the extremes 
is greater than the sum of the squares of the means. 

(58.) Sum In terms of a series in which every even term is a times the 
term before it, and every odd term c times the term before it, tire first term 

being 1. 

(59.) If x = a + a/r + a/r 2 + . . . ad oo , 

y = b- b/r+ bjr 2 - . . . ad oo , 
z = c + clr 2 +cjr i + . . . adoo, 
then xylz = ab/c. 

(60. ) Find the sum of all the products three and three of the terms of an 
infinite G.P., and if this be one-third the sum of the cubes of the terms, show 
that r=\. 

Exercises XLII. 

(1.) Insert two harmonic means between 1 and 3, and five between 6 and 8. 

(2.) Find the harmonic mean of 1 and 10, and also the harmonic mean of 
1, 2, 3, 4, 5. 

(3.) Show that 4, 6, 12 are in H.P., and continue the progression both 
ways. 

(4.) Find the H.P. whose 3rd term is 5 and whose 5th term is 9. 

(5.) Find the H.P. whose joth term is P and whose gill term is Q. 

(6. ) Show that the harmonic mean between the arithmetic and geometric 
means of a and b is 2(a + b)/ {{a/b^ + (b/a) i } n : 

(7.) Four numbers are proportionals ; show that, if the first three are in 
G.P., the last three are in G.P. 

(8.) Three numbers are in G.P. ; if each be increased by 15, they are in 
H.P. : find them. 

(9. ) Between two quantities a harmonic mean is inserted ; and between 
each adjacent pair of the three thus obtained is inserted a geometric mean. 



508 EXERCISES XLII chap, xx 

It is now found that the three inserted means are in A. P. : show that the ratio 
of the two quantities is unity. 

(10.) The sides of a right-angled triangle are in A. P. : show that they are 
proportional to 3, 4, 5. 

(11.) a, b, c are in A.P., and a, b, d in H.P. : show that c/d = 
1 - 2(o - bflab. 

(12.) If x be any term in an A. P. whose two first terms are a, b, y, the 
term of the same order in a H.P. commencing with the same two terms, then 
(x-a)/(y-x) = b/(y-b). 

(13.) If a 2 , b\ c 2 be in A. P., then 1/(5 + c), l/(c + a), lfta + b) are in A. P. 

(14.) If P be the product of n quantities in G.P., S their sum, and S' the 
sum of their reciprocals, then P 2 =(S/S') n . 

(15.) If a, b, c be the pth, qth, and rth terms both of an A.P. and of a 
G.P., then a*-" &*-»«»-» =1. 

(16.) If P, Q, R be the .pth, qth., rth terms of a H.P., then 2{PQ(j3- q)} 
= 0, and {S(g 2 -r 2 )/P 2 } 2 = 4{2to-r)/P 2 }{2 ? r( ? -r)/F}. 

/17.) If the sum of m terms of an A.P. be equal to the sum of the next n, 
and also to the sum of the next^J, then (m + n) (1/n - l/p) = (n +p) (1/m - I/n). 

(18.) If the squared differences of p, <?> r be in A.P., then the differences 
taken in cyclical order are in H.P. 

(19.) If a + b + c, a 2 + & 2 + c 2 , a? + b z + <? be in G.P., prove that the common 
ratio is ^2a - 3a&c/226c. 

(20.) If x, y, z be in A. P., ax, by, cz in G.P., and a, b, c in H.P., then 
H.M. of a, c:G.M. of a, c = H.M. oix, z:G.M. of a, z. 

(21.) Ifa 2 + 6 2 -c 2 , P + c--a", c 2 -i-a 2 -& 2 be in G.P., then a 2 /c 2 + c 2 /i 2 , 
& 2 /c 2 + c 2 /6 2 , a 2 /6 2 + 6 2 /c 2 are in A.P. 

(22.) If a, b, c, d be in G.P., then abcdl 2 - ] = (2a) 2 , and 

V(^ + ^) + V(6* + c*) + V(c« + * )_&«■ ^ „ 4 m 

(23.) The sum of the n geometric means between a and k is 
{aVWik - aWl»W)l(tM n + 1 ) - a 1 ( n +V). 

(24.) If Ai, A 2 , . . ., A„ be the n arithmetic means, and H^ H 2 , . . ., 
H„ the n harmonic means, between a and c, sum to n terms the series whose 
rth term is (A r -a)(H r -«)/H r . 

(25.) If«i,a 2 , . . ., a n be in G. P. , then 
(^ + a, 2 + 03)- + {d2 + a 3 + en) 2 + . . . + (a„_n + «.„_;, + a n f 

= («! 2 + «!«-, + «.o 2 ) 2 (a! 2 «- 4 - «.2 2 "- 4 )/(«l 2 - «2>1 5 "- 4 . 

(26.) If a r , b r be the arithmetic and geometric means respectively between 
« r _i and b r -i, show that 

a„- 2 - {a n *±(a„ 2 -&„ 2 ) } 2 , 
Z>„_ 2 = {aj 2f{a n 2 -b n -) } 2 . 
(27.) If «j, a 2 , . . ., a n be real, and if 
(ai 2 + 02 2 +. . . +a„-i 2 )(a 2 2 + (T 3 2 + . . . + a„ 2 ) = {aiao + a- 2 a i + . . . + a„-ia„), 2 
then ai, a- 2 , • ■ -, cin are in G.P. 



CHAPTEE XXI. 
Logarithms. 

§ 1.] It is necessary for the purposes of this chapter to define 
and discuss more closely than we have yet done the properties 
of the exponential function a x . For the present we shall sup- 
pose that a is a positive real quantity greater than 1. AY hat- 
ever positive value, commensurable or incommensurable, we give 
to x, we can always find two commensurable values, mjn and 
(m + l)/n (where m and n are positive integers), between which 
x lies, and which differ from one another as little as we please, 
see chap, xiii., § 15. In defining a x for positive values of x, we 
suppose x replaced by one (say m/n) of these two values, which 
we may suppose chosen so close together that, for the purpose 
in hand, it is indifferent which we use. We thus have merely 
to consider a'" ,ln ; and the understanding is that, as in the 
chapter on fractional indices, we regard only the real positive 
value of the nth root ; so that a m,n may be read indifferently as 
( Z/a) m , or as ya m . 

For negative values of x we define a x by the equation 
a x = lja~ x , in accordance with the laws of negative indices. 

§ 2.] We shall now show that a x , defined as above, is a continu- 
ous function of x susceptible of all positive values between and + oo . 

1st. Let y be any positive quantity greater than 1, and let 
n be any positive integer. Since a > 1 , a lln > 1 ; but, by suffi- 
ciently increasing n, we may make a 1!n exceed 1 by as little as 
we please. Also, when n is given, we can, by sufficiently in- 
creasing w, make a m!n as great as we please.* Hence, whatever 

* See chap, xi., § 14. 



510 



CC CONTINUOUS ITS GRAPH 



CHAP. 



may be the value of y, we can so choose n that a 1/n <y ; and then 
y will lie between two consecutive integral powers of a l!n ; say 
a min < y < a (»»+iy» Now the difference between these two values 
of a x is a mln (a 1/n - 1); and this, by sufficiently increasing n, we 
may make as small as we please. Hence, given any positive 
quantity y > \, we can find a value of x such that a x shall be as 
nearly equal to y as we please. 

2nd. Let y be positive and < 1 ; then 1/y is positive and 
greater than 1. Hence we can find a value of x, say x', such 
that a x ' = \jy as nearly as we please. Hence a~ x ' = y. 

We may make y pass continuously through all possible values 
from to + oo . Hence a x is susceptible of all positive values 
from to + oo . It is obviously a continuous function, since 
the difference of two finite values corresponding to x = m 'n and 
x = (wi + 1 )/n is a m!n (a 1/n - 1 ), which can be made as small as 
we please by sufficiently increasing n. 

Cor. We have the following set of corresponding values : — 
X= -oo, -, -1, 0, +, 1, +co; 
y = a x = 0. < 1, 1/a, 1, > 1, a, + oo . 




Fio. 1. 



In Fig. 1 the full-drawn curve is the graph of the function 
y ~ lO* ; the dotted curve is the graph of y = 100*. 



xxi DEFINITION OF LOGARITHMIC FUNCTION 511 

It will be observed that the two curves cross the axis of y 
at the same point B, whose ordinate is + 1 ; and that for one 
and the same value of y the abscissa of the one curve is double 
that of the other. 

§ 3.] The reasoning by Avhich we showed that the equation 
y = a x , certain restrictions being understood, determines y as a 
continuous function of x, shows that the same equation, under 
the same restrictions, determines a; as a contiimous function of y. 
This point will perhaps be made clearer by graphical considera- 
tions. If we obtain the graph of y as a function of x from the 
equation y = a x , the curve so obtained enables us to calculate x 
when y is given ; that is to say, is the graph of x regarded as a 
function of y. Thus, if we look at the matter from a graphical 
point of view, we see that the continuity of the graph means the 
continuity of y as a function of x, and also the continuity of x as 
a function of y. 

When we determine x as a function of y by means of the 
equation y = a x , we obviously introduce a new kind of transcend- 
ental function into algebra, and some additional nomenclature 
becomes necessary to enable us to speak of it with brevity and 
clearness. 

The constant quantity a is called the base. 

y is called the exponential of x to base a (and is sometimes 
written exp a ;r).* 

x is called the logarithm of y to base a, and is usually written 

log a y- 

The two equations 

y = a* (1), 

z = log a y (2), 

are thus merely different ways of writing the same functional 
relation. It follows, therefore, that all the properties of our new 
logarithmic function must be derivable from the properties of our old 
exponential function, that is to say, from the laws of indices. 

