I. Put x = 1 in the above formula; and we have

Thus the sum of the coefficients in the expansion of (1 +x)”

is 2".

II.

Again, put x=- 1; and we have

(1 − 1)" = c − c1 + C2 − C3 +

...

;

.. 0 = (C2+ C2+ C + ...) (C1 + Cg + Cs+...).

Thus the sum of the coefficients of the odd terms of a binomial expansion is equal to the sum of the coefficients of the

even terms.

III. Since c, Cn-r, [Art. 246] we may write the binomial theorem in either of the following ways:

The coefficient of x" in the product of the two series on the right is equal to

2

Hence* c2 + c12 +

...

2

...

2

+ Cr2 + c is equal to the coefficient of an in (1+x)" × (1+x)", that is in (1+x), and this

coefficient is

2n

nn

Hence the sum of the squares of the coefficients in the

356 MISCELLANEOUS THEOREMS AND EXAMPLES.

The values of A, B, C, ... can, however, be obtained separately in the following manner. Since (i) is to be true for all values of x, it must be true when x = a; and putting x=a, we have F(a) A (a - b) (a-c)...; and therefore A = F (a)/(a − b) (a–c).... Similarly we have

and so for C, D,

H

We have thus found values of A, B, C, ... which make the relation (i) true for n values a, b, c,... of x; and as the expressions on the two sides of (i) are of not higher degree than the (n − 1)th, it follows [Art. 272] that the relation is true for all values of x.

Thus

F(x)

Ex. 1. Resolve

(x − 1) (x − 3)

In this identity put x=1; then 2=4(-2). Now put x=3; then 6 B.2.

Or, by equating the coefficients of the different powers of x, we have

2=4+B, and 0=-3A-B;

whence, as before,

A =

- 1, B=3.

Ex. 2. Find the coefficient of x" in the expansion of

according to ascending powers of x.

2x x2-4x+3

MISCELLANEOUS EXAMPLES. VII.

391

144. A cask was filled with wine and water mixed together in the ratio of 5: 3. When 16 gallons of the mixture had been drawn off and the cask filled up with water, the ratio of the wine to the water became 3: 5. How many gallons did the cask hold?

146. To each of three quantities in geometrical progression the second is added. Shew that the three resulting quantities are in harmonical progression.

147. Shew that, if a, b, c are in Arithmetical Progression and a, b, c +1 in Geometrical Progression, then will

148. Shew that, if x and y are both positive and x<

1-y 1+y'

150. Expand, in a series of ascending powers of x, each of the expressions (1-x+x2)−1 and (1 − x − 2x2)−1, and shew that, if pn, In be the respective coefficients of an in the two series, then

151. Shew that, if 2s=a+b+c and 203=a3+b3+ c3, then will

(03-a3) (sa) + (03 — b3) (8 − b ) + (σ3 — c3) (8 −c)=a+b+c1-803.

1

EUCLID'S ELEMENTS

INCLUDING ALTERNATIVE PROOFS, TOGETHER WITH
ADDITIONAL THEOREMS AND EXERCISES
CLASSIFIED AND ARRANGED

BY

H. S. HALL, M.A., AND F. H. STEVENS, M.A.

Book I. covers the First Stage of the South Kensington Syllabus. Books I.-IV. cover the First and Second Stages of the Syllabus.

PRESS OPINIONS

Cambridge Review-"To teachers and students alike we can heartily recommend this little edition of Euclid's Elements. The proofs of Euclid are with very few exceptions retained, but the unnecessarily complicated expression is avoided, and the steps of the proofs are so arranged as readily to catch the eye. Prop. 10, Book. IV., is a good example of how a long proposition ought to be written out. Every now and then alternative proofs, allowable in the Previous and similar examinations, are given or suggested. The book is not, however, a mere text-book for pass examinations. The candidate for mathematical honours will find introduced in their proper places short sketches of such subjects as the Pedal Line, Maxima and Minima, Harmonic Division, Concurrent Lines, etc., quite enough of each for all ordinary requirements. Useful notes and easy examples are scattered throughout each book, and sets of hard examples are given at the end. The whole is so evidently the work of practical teachers, that we feel sure it must soon displace every other Euclid."

