REVIEWS 151 



p. 414) of Appell's equations or the fact that Gauss's principle of least constraint 

 leads to these equations, which are more general than Lagrange's equations in 

 that they apply to non-holonomous systems. Ferrers's generalisation of Lagrange's 

 equations is given (p. 412), but Ferrers's name is not mentioned. 



Philip E. B. Jourdain. 



An Introduction to the Algebra of Quantics. By Edwin Bailey Elliott, 

 M.A., F.R.S., Waynflete Professor of Pure Mathematics, and Fellow of 

 Magdalen College, Oxford. Second Edition. [Pp. xvi + 416.] (Oxford : 

 Clarendon Press, 1913. Price 155. net.] 

 This well-known, lucid, and well-written book has two objects : the first is didactic, 

 and the second is to help the investigator in his researches. The second object is 

 intended to be fulfilled especially by the chapters near the middle of the book 

 (p. v). The first edition was published in 1895, and both the first and this edition 

 are characterised by the use of methods which are distinctively English. Very 

 little attention has been given to the symbolical methods of Aronhold, Clebsch, 

 Gordan, and others ; and, now that an introduction to these methods has been 

 published in English in Grace and Young's Algebra of Invariants, this omission is 

 certainly excusable (pp. vi, viii). Thus Gordan's own proof that the number of 

 irreducible invariants of a binary p-\c is finite for any value of p is not given. 

 The proofs given are those due to Hilbert (p. 182). 



The sixteen chapters deal respectively with principles and direct methods ; 

 essential qualities of invariants ; essential of covariants ; cogredient and contra- 

 gredient quantities ; binary quantics ; invariants and covariants as functions of 

 differences ; annihilators and seminvariants ; further theory of the annihilators and 

 reciprocity ; generating functions ; Hilbert's proofs of Gordan's theorem ; proto- 

 morphs, and so on ; further theory of seminvariants and the binary quantic of 

 infinite order; canonical forms, etc.; invariants and covariants of the binary 

 quantic and sextic ; several binary quantics ; binary quantics in Cartesian geometry 

 and restricted substitutions ; and ternary quantics and the quadratic and cubic. 



The theory discussed in this book cannot, in spite of Sylvester's extravagant 

 claims for the almost universal character of the theory of algebraic invariants, be 

 regarded as more than a very small part of algebra. It seems that writers should 

 point this out, for students are little apt for independence of thought. The 

 apparent importance of what is treated in such books as the present one and a 

 well-known work of Salmon is largely due to historical circumstances. It is, 

 therefore, particularly unfortunate that history is usually suppressed in them. 

 Thus Boole very early discovered (cf. p. 344) an interesting invariant for restricted 

 substitutions. In textbook order such invariants are treated at a fairly late stage, 

 and yet they contain the germ of rational views on the scope of invariant algebra 

 as a whole. Philip E. B. Jourdain. 



Integral Calculus. By H. B. Phillips, Ph.D., Assistant Professor of Mathe- 

 matics in the Massachusetts Institute of Technology. [Pp. vi + 194.] 

 (New York : John Wiley & Sons ; London : Chapman & Hall, 1917. 

 Price Si. 25, or 6.?. net.) 

 Dr. Phillips's preliminary volume on the Differential Calculus has already been 

 reviewed in SCIENCE PROGRESS (1917, 12, 343), and both that volume and the 

 present one can be had bound up together from the same publishers ($2, or gs. 6d. 

 net). " Throughout this course," says the author (p. iii), " I have endeavoured to 