* This notation is little used in elementary text-books, but it is con- 
venient when in place of x we have some complicated function of x. Thus 
exp a (l +X + X 2 ) is easier to print than a 1+x+x2 . 



512 FUNDAMENTAL PEOPERTIES OF LOGARITHMS chap. 

The student should also notice that it follows from (1) and 
(2) that the equation 

y = a 1 **** (3) 

is an identity. 

§ 4.] If the same hase a be understood throughout, we have 
the following leading properties of the logarithmic function : — 

I. The logarithm of a product of positive numbers is the sum of 
the logarithms of the separate factors. 

II. The logarithm of the quotient of hvo positive numbers is the 
excess of the logarithm of the dividend over the logarithm of the divisor. 

III. The logarithm of any power {positive or negative, integral 
or fractional) of a positive number is equal to the logarithm of the 
number multiplied by the power. 

Let ?/,, y 2 , . . ., y n be n positive numbers, x lt x 3 , . . ., x n their 
respective logarithms to base a, so that 

«i = log a ?/ 1) x 2 = \og a y 2 , . . ., x n = log a y n . 
By the definition of a logarithm we have 

y x = a x \ y 2 = a x \ . . ., y n = a x \ 
Hence y^Ji- ■ ■ II n = a Xl a x - . . . a Xn - a x i +x *+ • ■ • + Xn , 

by the laws of indices. 
Hence, by the definition of a logarithm, 

x x + x. 2 + . . . + x n = logafay, . . . y n ), 
that is, log^ + log a ?/ 2 + . . . + log a y n = log (y,y 9 . . . y n ). 
We have thus established I. 

Again, y 1 /y 2 = a x i/a x ~ = a x ^ _a ' 2 , 

by the laws of indices. 
Hence, by the definition of a logarithm, 

x x -x a = logaiyM, 
that is, log a y, - \og a y, = logatyjyj), 

which is the analytical expression of II. 

Again, y* = (a x i) r = a rx \ 

by the laws of indices. 



xxi EXAMPLES 513 

Hence, by the definition of a logarithm, 

ra, = \og a y x \ 

that is, r log a y 1 = log*^, 

which is the analytical expression of III. 

Example 1. 

log 21 = log (7x3) = log 7 + log 3. 

As the equation is true for any base, provided all the logarithms have the 
same base, it is needless to indicate the base by writing the suffix. 

Example 2. 

log (113/29)= log 113 -log 29. 

Example 3. 

log (540/539) = log (2 2 . 3 3 . 5/7 2 . 11), 

= 2 log2 + 3 log3 + log5-2 log7-logll. 
Example 4. 

log 4/(49)/ 7/(21) = log (7 2 '*/3^7n 

= #log7-f log3-f log 7, 
=Hlog7-f log 3. 



COMPUTATION AND TABULATION OF LOGARITHMS. 

§ 5.] If the base of a system of logarithms be greater than 
unity, we have seen that the logarithm of any positive number 
greater than unity is positive ; and the logarithm of any positive 
number less than unity is negative. 

The logarithm of unity itself is always zero, whatever the base 
may be. 

The logarithm of the base itself is of course unity, since a = a 1 . 

The logarithm of any power of the base, say a r , is r ; and, in 
particular, the logarithm of the reciprocal of the base is - 1. 

The logarithm of + oo is + cc , and the logarithm of is - oo . 

It is further obvious that the logarithm of a negative number 
could not (with our present understanding) be any real quantity. 
With such, however, we are not at present concerned. 

The logarithm of any number which is not an integral power 

of the base will be some fractional number, positive or negative, 

as the case may be. For reasons that will appear presently, it is 

usual so to arrange a logarithm that it consists of a positive 

VOL. I 2 L 



514 



DETERMINATION OF CHARACTERISTIC 



CHAP. 



fractional part less than unity, and an integral part, positive or 
negative, as the case may be. 

The positive fractional part is called the mantissa. 

The integral part is called the cliaracteristic. 

For example, the logarithm of "0451 to base 10 is the negative 
number - 1 '3458235. In accordance with the above understand- 



ing, we should write 



log 10 -0451 



= - 1-3458235 = - 2 + (1 - -3458235), 
- - 2 + '6541765. 



For the sake of compactness, and at the same time to prevent 
confusion, this is usually written 

log 10 -0451 = 2-6541765. 

In this case then the characteristic is 2 (that is, - 2), and the 
mantissa is -6541765. 

§ 6.] To find the logarithm of a given number y to a given 
base a is the same problem as to solve the equation 

a* = y, 

where a and y are given and x is the unknown quantity. 

There are various ways in which this may be done ; and it 
will be instructive to describe here some of the more elementary, 
although at the same time more laborious, approximative methods 
that might be used. 

In the first place, it is always easy to find the characteristic 
or integral part of the logarithm of any given number y. We 
have simply to find by trial hvo consecutive integral powers of the 
base between which the given number y lies. The algebraically less of 
these two is the characteristic. 

Example 1. 

To find the characteristic of log 3 451. 

We have the following values for consecutive integral powers of 3 : — 



Power 


1 


2 


3 


4 


5 


6 


Value 


3 


9 


27 


81 


243 


729 



XXI 



LOGARITHMS TO BASK 10 



515 



Hence 3 5 <451<3 fi . Hence log 3 451 lies between 5 am! 6. 

log 3 451 = 5+a proper fraction. 
Hence char. log 3 451=5. 

Example 2. 

Find the characteristic of log3 , 0451. We have 



Therefore 



Powers of Base 





-1 


-2 


-3 


Values 


1 


•333 . . . 


•111. . . 


•037 . . . 



Hence 3~ 3 < '0451 <3~ 2 
Hence 



; that is to say, 
log3 - 0451 = - 3 + a proper fraction, 
char. log 3 -0451=3. 



When the base of the system of logarithms is the radix of 
the scale of numerical notation, the characteristic can always be 
obtained by merely counting the digits. 

For example, if the radix and base be each 10, then 

If the number have an integral part, the characteristic of its 
logarithm is + {one less than the number of digits in the integral 
part). 

If the number have no integral part, the characteristic is - {one 
more than the number of zeros that follow the decimal point). 

The proof of these rules consists simply in the fact that, if a 
number lie between 10 M and 10 n+1 , the number of digits by 
which it is expressed is n + 1 ; and, if a number lie between 
10 -(*+!) and 10 _ri , the number of zeros after the decimal point 
is n. 

For example, 351 lies between 10 2 andl0 3 . Heucechar. logi 351 = 2 = 3 - 1, 
according to the rule. 

Again, -0351 lies between '01 and T, that is, between 10 -2 and 10' 1 . 
Hence char, log w '0351= - 2~ - (1 + 1), which agrees with the rule. 

The rule suggests at once that, if 1 be adopted as the base of 
our system of logarithms, then the characteristic of a logarithm depends 
merely on the position of the decimal point ; and the mantissa is in- 
dependent of the position of the decimal point, but depends merely on 
the succession of digits. 



516 COMPUTATION OF LOGARITHMS BY SOLUTION OF a x = y chap. 

We may formally prove this important proposition as 
follows : — 

Let N be any number formed by a given succession of digits, 
c the characteristic, and m the mantissa of its logarithm. Then 
any other number which has the same succession of digits as N, 
but has the decimal point placed differently, will have the form 
10*N, where i is an integer, positive or negative, as the case may be. 
But log 10 10*N = log.JO 1 ' + log 10 N, by § 4, = i + log 10 N = (i + c) + m. 
Now, since by hypothesis m is a positive proper fraction, and c 
and i are integers, the mantissa of log I0 10 f N is m, and the 
characteristic is i + c. In other words, the characteristic alone 
is altered by shifting the decimal point. 

§ 7.] The process used in § 6 for finding the characteristic 
of a logarithm can be extended into a method for finding the 
mantissa digit by digit. 

Example. 

To calculate log 10 4-217 to three places of decimals. 

The characteristic is obviously 0. Let the three first digits of the mantissa 
be xyz. Then we have 

4-217 = 10°-^, hence (4-217) 10 =10*-3«. 

We must now calculate the 10th power of 4-217. In so doing, however, 
there is no need to find all the significant figures— a few of the highest will 
suffice. We thus find 

1778400 = 10^-^. 

We now see that x is the characteristic of logi 1778400. Hencea:=6. 
Dividing by 10 6 , and raising both sides of the resulting equation to the 10th 
power, we find 

(l-77S) in = 10J'- 2 ; hence 315'7 = 10^. 
Hence y = 2. Dividing by 10 2 , and raising to the 10th power, we now find 

(3-16) 10 =10 2 ; hence 99280 = 10*. 
Hence z = 5 very nearly. 

Wc conclude, therefore, that 

log 10 4-217=-625 nearly. 

This method of computing logarithms is far too laborious to 
be of any practical use, even if it were made complete by the 
addition of a test to ascertain what effect the figures neglected 
in the calculation of the 10th powers produce on a given decimal 
place of the logarithm ; it has, however, a certain theoretical 



xxi COMPUTATION BY INSERTING GEOMETRIC MEANS 517 

interest on account of its direct connection with the definition of 
a logarithm. 

By a somewhat similar process a logarithm can he expressed 
as a continued fraction. 

§ 8.] If a series of numbers be in geometric progression, their 
logarithms are in arithmetic progression. 

