680 SCIENCE PROGRESS 



mathematics and sustain it, but that is done in this book. The book consists 

 of nine chapters, and each chapter is divided into sections. The first chapter is 

 on the principles of invariant-theory, and the sections are concerned with illustra- 

 tions of the nature of an invariant ; terminology, definitions, and transformations ; 

 and special invariant formations. The second chapter deals with properties of 

 invariants, and the sections are on the homogeneity of a binary concomitant ; 

 index, order, degree, and weight ; simultaneous concomitants ; and symmetry 

 and the fundamental existence-theorem. The third chapter is on the processes of 

 invariant-theory, and the sections deal with invariant operators ; the Aronhold 

 symbolism and symbolical processes ; reducibility and elementary irreducible 

 systems ; concomitants in terms of the roots ; and geometrical interpretations. 

 The fourth chapter is on reduction, and the sections deal with Gordan's series and 

 the quartic ; theorems on transvectants ; reduction of transvectant systems ; 

 syzygies ; Hilbert's theorem ; Jordan's lemma ; and grade. The fifth chapter is 

 on Gordan's theorem ; the sixth chapter is on fundamental systems ; the seventh 

 chapter is on combinants and rational curves ; the eighth chapter is on sem- 

 invariants, modular invariants, and covariants ; and the ninth chapter is on 

 invariants of ternary forms. At the end of the book there is an appendix of 

 good and graduated exercises and theorems, and an index. 



Philip E. B. Jourdain. 



ASTRONOMY 



Collected Papers on Spectroscopy. By G. D. Liveing and Sir J. Dewar. 

 [Pp. xvi -f 566, with numerous diagrams and tables.] (Cambridge : at the 

 University Press, 191 5. Price 30^. net.) 



Scientists in general and spectroscopists in particular will be under a debt of 

 gratitude to the Cambridge University Press for making possible, by undertaking 

 almost the whole of the expense, this republication in collected form of the spectro- 

 scopic papers by G. D. Liveing and Sir James Dewar. The volume is handsomely 

 produced, and in all respects worthy of the best traditions of the Cambridge Press. 

 Such reproductions of collected papers as this fulfil a very definite aim ; no doubt, 

 as the authors remark in their preface, all the contributions to knowledge contained 

 in these papers have already been culled and classified in Kayser's great work, the 

 Handbuch der Spectroscopic ; but whilst such a record is useful as an encyclopaedia 

 of knowledge, to the investigator who is seeking to retrace the steps by which the 

 authors obtained their results, and to follow the train of their thought and argu- 

 ment, it is useless. For such a purpose it is essential to refer to the original 

 papers, which generally involves the necessity of having to refer to papers scattered 

 amongst many different volumes of the proceedings and transactions of various 

 societies, to which oftentimes it is difficult or impossible for the investigator to 

 obtain access. The collection of papers together in this manner avoids this 

 necessity and at the same time enables the author's work to be viewed in a proper 

 perspective. 



The papers in this volume extend in time over a period of twenty-seven years 

 —from 1877 to 1904. The collaboration of two scientists extending over so long 

 a period is in itself remarkable. Their work is too well known for it to be neces- 

 sary to refer in any detail to the subject-matter of these papers. A glance through 

 the subject index makes apparent at a glance the extraordinary diversity of the 

 subjects touched upon. At the present time, when the existence of series relations 

 amongst the lines in the spectra of many elements is assuming so prominent a 

 place in theories of the structure of the atom, it is interesting to note that Liveing 



