220 Notices respecting JVeiv Books. 



length the transformation of the Quotient of two integral functions, 

 treating under this head of the Remainder theorem and its appli- 

 cation to Factorisation, of a new basis for the principle of Indeter- 

 minate Coefficients and of Continued Division. 6 gives much 

 useful matter under the heads of Greatest Common Measure and 

 Least Common Multiple. 7, on the Factorisation of Integral 

 Functions, introduces the consideration of surd and imaginary 

 Quantity and of Complex Quantity. In 8 are discussed Rational 

 Fractions ; in 9 we have a continuation of Theory of Numbers, com- 

 prising Scales of Notation and Lambert's Theorem ; in 10 we have 

 a general discussion of irrational functions . . interpretations oixPli, 

 oc°, w 2m , and a general theory of rationalization; and in 11 is an 

 account of the Arithmetical theory of Surds. One of the best 

 chapters to our thinking is 12, which gives an excellent account of 

 complex numbers and herein of Argand's diagrams and of De- 

 moivre's theorem. 13 is good on Ratio and Proportion. 14 at some 

 length discusses conditional equations in general; 15 at equal 

 length treats of the Variation of a function (a good introduction to 

 the theory of Maxima and Minima for a student of the Calculus) ; 

 and 16, 17 are concerned with equations of the first and second 

 degree respectively. In 18 is an account of a general theory of 

 Integral Functions, in which figure Symmetric functions of the 

 roots of an Equation, Newton's theorem regarding sums of powers 

 of roots, special properties of Quadratic Functions (including La- 

 grange's Interpolation formula), and Variation of a Quadratic 

 Function for real values of its Variable (analytical and graphical 

 discussion of three fundamental cases, Maxima and Minima). 19 

 is devoted to the Solution of Problems by means of Equations. 20 

 discusses the Arithmetic, Geometric, and allied series ; 21 is occu- 

 pied with Logarithms (interpolation by first differences); and 22 

 closes the work with an account of the Theory of Interest and 

 Annuities. Numerous historical notes impart considerable interest 

 to the perusal of the text. 



It is evident that there are many subjects handled which do not 

 come within the range of an elementary student's reading ; but 

 these are all handled in such a way as to be most valuable to more 

 advanced students and to teachers. The author warns us on the 

 very threshold, that his work is not intended for the use of 

 absolute beginners. His great object in laying down the three 

 fundamental laws is to introduce the idea of Algebraic Form, 

 " which is the foundation of all the modern developments of Algebra 

 and the secret of analytical Geometry, the most beautiful of all its 

 applications." "We advise higher-form boys and others, who have a 

 desire to go more deeply into the subject than they can do with the 

 aid of ordinary textbooks, to get or borrow Prof. Chrystal's splendid 

 treatise and make a careful study of its well-arranged contents. 

 It is the pure Mathematical Elementary textbook of the year 1886. 



Elements of the Theory of the Newtonian Potential Function. 



By Dr. B. O. Peirce. [Boston : Ginn & Co., 1886.] 



The Compiler of these Lecture Notes adopts the term employed 