Let the numbers in question be y 1} y. 2 , y a , . . ., y n . Let the 
logarithm of the first to base a be a, and the logarithm of the 
common ratio of the G.P. y l} y 2 , y.,, . . ., y n to the same base be 
(3. Then we have the following series of corresponding values : — 

Vi, y-2, y*, • • •, Vu, 

ii ii ii ii 

a a , a x +?, a x+ -<\ a *+(».-i)^ 

from which the truth of the proposition is manifest. 

As a matter of history, it was this idea of comparing two 
series of numbers, one in geometric, the other in arithmetic pro- 
gression, that led to the invention of logarithms ; and it was on 
this comparison that most of the early methods of computing 
them were founded. 

The following may be taken as an example. Let us suppose 
that we know the logarithms x x and a" 9 of two given numbers, y, 
and y 9 ; then we can find the logarithms of as many numbers 
lying between y x and y 9 as we please. We have 

Vi = a Xl , ft = a* 3 - 
Let us insert a geometric mean, y 6 , between y l and y g , then 

% = (Mi.)* = « ( * 1+a " 9)/2 = «*, say, 
where x b is the arithmetic mean between x^ and x a . We have 
now the following system : — 

Logarithm x l x s x ti , 

Number y x y, y 9 . 
Next insert geometric means between y } , y B and y u y 9 . The 
logarithms of the corresponding numbers will be the arithmetic 
means between x u x b and x b , x y . We thus have the system — 

Logarithms x 1 , x s , x b , x 7 , x 9 
Numbers y u y a , y s , y 7 , y 9 . 



^a > 



518 



COMPUTATION BY INSERTING GEOMETRIC MEANS chai\ 



Proceeding in like manner, we derive the system — 



Logarithms 



wo 



/y* /y rv* /■>• iy iy* <y* • 

[q, a, 2? «^ 3 j w 4 , WfiJ rt 6 , .(' 7 , «^g, «*/g J 

Numbers y„ y 9J y 3J y i} y„ y 6 , y 7 , y 8 , y 9 ; 

and so on. Each step in this calculation requires merely a multi- 
plication, the extraction of a square root, an addition accompanied 
by division by 2, and each step furnishes us with a new number 
and the corresponding logarithm. 

Since x 1} x 2 , . . ., x n form an A.P., the logarithms are spaced 
out equally, but the same is not true of the corresponding num- 
bers which are in G.P. It is therefore a table of antilogarithms * 
that we should calculate most readily by this method. It will 
be observed, however, that by inserting a sufficient number of 
means we can make the successive numbers differ from each other 
as little as we please ; and by means of the method of interpola- 
tion by first differences, explained in the last section of this 
chapter, we could space out the numbers equally, and thus con- 
vert our table of antilogarithms into a table of logarithms of the 
ordinary kind. 

As a numerical example we may put a = 10, y 1 = l,y s = 10; thena^=0, a'9=l. 
Proceeding as above indicated, we should arrive at the following table : — 



Number. 


Logarithm. 


Number. 


Logarithm. 


1-0000 


o-oooo 


4-2170 


0-6250 


1-3336 


0-1250 


5-6235 


0-7500 


1-7783 


0-2500 


7-4990 


0-8750 


2-3714 


0-3750 


10-0000 


1 -oooo 


3-1622 


0-5000 







§ 9.] In computing logarithms, by whatever method, it is 
obvious that it is not necessary to calculate independently the 
logarithms of composite integers after we have found to a suffi- 
cient degree of accuracy the logarithms of all primes up to a 
certain magnitude. Thus, for example, log 4851 = log 3 2 .7 2 11 
= 2 log 3 + 2 log 7 + log 11. Hence log 4851 can be found when 
the logarithms of 3, 7, and 1 1 are known. 

* By the antilogarithm of any number N is meant the number of which 
X is the logarithm. 



XXI 



CHANGE OF BASE 519 



Again, having computed a system of logarithms to any one base 

a, we can without difficulty deduce therefrom a system to any other base 

b. All we have to do is to multiply all the logarithms of the former 
system by the number fx = l/\og a b. 

For, if x = log b y, then y = L x . 

Hence log rt y = log a ^, 

= ajlog a &, by § 4. 
Hence \og b y = x = log a ?//log a & (1). 

The number //, is often called the modulus of the system 
whose base is b with respect to the system whose base is a. 

Cor. 1. If in the equation (1) we put y=a, we get the 
following equation, which could easily be deduced more directly 
from the definition of a logarithm : — 

log & «= l/log a 6, 

or log a Hog 6 a=l (2). 

Cor. 2. The equation y = b x may be written 

y = a xlos « b or y = a x!l0Bba . 

Hence the graph of the exponential b x can be deduced from the 
graph of the exponential a x by shortening or lengthening all the abscissce 
of the latter in the same ratio 1 : log a b. 

This is the general theorem corresponding to a remark in § 2. 

We may also express this result as follows : — 

Given any two exponential graphs A and B, then either A is the 
orthogonal projection of B, or B is the orthogonal projection of A, on a 
plane passing through the axis of y. 

USE OF LOGARITHMS IN ARITHMETICAL CALCULATIONS. 

§ 10.] We have seen that, if we use the ordinary decimal 
notation, the system of logarithms to base 10 possesses the im- 
portant advantages that the characteristic can be determined by 
inspection, and that the mantissa is independent of the position 
of the decimal point. On account of these advantages this 
system is used in practical calculations to the exclusion of all 
others. 



520 



SPECIMEN OF LOGARITHMIC TABLE. 



Oil A P. 



No. 


12 3 4 


5 6 7 8 9 


Diff. 




3050 


484 2998 3141 32S3 342G 3568 


3710 3853 3995 4137 4280 






51 


4422 4564 4707 4849 4991 


5134 5276 5418 5561 5703 






52 


5845 5988 6130 6272 6414 


6557 6699 6841 6984 7126 






53 


7268 7410 7553 7695 7837 


7979 8121 8264 8406 8548 






54 


8690 8833 8975 9117 9259 


9401 9543 9686 9828 9970 






55 


485 0112 0254 0396 0539 0GS1 


0823 0965 1107 1249 1391 






56 


1533 1676 1818 1960 2102 


2244 2386 2528 2670 2812 






57 


2954 3096 3239 33S1 3523 


3665 3807 3949 4091 4233 


142 




58 


4375 4517 4659 4801 4913 


5085 5227 5369 5511 5053 


1 14 




59 


5795 5937 6079 6221 6363 


6505 6647 6788 6930 7072 


2 28 

3 43 

4 57 




60 


7214 7356 7498 7640 7782 


7924 8066 8208 8350 8491 




3061 


8633 8775 8917 9059 9201 


9343 9484 9626 9768 9910 


5 71 




62 


4860052 0194 0336 0477 0619 


0761 0903 1045 1186 1328 


6 85 




63 


1470 1612 1754 1895 2037 


2179 2321 2462 2604 2746 


7 99 




64 


2888 3029 3171 3313 3455 


3596 3738 3880 4021 4163 


8 114 

9 128 




65 


4305 4446 4588 4730 4872 


5013 5155 5297 5438 5580 




66 


5722 5863 6005 6146 62S8 


6430 6571 6713 6855 6996 






67 


7138 7279 7421 7563 7704 


7846 7987 8129 8270 8412 






68 


8554 8695 8837 8978 9120 


9261 9403 9544 9686 9827 






69 


9969 0110 0252 0393 0535 


0676 0818 0959 1101 1242 






70 


4871384 1525 1667 1808 1950 


2091 2232 2374 2515 2657 






3071 


2798 2940 3081 3222 3364 


3505 3647 3788 3929 4071 






72 


4212 4353 4495 4636 4778 


4919 5060 5202 5343 5484 






73 


5626 5767 5908 6050 6191 


6332 6473 6615 6756 6897 






74 


7039 7180 7321 7462 7604 


7745 7886 8027 8169 8310 






75 

76 


8451 8592 8734 8875 9016 
9863 0004 0146 0287 042S 


9157 9299 9440 9581 9722 






0569 0710 0852 0993 1134 


77 


488 1275 1416 1557 1698 1839 


1981 2122 2263 2404 2545 






78 


2686 2827 2968 3109 3251 


3392 3533 3674 3815 3956 






79 


4097 4238 4379 4520 4661 


4802 4943 5084 5225 5366 






80 


5507 5648 5789 5930 6071 


6212 6353 6494 6635 6776 


141 

1 14 

2 28 




3081 


6917 7058 7199 7340 7481 


7622 7763 7904 8045 8185 




82 


8326 8467 860S 8749 8890 


9031 9172 9313 9454 9594 


3 42 




83 


9735 9876 0017 0158 0299 


0440 0580 0721 0862 1003 


4 56 




84 


4891144 1285 1425 1566 1707 


1848 1989 2129 2270 2411 


5 71 




85 


2552 2692 2833 2974 3115 


3256 3396 3537 3678 3818 


6 85 

7 99 

8 113 




86 


3959 4100 4241 4381 4522 


4663 4804 4944 5085 5226 




87 


5366 5507 5648 5788 5929 


6070 6210 6351 6492 6632 


9 127 




88 


6773 6914 7054 7195 7335 


7476 7617 7757 7898 8038 






89 


8179 8320 8460 8C01 8741 


8882 9023 9163 9304 9444 






90 


9585 9725 9866 0006 0147 


0287 0428 0569 0709 0850 






3091 


4900990 1131 1271 1412 1552 


1693 1833 1973 2114 2254 






92 


2395 2535 2676 2816 2957 


3097 3238 3378 3518 3659 






93 


3799 3940 4080 4220 4361 


4501 4642 4782 4922 5063 






94 


5203 5343 5484 5624 5765 


5905 6045 6186 6326 6466 






95 


6G07 6747 6887 7027 7168 


7308 7448 7589 7729 7869 






96 


8010 8150 8290 8430 8571 


8711 8851 8991 9132 9272 






97 


9412 9552 9693 9833 9973 


0113 0253 0394 0534 0674 






98 


4910S14 0954 1094 1235 1375 


1515 1655 1795 1935 2076 






99 


2216 2356 2496 2636 2776 


2916 3057 3197 3337 3477 






3100 


3617 3757 3897 4037 4177 


4317 4457 4597 4738 4878 







xxl USE OF LOGARITHMIC TABLE 521 

la printing a table of logarithms to base 10 it is quite un- 
necessary, even if it were practicable, to print characteristics. 
The mantissa? alone are given, corresponding to a succession of 
five digits, ranging usually from 10000 to 99999.* 

A glance at p. 520, which is a facsimile of a page of the 
logarithmic table in Chambers's Mathematical Tables, will show 
the arrangement of such a table. To take out the logarithm of 
30715 from the table, we run down the column headed "No." 
until we come to 3071 ; the first three figures of the mantissa 
are 487 (standing over the blank in the first half column) ; the 
last four are found by running along the line till Ave reach the 
column headed 5, they are 3505. The characteristic is seen by 
inspection to be 4. Hence log 30715 = 4-4873505. 