Saturday Review-"Amongst the many editions of Euclid's Elements of Plane Geometry produced recently, that of Messrs. Hall and Stevens will deservedly occupy a prominent place as a class-book."

Literary World-" By bringing out their new edition of Euclid Messrs. Hall and Stevens deserve the thanks of every one who is interested in geometrical teaching, and it is a distinct advance on all previous editions."

Journal of Education-"The most complete introduction to plane geometry based on Euclid's Elements that we have yet seen."

Irish Teachers' Journal-"It must rank as one of the very best editions of Euclid in the language.'

...

Athenæum "The joint authors of this book, though adhering pretty closely to Euclid's arrangement and reasoning, have nevertheless ventured to make several important alterations. These changes will, almost without exception, obtain entire approval. Exercises and notes are abundant, and there are several interesting and useful propositions which, though strictly belonging to more modern and more advanced geometry, may be perfectly understood by any student who has mastered the ordinary text. Lest timid teachers, with the fear of conservative examiners before their eyes, should imagine that the book contains too many or too sweeping innovations, we may state that wherever any serious departure is made from Euclid's method, it is put into the form of an alternative proof, Euclid's proof being given as well. The work will be received with favour by practical teachers."

Nature-"We have here the completion of a work which in its first instalment (Books I. and II.) has already won a considerable amount of favourable notice from teachers. The end has crowned the work in a similar satisfactory manner. . . . Great attention has been paid to the arrangement and composition of the text, and the difficulties which delay beginners have been carefully smoothed and explained. The ordinary proofs have been adhered to as much as possible, and... changes have been adopted only where the old text has been generally found the cause of difficulty.... The subject of proportion has been treated on the system advocated by De Morgan, and here great use has been made of the admirable exposition of it given in the Association's (A.I.G.T.) text-book. The principal propositions have been established in a clear manner, both from the algebraical and geometrical definitions of ratio and proportion, and the distinction between the two modes of treatment is well brought about. The whole of this part forms a good introduction to the sixth book.... The explanatory matter and additional sections contain all, or nearly all, that is looked for nowadays, and include articles on harmonic section, centres of similarity and similitude, pole andpolar, radical axes and transversals. The exercises in the text are well graduated, and should bring out the pupils' acquaintance with, and mastery over, the propositions to which they are appended. More difficult problems are led up to by the solution of typical examples. In conclusion, we need only say the work before us contains all that is needful to a student, who, if he has this, will require no other text-book to become an expert geometer."

ΚΕΥ ΤΟ

"A TEXT-BOOK OF EUCLID'S ELEMENTS." Books I.-IV. Crown 8vo. 6s. 6d.

Books VI. and XI. Crown 8vo.

35. 6d.

Books I.-VI. and XI. Crown 8vo. 8s. 6d.

Of the 530 Public Schools, Grammar Schools, Training Colleges, and Girls' High Schools using Messrs. Hall & Knight's Elementary Algebra, 295 use Messrs. Hall & Stevens' Elements of Euclid, besides the following:-King's College; Glasgow, Girls' Grammar School; Norwich; Maidstone, Girls' Grammar School; Liverpool, Merchant Taylors' Girls' School; Bow, E., Coborn School for Girls; Tunbridge Wells, High School; Abingdon, Grammar School; Alford, Grammar School; Andover, Grammar School; Aston, King Edward's School; Audlem, Grammar School; Aysgarth School, Yorks.; Bangor, Baptist College; Barnsley, Grammar School; Beaumaris, Grammar School; Bewdley, Grammar School; Birkenhead, Clifton Park School; Birmingham, Blue Coat School; Birmingham, School Board; Blairs, St. Mary's College; Bloxham, All Saints' School; Brecon, Christ's College; Bridge of Allan, Stanley House School; Brigg, Grammar School; Brighton, St. Aubyn House School; Brighton, A. H. Thomas, Montpelier Crescent; Bristol, Grammar School; Brockley, High School; Bromsgrove, Grammar School; Bromyard, Grammar School; Bruton, Grammar School;