To find the number corresponding to any given logarithm 
we have of course simply to reverse the process. 

To find the logarithm of '030715 Ave have to proceed exactly 
as before, only a different characteristic, namely 2, must be pre- 
fixed to the mantissa. We thus find log -030715 = 2-4873505. 

If we Avish to find the logarithm of a number, say 3 - 083279, 
Avhere we have more digits than are given in the table, then we 
must take the nearest number whose logarithm can be found 
by means of the table, that is to sa} r , 3-0833. "We thus find 
log 3-0833 = 0-4890158f nearly. Greater accuracy can be at- 
tained by using the column headed c 'Diff," as will be explained 
presently. 

Conversely, if a logarithm be given Avhich is not exactly 
coincident with one given in the table, Ave take the one in the 
table that is nearest to it, and take the corresponding number as 
an approximation to the number required. Greater accuracy 
can be obtained by using the difference column. Thus the 
number whose logarithm is 1 -4872191 has for its first five 

* For some purposes an extension of the table is required, and such ex- 
tensions are supplied in various ways, which need not be described here. For 
rapidity of reference in calculations that require no great exactness a short 
table for a succession of 3 digits, ranging from 100 to 999, is also usually 
given. 

+ The bar over 0158 indicates that these digits follow 489, and not 48S. 



522 NUMBER OF FIGURES REQUIRED chap. 

significant digits 30705 ; but, if we wish the best approximation 
with five digits, we ought to take 30706. Since the character- 
istic is 1, the actual number in question has two integral digits. 
Hence the required number is 30'706, the error being certainly 
less than '0005. 

§ 11.] The principle which underlies the application of 
logarithms to arithmetical calculation is the very simple one 
that, since to any number there corresponds one and only one 
logarithm, a number can be identified by means of its logarithm. 

It is this principle which settles how many digits of the 
mantissa of a logarithm it is necessary to use in calculations 
which require a given degree of accuracy. 

Suppose, for example, that it is necessary to be accurate 
down to the fifth significant figure ; and let us inquire whether 
a table of logarithms in which the mantissae are given to four 
places would be sufficient. In such a table we should find log 
3-0701 =0-4871, log 3-0702 = 0-4871 ; the table is therefore 
not sufficiently extended to distinguish numbers to the degree 
of accuracy required. Five places in the mantissa would, in the 
present instance, be sufficient for the purpose; for log 3-0701 
= 0-48715, log 3-0702 = 0-48716. Towards the end of the 
table, however, five places would scarcely be sufficient ; for log 
9-4910 = 0-97731 and log 9-4911 = 0-97731. 

§ 12.] The great advantage of using in any calculation 
logarithms instead of the actual numbers is that we can, in 
accordance with the rules of § 4, replace every multiplication by 
an addition, every division by a subtraction, and every operation 
of raising to a power or extracting a root by a multiplication or 
division. 

The following examples will illustrate some of the leading 
cases. We suppose that the student has a table of the loga- 
rithms of all numbers from 10000 to 100000, giving mantissa? 
to seven places. 
Example 1. 
Calculate the value of 1-6843 x -00132 -J- '3692. 

If A = 1-6843 x -00132H--3692, 
log A = log 1 -6843 + log -00132 - log -3692, 



XXI 



EXAMPLES 523 



log 1-6843 = -2264194 
log -00132 = 3-1205739 



3*3469933 
log -3692 =1-5672617 
logA = 3-7797316. 
Hence A= -0060219. 

Observe that the negative characteristics must be dealt with according to 
algebraic rules. 

Example 2. 

To extract the cube root of -016843. 

Let A = (-016843) 1 / 3 , then 

log A= | log -016843, 

= i(2 -2264194), 

= 4(3 + 1-2264194), 

= 1-4088065. 

A= -25633. 
Example 3. 

Calculate the value of A = (368)"' 3 /(439) 5;9 . 
Log A =1 log 368 -flog 439. 

I log 368 = |(2 -5658478) = 5-9869782 
$ log 439 = f (2 -6424645) = 1 "4680358 
log A = 4-5189424 
A= 33033. 
Example 4. 

Find how many digits there are in A = (l-01) 10000 . 
log A= 10000 log 1-01, 

= 10000 x -0043214, 
= 43-214. 
Hence the number of digits in A is 44. 

Example 5. 

To solve the exponential equation 1-2*=1-1 by means of logarithms. 
AVe have log 1 2* = log 1 • 1 . 

Therefore a?logl'2=logl , L 

„ _ log 1-1 _ -0413927 

Ce ""log l-2~ -0791812 " 

Hence log.r = l_og -0413927 -log -0791812, 

= 1-7183059. 
Therefore x — -52276. 

Remark.— It is obvious that we can solve any such equation as a* 2- *"** = &, 
where;, q, a, b are all given. For, taking logarithms of both sides, we have 

(x 2 -px + q) log a = log b. 
We can now obtain the value of a: by solving a quadratic equation. 



524 INTERPOLATION BY FIRST DIFFERENCES chap. 



INTERPOLATION BY FIRST DIFFERENCES. 

§ 13.] The method by which it is usual to find (or "interpo- 
late ") the value of the logarithm of a number which does not 
happen to occur in the table is one which is applicable to any 
function whose values have been tabulated for a series of equi- 
different values of its independent variable (or "argument"). 

The general subject of interpolation belongs to the calculus 
of finite differences, but the special case where first differences 
alone are used can be explained in an elementary way by means 
of graphical considerations. 

We have already seen that the increment of an integral 
function of x of the 1st degree, y = A.?; + B say, is proportional 
to the increment of its argument ; or, what comes to the same 
thing, if we give to the argument x a series of ecpiidifferent 
values, a, a + h, a + 2h, a + 3/j, &c, the function y will assume a 
series of equidifferent values Aa + B, Aa + B + Ah, Aa + B + 2 Ah, 
Aa + B + 3Ah, &c. 

If, therefore, we were to tabulate the values of Ax + B for a 
series of equidifferent values of x, the differences between suc- 
cessive values of y (" tabular differences ") would be constant, no 
matter to how many places we calculated y. 

Conversely, a function of x which has this property, that the 
differences between the successive values of y corresponding to 
equidifferent values of x are absolutely constant, must be an in- 
tegral function of x of the 1st degree. 

If, however, we take the difference, h, of the argument small 
enough, and do not insist on accuracy in the value of y beyond 
a certain significant figure, then, for a limited extent of the table 
of any function, it will be found that the tabular differences are 
constant. 

Bef erring, for example, to p. 520, it will be seen that the 
difference of two consecutive logarithms is constant, and equal 
to -0000141, from log 30660 up to log 30S99, or that there is 
merely an accidental difference of a unit in the last place ; that 
is to say, the difference remains constant for about 240 entries. 



XXI 



LIMITS OF THE METHOD 



525 



A similar phenomenon will be seen in the following extract 
from Barlow's Tables, provided we do not go beyond the 7th 



significant figure :- 



Number. 


Cube Root. 


Diff. 


2301 


13-2019740 


19122 


2302 


13-2038862 


19117 


2303 


13-2057979 


19111 


2304 


13-2077090 


19105 


2305 


13-2096195 





Let us now look at the matter graphically. Let ACSDQB 
be a portion of the graph of a function y = /(•>') ; and let us 
suppose that up to the nth significant figure the differences of y 
are constant for equidifferent 
values of x, lying between 
OE and OH. This means 
that in calculating (up to the 
nth significant figure) values 
of y corresponding to values 
of x between OE and OH we 
may replace the graph by the 
straight line AB. Thus, for 
example, if x - OM, then 
/(OM) = MQ ; and PM is the 
value calculated by means of the straight line AB. Our state- 
ment then is that PM - QM, that is PQ, is less than a unit in 
the n significant place. 

If this be so, then, a fortiori, it will be so if Ave replace a 
portion of the graph, say CD, lying between A and B by a 
straight line joining C and D. 

In other words, if up to the nth place the increment of the func- 
tion for eqiudifferent values of x be constant, between certain limits, 
then, to that degree of accuracy at least, the increment of the function 
will be proportional to the increment of the argument fcrr all values 
within those limits. 

§ 1 4.] Let us now state the conclusion of last article under 




Fig. 2. 



526 RULE OF PROPORTIONAL PARTS chap. 

an analytical form, all the limitations before mentioned as to 
constancy of tabular (or first) difference being supposed fulfilled. 
Let h be the difference of the arguments as they are entered 
in the table, D the tabular difference f(a + h) -/(«), a + h' a value 
of the argument, which does not occur in the table, but which 
lies between the values a and a + h, which do occur, so that h'<h. 
Then, by last paragraph, 

f(a + h') -f (a) _ (a + h') - a 
f(a + h) -f(a) (a + h) - a 

Hence f{a + h')-f(a) K 

D h 

Since in (1) f(a), D, h are all known, it gives us a relation 
between h' and f(a + h'). When, therefore, one or other of these 
is given, we can calculate the other. We have in fact 

f(a + h')=f(a) + ^J) (2), 

and a + h^a + f(a + h ^- f(a) h (3). 

From (2) we find a value of the function corresponding to a 
given intermediate value of the argument. From (3) we find an 
intermediate value of the argument corresponding to a given 
intermediate value of the function. 

The equations (2) and (3) are sometimes called the Rule of 
Proportional Parts. 

Example 1. 

To find by means of first differences the value of ^/(2303 , 45) as accurately 

as Barlow's Table will allow. 

By the rule of proportional parts, we have 

^'(2303 -45) = ^(2303 -00) + T <&( -00191), 
= 13-20580+ -00086, 
= 13-20666, 
which will be found correct down to the last figure. 

The only labour in the above calculation consists in working 
out the fraction 45/100 of the tabular difference. In tables of 



XXI 



EXAMPLES 527 



logarithms even this labour is spared the calculator ; for under 
each difference there is a small table of proportional parts, giving 
the values of 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 
9/10 of the difference in question (see the last column on p. 520). 
It will be observed that, if we strike the last figure off each of 
the proportional parts (increasing the last of those left if the 
one removed exceeds 5), we have a table of the various hun- 
dredths, and so on. Hence Ave can use the table twice over (in 
some cases it might be oftener), as in the following example : — 

Example 2. 
To find log 30 -81 345. 

We may arrange the corresponding contributions as follows ' — 

30-81300 1-4887340 

40 56 

5 7 



log 30 -81345 = 1-4887403 
Example 3. 

To find the number whose logarithm is 1-4871763. 

1-4871763 

1-4871637 -3070200 
96 

85 60 

11 8 



antilog 1 -4871763 = -3070268 

Here we set down under the given logarithm the next lowest in the table, 
and opposite to it the corresponding number -30702. 

Next, we write down -0000096, the difference of these two logarithms, and 
look for the greatest number in the table of proportional parts which does not 
exceed 96— this is 85. We set down 85, and opposite to it the corresponding 
figure 6. 

Lastly, we subtract 85 from 96, the result being -0000011. We then 
imagine a figure struck off every number in the table of proportional parts, 
look for the remaining one which stands nearest to 11, and set down the 
figure, namely 8, corresponding to it, as the last digit of the number we are 
seeking. 

Exercises XLIII. 
(1.) Find the characteristics of log 10 36983, logi 5 8 , logi 5- 3 , log 10 '00068 
(2.) Find the characteristics of log 5 1067, log 5 -0138, log^, logvAjl/8 
(3.) Find log 2 8 ^2. 
(4.) Calculate log 2 36'432 to two places of decimals. 



528 EXERCISES XLIII 



CHAP. 



Calculate out the values of tlio following as accurately as your tables will 
allow: — 

(5.) 4163x7-835. (6.) *3068 x -0015^-0579. 

(7.) (5-0063745) 5 . (8.) ^/(5 -0063745). 

(9.) (-01369) 12 . (10.) (-001369)*. 

(11.) {15(*318)fyl6}^. (12.) {(1-035) 7 - 1}/ {1-035 - 1}. 

(13.) The population of a country increases each year by *13 % 0I " its 
amount at the beginning of the year. By how much °/ will it have increased 
altogether after 250 years ? 

(14.) If the number of births and deaths per annum be 3*5 and 1*2 °/ 
respectively of the population at the beginning of each year, after how many 
years will the population be trebled ? 

(15.) Calculate the value of ^(32 6 - 8 +55 3 - 6 ). 

(16.) Calculate the value of 1 + c + c- + . . . +e 19 , where e= 2 71828. 

(17.) Find a mean proportional between 3*17934 and 3*987636. 

(18.) Insert three mean proportionals between 65"342 and 88*63. 

(19.) The 1st and 13th terms of a geometric progression are 3 and 65 
respectively : find the common ratio. 

(20.) The 4th and 7th terms of a geometric progression are 31 and 52 
respectively : find the 5 th term. 

(21.) How many terms of the series s + p+p+ ■ • • must be taken in 

order that its sum may differ from unity by less than a millionth ? 

(22.) Given log 10 5= *69897, find the number of digits in (\/5) 9a . 

(23.) Given log 2673 = 3*427, and log 3267 = 3*51415, find log 11. 

(24.) Find the first four significant figures and the number of digits in 
1.2.3.4. . . 20. 

(25.) How many terms of 3 1 , 3-, 3 3 , . . . must I take in order that the 
product may just exceed 100000 ? 

(26.) Given log x 36 = 1*3678, find a:. 

(27.) Given log a: 6£ = 2, find x. 

Solve the following equations : — 

(28.) 21* = 20. (29.) 2* 15*= 5. (30.) 2* 2 = 5 x 2*. 

(31.) 6* = 5y, 7*=3y. (32.) 3*-3-*=5. 

(33.) 2 3 *+ 2 »=5, 4 2 *=2 2 <'+ ;! . (34.) &*&*-* ->;»-\ip-x m 

(35. ) (a + b)^(a 4 - 2a-u- + b*)*- 1 = {a- bf*. 
(36.) x x +y = y ia , y x +« = x a . 

(37.) Find by means of a table of common logarithms log e 16.345, where 
« = 2*71828. 

(38.) Show that x—a ; and that x = a 

(39.) Show that 

log„(log„N) _ h^(k>gj^ 

V(l0ga"i) V(log*'0 : V(l<>go&) \/(l0g»rt) 

(40.) Show that the logarithm of any number to base a n is a mean pro- 

g 
portional between its logarithms to the bases a and a" . 



xxr HISTORICAL NOTE 529 

(41.) If P, Q, R be the pth, qth, rth terms of a geometric progression, 
show that 2(y - r) log P = 0. 

(42.) If ABC be in harmonic progression, show that log(A + C) 
+ log(A + C-2B) = 2log(A-C). 

(43.) If a, b, c be in G.P., show that S{log (6/c)}- 1 =-3{log 4 (c/a)}- 1 . 

(44.) If a, b, c be in G.P., and log c «, log 6 c, log a b in A. P., then the 
common difference of the latter is 3/2. 

(45.) If a 2 + b 2 = c-, then log 6+c « + log c _ & a = 2iog 6+c relog c _ 6 «. 

/ a /» \ re x (v +a — x) y(z + x - y) z(x + y - z) . , • „ „ . „ . 

(46.) If- 5 ^. =^-r -=-■ , -, then y*0'=f? ! x*=x*y*. 

log a; logy logs ' J a 

(47.) If :c 2 = \ogx n Xi, x z =\og Xl x 2) K 4 =logx 2 a! s , . . ., av,=logr B _ 2 a3„-i, x 1 = 

l°S*n-\ x »> then XlX -' ■ ■ ' T » = 1 - 

Historical Note. — The honour of devising the use of logarithms as a means of 
abbreviating arithmetical calculations, and of publishing the first logarithmic 
table, belongs to John Napier (1550-1617) of Merchiston (in Napier's day near, 
iu our day in, Edinburgh). This invention was not the result of a casual inspira- 
tion, for we learn from Napier's Rabdolorjla (1617), in which he describes three 
other methods for facilitating arithmetical calculations, among them his calculat- 
ing rods, which, uuder the name of " Napier's Bones," were for long nearly as 
famous as his logarithms, that he had devoted a great part of his life to the con- 
sideration of methods for increasing the power and diminishing the labour of 
arithmetical calculation. Napier published his invention in a treatise entitled 
"Mirifici Logarithmorum Cauonis Descriptio, ejusque usus, in utraque Trigo- 
nometria ut etiam in omni Logistica Mathematica, Amplissimi, facillimi, et 
expeditissimi explicatio. Authore ac Inventore Ioanne Nepero 5 Barone Merchis- 
tonii, &c, Scoto, Edinburgi (1614)." In this work he explains the use of 
logarithms ; and gives a table of logarithmic sines to 7 figures for every minute 
of the quadrant. In the Canon Mirificus the base used was neither 10 nor what 
is now called Napier's base (see the chapter on logarithmic series in the second 
part of this work). Napier himself appears to have been aware of the advantages 
of 10 as a base, and to have projected the calculation of tables on the improved 
plan ; but his infirm health prevented him from carrying out this idea ; and his 
death three years after the publication of the Canon Mirificus prevented him from 
even publishing a description of his methods for calculating logarithms. This 
work, entitled Mirifici Logarithmorum Canonis Constructio, &c. , was edited by 
one of Napier's sons, assisted by Henry Briggs. 

To Henry Briggs (1556-1630), Professor of Geometry at Gresham College, and 
afterwards Savilian Professor at Oxford, belongs the place of honour next to 
Napier in the development of logarithms. He recognised at once the merit and 
seized the spirit of Napier's invention. The idea of the superior advantages 
of a decimal base occurred to him independently ; and he visited Napier in 
Scotland in order to consult with him regarding a scheme for the calculation 
of a logarithmic table of ordinary numbers on the improved plan. Finding 
Napier in possession of the same idea in a slightly better form, he adopted his 
suggestions, and the result of the visit was that Briggs undertook the work which 
Napier's declining health had interrupted. Briggs published the first thousand 
of his logarithms in 1617 ; and, in his Arithmetica Logarithmica, gave to 14 
places of decimals the logarithms of all integers from 1 to 20,000, and from 
90,000 to 100,000. In the preface to the last-mentioned work he explains the 
methods used for calculating the logarithms themselves, and the rules for using 
them in arithmetical calculation. 

VOL. I 2 M 



530 HISTOKICAL NOTE chap, xxi 

While Briggs was engaged in filling up the gap left in his table, the work of 
calculating logarithms was taken up in Holland by Adrian Vlacq, a bookseller of 
Gouda. He calculated the 70,000 logarithms which were wanting in Briggs' 
Table ; and published, in 1628, a table containing the logarithms to 10 places of 
decimals of all numbers from 1 to 100,000. The work of Briggs and Vlacq has 
been the basis of all the tables published since their day (with the exception of 
the tables of Sang, 1871) ; so that it forms for its authors a monument both 
lasting and great. 

In order fully to appreciate the brilliancy of Napier's invention and the merit 
of the work of Briggs and Vlacq, the reader must bear in mind that even the 
exponential notation and the idea of an exponential function, not to speak of the 
inverse exponential function, did not form a part of the stock-in-trade of mathe- 
maticians till long afterwards. The fundamental idea of the correspondence of 
two series of numbers, one in arithmetic, the other in geometric progression, which 
is so easily represented by means of indices, was explained by Napier through 
the conception of two points moving on sejjarate straight lines, the one with 
uniform, the other with accelerated velocity. If the reader, with all his acquired 
modern knowledge of the results to be arrived at, will attempt to obtain for him- 
self in this way a demonstration of the fundamental rules of logarithmic calcula- 
tion, he will rise from the exercise with an adequate conception of the penetrating 
genius of the inventor of logarithms. 

For the full details of this interesting part of mathematical history, and 
in particular for a statement of the claims of Jost Biirgi, a Swiss contemporary 
of Napier's, to credit as an independent inventor of logarithms, we refer the 
student to the admirable articles "Logarithms" and "Napier," by J. W. L. Glaisher, 
in the Encyclopaedia Britannica (9th ed.). An English translation of the 
Constnictio, with valuable bibliographical notes, has been published by Mr. W. 
R. Macdonald, F.F.A. (Edinb. 1889). 



CHAPTEE XXII. 

Theory of Interest and Annuities Certain. 

§ 1.] Since the mathematical theory of interest and annuities 
affords the best illustration of the principles we have been dis- 
cussing in the last two chapters, we devote the present chapter 
to a few of the more elementary propositions of this important 
practical subject. What we shall give will be sufficient to enable 
the reader to form a general idea of the principles involved. 
Those whose business requires a detailed knowledge of the 
matter must consult special text-books, such as the Text-Book of 
the Institute of Actuaries, Part I., by Sutton.* 

SIMPLE AND COMPOUND INTEREST. 

§ 2.] When a sum of money is lent for a time, the borrower 
pays to the lender a certain sum for the use of it. The sum 
lent is spoken of as the capital or principal ; the payment for 
the privilege of using it as interest. There are various ways of 
arranging such a transaction ; one of the commonest is that the 
borrower repays after a certain time the capital lent, and pays 
also at regular intervals during the time a stated sum by way of 
interest. This is called paying simple interest on the borrowed 
capital. The amount to be paid by way of interest is usually 
stated as so much per cent per annum. Thus 5 per cent (5 %) 
per annum means £5 to be paid on every £100 of capital, for 



* Full references to the various sources of information will be found in 
the article "Annuities" (by Sprague), Encyclopaedia Britamica, 9th edition, 
vol. ii. 



532 AMOUNT PRESENT VALUE DISCOUNT chap. 

every year that the capital is lent. In the case of simple 
interest, the interest payable is sometimes reckoned strictly in 
proportion to the time ; that is to say, allowance is made not 
only for whole years or other periods, but also for fractions 
of a period. Sometimes interest is allowed only for integral 
multiples of a period mutually agreed on. We shall suppose 
that the former is the understanding. If then r denote the 
interest on £1 for one year, that is to say, one-hundredth of 
the named rate per cent, n the time reckoned in years and 
fractions of a year, P the principal, I the whole interest paid, A 
the amount, that is, the sum of the principal and interest, both 
reckoned in pounds, we have 

I = nrP (1); A = I + P = P(1 + nr) (2). 

These formulae enable us to solve all the ordinary problems of 
simple interest. 

If any three of the four I, n, r, P, or of the four A, n, r, P, 
be given, (1) or (2) enables us to find the fourth. 

Of the various problems that thus arise, that of finding P 
when A, n, r are given is the most interesting. We suppose 
that a sum of money A is due n years hence, and it is required 
to find what sum paid down at once would be an equitable 
equivalent for this debt. If simple interest is allowed, the 
answer is, such a sum P as would at simple interest amount in n 
years to A. In this case P = A/(l + nr) is called the present 
value of A, and the difference A-P = A{1- 1/(1 + nr)} 
= A?w/(1 + nr) is called the discount. Discount is therefore the 
deduction allowed for immediate payment of a sum due at some 
future time. The discount is less than the simple interest 
(namely Anr) on the sum for the period in question. When n 
is not large, this difference is slight. 

Example. 

Find the difference between the interest and the discount on £1525 
due nine months hence, reckoning simple interest at 3£ %• The difference 
in question is given by 

Anr - An?i(l + nr) = An-r-/(l +nr). 



XXH COMPOUND INTEREST 533 

In the present case A *= 1525, n = 9/12 = 3/4, r= 3-5/100= -035. Hence 
Difference = 1525 x ( -02625) 2 -f-(l -02625), 
= £1 :0:5£. 

§ 3.] In last paragraph we supposed that the borrower paid 
up the interest at the end of each period as it became due. In 
many cases that occur in practice this is not done ; but, instead, 
the borrower pays at the end of the whole- time for which the 
money was lent a single sum to cover both principal and 
interest. In this case, since the lender loses for a time the use 
of the sums accruing as interest, it is clearly equitable that the 
borrower should pay interest on the interest ; in other words, 
that the interest should be added to the principal as it becomes 
due. In this case the principal or interest-bearing capital 
periodically increases, and the borrower is said to pay compound 
interest. It is important to attend carefully here to the under- 
standing as to the period at which the interest is supposed to 
become due, or, as it is put technically, to be convertible (into 
principal) ; for it is clear that £100 will mount up more rapidly 
at 5 % compound interest convertible half-yearly than it will at 
5 % compound interest payable annually. In one year, for in- 
stance, the amount on the latter hypothesis will be £105, on the 
former £105 plus the interest on £2 : 10s. for a half-year, that 
is, £105 : 1 :3. 

In what follows we shall suppose that no interest is allowed 
for fractions of the interval (conversion -period) between the 
terms at which the interest is convertible, and we shall take the 
conversion-period as unit of time. Let P denote the principal, A 
the accumulated value of P, that is, the principal together with 
the compound interest, in n periods ; r the interest on £1 for 
one period ; 1 + r = R the amount of £1 at the end of one period. 

At the end of the first period P will have accumulated to 
P + Pr, that is, to PR. The interest-bearing capital or principal 
during the second period is PR ; and this at the end of the 
second period will have accumulated to PR + PRr, that is, to 
PR 2 . The principal during the third period is PR 2 , and the 
amount at the end of that period PR 3 , and so on. In short, the 



534 EXAMPLES CHA1\ 

amount increases in a geometrical progression whose common ratio is 
R ; and at the end of n periods we shall liave 

A = PR» (1). 

By means of this equation we can solve all the ordinary 
problems of compound interest ; for it enables us, when any three 
of the four quantities A, P, R, n are given, to determine the 
fourth. In most cases the calculation is greatly facilitated by 
the use of logarithms. See the examples worked below. 

Cor. I. If I denote the whole compound interest on P during the 
n periods, we have 

I=A-P = P(R"-1) (2). 

Cor. 2. If P denote the present value of a sum A due n periods 

hence, compound interest being allowed, then, since P must in n periods 

amount to A, we have 

A = PR", 

so that P = A/R Jl (3). 

The discount on the present understanding is therefore 

A(l - 1/R' 1 ) (4). 

Example 1. 

Find the amount in two years of £2350 : 5 : 9 at Z\ % compound interest, 
convertible quarterly. 

Here P = 2350-2875, n = S, r = 3-5/400 =-00875, R = l-00875. 
log A = log P + n log R, 
log P = 3-3711210 
?ilogR* = -0302684 



3-4013894 
A = £2519-936 = £2519 : 18 : 8. 
Example 2. 

How long will it take £186 : 14 : 9 to amount to £216 : 9 : 7 at 6 % com- 
pound interest, convertible half-yearly. 

* When n is very large, the seven figures given in ordinary tables hardly 
afford the necessary accuracy in the product n log R. To remedy this defect, 
supplementary tables are usually given, which enable the computer to find 
very readily to 9 or 10 places the logarithms of numbers (such as R) which 
differ little from unity. 



xxn NOMINAL AND EFFECTIVE RATE .535 

Here P= 186 -7375, A = 216"4792, R=1'03. 
»=(logA-logP)/logR 

•0641847 



= 5-00 
•0128372 



Hence the required time is five half-years, that is, 1\ years. 

Example 3. 

To find the present value of £1000 due 50 years hence, allowing compound 
interest at 4 %> convertible half-yearly. 

HereA = 1000, ?i = 100, R = l-02. We have P=A/R». 
log P = log 1000 - 100 log 1-02, 
= 3 - 100 x -0086002, 
= 2-1399800. 
P = £138-032 = £138 : : 8. 

§ 4.] In reckoning compound interest it is very usual in 
practice to reckon by the year instead of by the conversion- 
period, as we have done above, the reason being that different 
rates of interest are thus more readily comparable. It must be 
noticed, however, that when this is done the rate of interest to 
be used must not be the nominal rate at which the interest due 
at each period is reckoned, but such a rate (commonly called the 
effective rate) as would, if convertible annually, be equivalent to 
the nominal rate convertible as given. 

Let r n denote the effective rate of interest per pound which 
is equivalent to the nominal rate r convertible every l/?ith part 
of a year ; then, since the amount of £ 1 in one year at the two 
rates must be the same, we have 

(l+r) n =l+r n , 
that is, r n = (\+r)' l -l (1), 

and r = (l+r n ) 1 / B -l (2). 

The equations (1) and (2) enable us to deduce the effective rate 
from the nominal rate, and vice versa. 

Example. 

The nominal rate of interest is 5 %> convertible monthly, to find the 
effective rate. 

Here r= -05/12= -004166. 

Hence ?- 12 =(l-004166) 12 - 1, 

= 1-05114-1. 
r 12 = -05114. 
Hence the effective rate is 5 "11 4 °/ . 



536 ANNUITIES TERMINABLE OE PERPETUAL CHAP. 



ANNUITIES CERTAIN. 



§ 5.] When a person has the right to receive every year a 
certain sum of money, say £A, he is said to possess an annuity 
of £A. This right may continue for a fixed number of years and 
then lapse, or it may be vested in the individual and his heirs 
for ever ; in the former case the annuity is said to be terminable, 
in the latter perpetual. A good example of a terminable annuity 
is the not uncommon arrangement in lending money where B 
lends C a certain sum, and C repays by a certain number of 
equal annual instalments, which are so adjusted as to cover both 
principal and interest. The simplest example of a perpetual 
annuity is the case of a freehold estate which brings its owner a 
fixed income of £A per annum. 

Although in valuing annuities it is usual to speak of the 
whole sum which is paid yearly, yet, as a matter of practice, the 
payment may be by half-yearly, quarterly, &c. instalments ; and 
this must be attended to in annuity calculations. Just as in 
compound interest, the simplest plan is to take the interval 
between two consecutive payments, or the conversion-period, as 
the unit of time, and adjust the rate of interest accordingly. 

In many cases an annuity lasts only during the life of a cer- 
tain named individual, called the nominee, who may or may not 
be the annuitant. In this and in similar cases an estimate of 
the probable duration of human life enters into the calculations, 
and the annuity is said to be contingent. In the second part of 
this work we shall discuss this kind of annuities. For the 
present we confine ourselves to cases where the annual payment 
is certainly due either for a definite succession of years or in 
perpetuity. 

§ 6.] One very commonly occurring annuity problem is to 
find the accumulated value of a FORBORNE annuity. An annuitant 
B, who had the right to receive n successive payments at n suc- 
cessive equidistant terms, has for some reason or other not 
received these payments. The question is, What sum should he 
receive in compensation ? 



XXII 



ACCUMULATION OF FORBORNE ANNUITY 537 



To make the question general, let us suppose that the last of 
the n instalments was due m periods ago. 

It is clear that the whole accumulated value of the annuity 
is the sum of the accumulated values of the n instalments, 
and that compound interest must in equity be allowed on each 
instalment. 

Now the nth instalment, due for m periods, amounts to AR" ! , 
the n - 1th to AE" l+1 , the n - 2th to AR m + 2 ; and so on. Hence, 
if V denote the whole accumulated value, we have 

V = AR m + AR w+1 + . . . +AR m +»- 1 (1). 

Summing the geometric series, we have 

V = AR™(R n -l)/(R-l) (2). 

Cor. If the last instalment be only just due, m = 0, and the 
accumulated value of the forborne annuity is given by 

V = A(R»-1)/(R-1) (3). 
Example. 

A farmer's rent is £156 per annum, payable half-yearly. He was unable 
to pay for five successive years, the last half-year's rent having been due three 
years ago. Find how much he owes his landlord, allowing compound interest 
at 3%. 

HereA = 78, R = l-015, m = 6, n = 10. 

V = 78 x 1-015 6 (1-015 10 -1)/-015. 
10 log 1-015 =-0646600, 

1-015 10 = 1-16054. 

V = 78 x l-015 6 x -16054/-015. 
log 78 =1-8920946 

6 log 1-015 = -0387960 



log -16054 


= 1-2055833 


log -015 


1-1364739 
= 2-1760913 


logV 


= 2-9603826 
Y = £912-814 = £912 



16 : 3. 

§ 7.] Another fundamental problem is to calculate the purchase 
price of a given annuity. Let us suppose that B wishes, by paying 
down at once a sum £P, to acquire for himself and his heirs the 
right of receiving n periodic payments of £A each, the first pa}*- 
ment to be made m periods hence. We have to find P. 

P is obviously the sum of the present values of the n pay- 



538 PURCHASE PRICE OF AX ANNUITY chap. 

ments. Now the first of these is due m years hence ; its present 
value is therefore A/K" 1 . The second is due ra + 1 years hence ; 
A's present value is therefore A/K ,n+1 , and so on. Hence 

V-— A A . 



Hence 



P = K^( 1 "^)/( 1 "F. 

A — l (Rn_ l)/(R _ l) (2) . 



K m+r 

Cor. 1. The ratio of the purchase price of an annuity to the 
annual payment is often spoken of as the number of years' purchase 
which the annuity is worth. If the "period " understood in the 
above investigation be a year, and p be the number of years' purchase, 
then we have from (2) 

p = (R« - l)[R m + n - x (T& - 1) (3). 

If the period be 1/qth of a year, since the annual payment is then 

qA., we have 

p = (R n - 1 )/qR m + n -\R-l) (4 ) . 

Cor. 2. If the annuity be not " deferred" as it is called, but 
begin to run at once, that is to say, if the first payment be due one 
period hence* then m- 1, and we have 

P = A(R»-1)/R"(R-1), 
= A(l-R-»)/(R-l) (5). 

Also 

2> = (R"-1)/R»(R-1), 
= (l-R-»)/(R-l) (6); 

or p = (\-B- n )/q(R-l) (7), 

according as the period of conversion is a year or the qth part of a year. 
Cor. 3. To obtain the present value of a deferred perpetual 
annuity, or, as it is often put, the present value of the reversion of a 
perpetual annuity, we have merely to make n infinitely great in the 
equation (2). We thus obtain 

* This is the usual meaning of "heginning to run at once." In some 
cases the tir*t payment is made at once. In that case, of course. ?n. = 0. 



xxu NUMBER OF YEARS' PURCHASE OF A FREEHOLD 



539 



p-4/fi- 1 



= A/R'«- 1 (R-1) (8). 
Hence, for the number of years' purchase, we have 

p = l/E^-^R - 1) (9), 

or p=l/ ff R TO - 1 (E-l) (10), 

according as the period of conversion is a year or 1/qth of a yea?: 

Cor. 4. JFlien the perpetual annuity begins to run at once the 
formulce (8), (9), (10) become very simple. Putting m= 1 we have 

P = A/(R-1), 
= A/(l+r-l), 

= A/r (11). 

For the number of years' purchase 

p=\jr (12); 

or p=l/ qr (13), 

according as the period of conversion is a year or 1/qth of a year. 

If the period be a year, remembering that, if s be the rate 
per cent of interest allowed, then r = sj 100, we see that 

p=100/s (14). 

Hence the following very simple rule for the value of a perpetual 
annuity. To find the number of years' purcfoise, divide 100 by the 
rate per cent of interest corresponding to the kind of investment in 
question. This rule is much used by practical men. The following 
table will illustrate its application : — 



Rate % . . . 


3 


H 


4 


4| i 5 5i ' 6 

1 


No. of years' purchase . 


33 


28 


25 


22 


20 


18 


17 



Example. 

A sum of £30,000 is borrowed, to be repaid in 30 equal yearly instalments 
which are to cover both principal and interest. To find the yearly payment, 
allowing compound interest at 4| %. 

Let A be the annual payment, then £30,000 is the present value of an 



540 INTEREST AND ANNUITY TABLES CHAf. 

annuity of <£A payable yearly, the annuity to begin at once and run for 30 
years. Hence, by (5) above, 

30,000 = A(l - r045- 30 )/-045, 
A = 1350/(l-l-045- 30 ), 
- 30 log 1 -045 = 1 -4265110, 
l-045~ so = -267000. 
A = 1350/-733, 
= £1841 : 14 : 11. 

§ 8.] It would be easy, by assuming the periodic instalments 
or the periods of an annuity to vary according to given laws, to 
complicate the details of annuity calculations veiy seriously ; 
but, as we should in this way illustrate no general principle of 
any importance, it will be sufficient to refer the student to one 
or two instances of this kind given among the examples at the 
end of this chapter. 

It only remains to mention that in practice the calculation 
of interest and annuities is much facilitated by the use of tables 
(such as those of Turnbull, for example), in which the values of the 
functions (1 + r) n , (1 +r)~ n , {(1 + r) n - 1 }jr, {1 - (1 + r)- n }jr, 
rj{\ -(1 +r)~ n }, &c, are tabulated for various values of lOOr 
and ii. For further information on this subject see the Text-Book 
of the Institute of Actuaries, Part I., p. 151. 

Exercises XLIV. 

^1.) The difference between the true discount and the interest on £40,400 
for a period x is £4, simple interest being allowed at 4 % > fiud x. 

(2.) Find the present value of £15,000 due 50 years hence, allowing 4i °/ B 
compound interest, convertible yearly. 

(3.) Find the amount of £150 at the end of 14 years, allowing 3 °/ com- 
pound interest, convertible half-yearly, and deducting 6d. per £ for income- 
tax. 

(4.) How long will it take for a sum to double itself at 6 % compound 
interest, convertible annually ? 

(5.) How long will it take for one penny to amount to £1000 at 5 °/ com- 
pound interest, convertible annually ? 

(6.) On a salary of £100, what difference does it make whether it is paid 
quarterly or monthly ? Work out the result both for simple and for compound 
interest at the rate of 4 - 2 °/ . 

(7.) A sum £A is laid out at 10 % compound interest, convertible annually, 
arid a sum £2.V at 5 % compound interest, convertible half-yearly. After 
how many years will the amounts be ecpual ? 



XXII 



EXERCISES XLIV 541 



(8.) Show that the difference between bankers' discount and true discount, 
simple interest being supposed, is 

A?) V 2 {1 - nr + nh- 2 - iflr 3 + . . . ad oo } . 

(9.) If r> 5/100, n ■> 10, find an upper limit for the error in taking 
100(1 + n C i r + nC-2r 2 - + n C 3 r :] ) as the amount of £100 in n years at 100/- % com- 
pound interest, convertible annually. 

(10.) If £I C and £l s denote the whole compound and the whole simple 
interest on £P for n years at 100?- %> convertible annually, show that 
I r -I s = P(, 1 C 2 r 2 + „C 3 r ! + . . .+ '/■»). 

(11.) A man owes £P, on which he pays lOOr % annually, the principal 
to be paid up after n years. What sum must he invest, at 100r' °/ , so as to be 
just able to pay the interest annually, and the principal £P when it falls due ? 

(12.) B has a debenture bond of £500 on a railway. When the bond lias 
still five years to run, the company lower the interest from 5 %> which was 
the rate agreed upon, to 4 °/ , and, in compensation, increase the amount of 
B's bond by x °/ a . Find x, supposing that B can always invest his interest 
at 5 %. 

(13.) A person owes £20,122 payable 12 years hence, and offers £10,000 
down to liquidate the debt. What rate of compound interest, convertible 
annually, does he demand ? 

(14.) A testator directed that his trustees, in arranging his affairs, should 
set apart such sums for each of his three sons that each might receive the 
same amount when he came of age. When he died his estate was worth 
£150,000, and the ages of his sons were 8, 12, and 17 respectively. Find 
what sum was set apart for each, reckoning 4 % compound interest for 
accumulations. 

(15.) B owes to C the sums Ai, A- 2 , . . ., A r at dates Bi, n 2 , . . ., » r 
years hence. Find at what date B may equitably discharge his debt to C 
by paying all 'the sums together, supposing that they all bear the same rate 
of interest ; and 

1st. Allowing interest and interest in lieu of discount where discount is due. 

2nd. Allowing compound interest, and true discount at compound interest. 

(16.) Required the accumulated value at the end of 15 years of an annuity 
of £50, payable in quarterly instalments. Allow compound interest at 5 %• 

(17.) A loan of £100 is to be paid off in 10 equal monthly instalments. 
Find the monthly payment, reckoning compound interest at 6 %• 

(18.) I borrow £1000, and repay £10 at the end of every month for 10 
years. Find an equation for the rate of interest I pay. What kind of 
interest table would help you in practically solving such a question as this ? 

(19.) The reversion after 2 years of a freehold worth £168 : 2s. a year is to 
be sold : find its present value, allowing interest at 2 %> convertible annually. 

(20.) Find the present value of a freehold of £365 a year, reckoning com- 
pound interest at 3J %, convertible half-yearly, and deducting 6d. per £ of 
income-tax. 

(21.) If a perpetual annuity be worth 25 years' purchase, what annuity to 



542 EXERCISES XLIV chap, xxii 

continue for 3 years can be bought for £5000 so as to bring the same rate of 
interest ? 

(22. ) If 20 years' purchase be paid for an annuity to continue for a certain 
number of years, and 24 years' purchase for one to continue twice as long, find 
the rate of interest (convertible annually). 

(23.) Two proprietors have equal shares in an estate of £500 a year. One 
buys the other out by assigning him a terminable annuity to last for 20 years. 
Find the annuity, reckoning 3| % compound interest, convertible annually. 

(24.) The reversion of an estate of £150 a year is sold for £2000. How 
long ought the entry to be deferred if the rate of interest on the investment 
is to be 4| %> convertible annually ? 

(25.) If a lease of 19 years at a nominal rent be purchased for £1000, what 
ought the real rent to be in order that the purchaser may get 4 °/ on his 
investment (interest convertible half-yearly) ? 

(26.) B and C have equal interests in an annuity of £A for 2n years (pay- 
able annually). They agree to take the payments alternately, B taking the 
first. What ought B to pay to C for the privilege he thus receives ? 

(27. ) A farmer bought a lease for 20 years of his farm at a rent of £50, 
payable half-yearly. After 10 years had run he determined to buy the free- 
hold of the farm. "What ought he to pay the landlord if the full rent of the 
farm be £100 payable half-yearly, and 3 % be the rate of interest on invest- 
ments in land ? 

(28.) What annuity beginning n years hence and lasting for n years is 
equivalent to an annuity of £A, beginning now and lasting for n years ? 

(29.) A testator left £100,000 to be shared equally between two institu- 
tions B and C ; B to enjoy the interest for a certain number of years, C to 
have the reversion. How many years ought B to receive the interest if the 
rate be 3^ %> convertible annually ? 

(30.) If a man live m years, for how many years must he pay an annuity 
of £A in order that he may receive an annuity of the same amount for the 
rest of his life ? Show that, if the annuity to be acquired is to continue for 
ever, then the number of years is that in which a sum of money would doublo 
itself at the supposed rate of interest. 

(31.) A gentleman's estate was subject to an annual burden of £100. His 
expenses in any year varied as the number of years he had lived, and his 
income as the square of that number. In his 21st year he spent £10,458, 
and his income, before deducting the annual burden, was £4410. Show that 
he ran in debt every year till he was 50. 

(32.) A feu is sold for £1500, with a feu-duty of £18 payable annually, 
and a casualty of £100 payable every 50 years. What would have been the 
price of the feu if it had been bought outright ? Beckon interest at 4£ %• 

(33.) Find the accumulation and also the present value of an annuity 
when the annual payments increase in A.r. 

(34.) Solve the same problem when the increase is in G.P. 

(35.) The rental of an estate is £mA to begin with ; but at the end of 
every q years the rental is diminished by £A, owing to the incidence of fresh 
taxation. Find the present value of the estate. 



APPENDIX 

ON THE GENERAL SOLUTION OF CUBIC AND BIQUAD- 
RATIC EQUATIONS; AND ON THE CASES WHERE 
SUCH EQUATIONS CAN BE SOLVED BY MEANS OF 
QUADRATIC EQUATIONS. 

§ 1.] Since cubic and biquadratic equations are of frequent 
occurrence in elementary mathematics, and many interesting 
geometric problems can be made to depend on their solution, a 
brief account of their leading properties may be useful to readers 
of this book. Incidentally, we shall meet with some principles 
of importance in the General Theory of Equations. 



COMMENSURABLE ROOTS AND REDUCTIBILITY. 

§ 2.] We shall suppose in all that follows that the coefficients 
Po • • •> Pn °f an y equation, 

p x n +p,x n - 1 + . . . + p„, = (1), 

are all real commensurable numbers. If, as in chap, xv., § 21, 
we put x = £/m, we derive from (1) the equivalent equation 

Poi m +p l m$' 1 - 1 + . . . +p n -,m n ' 1 $ +p n m n = (2), 

each of whose roots is m times a corresponding root of (1). If 
we then choose m so that mp,jp , . . ., m n ~ 1 p n . 1 /p , m n p n fp are 
all integral — for example, by taking for m the L.C.M. of the de- 
nominators of the fractions p x jp m . . ., p n -i/Po> Pn/po — we shall 
reduce (2) to the form 

£■ + },£»-! + . . .+q n = (3), 

in which all the coefficients are positive or negative integers, and the 



544 APPENDIX 

coefficients of the highest term unity. We may call this the Special 
Integral Form. 

§ 3.] If, as in chap, xv., § 22, we put x = £ + a, we transform 
the equation (1) into 

i>o£" + fl , i£"~ 1 + . • - + <z» = o, 

where q 1 = np t) a+p 1 . Hence, if we take a= —pijnp m the trans- 
formed equation becomes 

&?+.?&-* + • • . + q n = o (4), 

wherein q 2 , . . ., q' n have now determinate values. It follows 
that 

By a proper linear transformation, we can always deprive an 
equation of the nth degree of its highest term but one. 

We can, of course, combine the transformations of §§ 2, 3, 
and reduce an equation to a special integral form wanting the 
highest term but one. 

§ 4.] If an equation of the special integral form has commensurable 
roots, these roots must be integral, and can only be exact divisors of 
its absolute term. For, suppose that the equation (3) has the 
fractional commensurable root a/b, where a may be supposed to 
be prime to b. Then we have the identity 

(a/b) n + qi (alb) n -i + . . .+q n = 0, 
whence 

a n /b= - q.a 11 - 1 - q a a n ~ 2 b - ... - q n b n ~ x , 

which is