n8 SCIENCE PROGRESS 



using them may be obvious, and that what are called negative 

 and imaginary solutions of problems may have as real and 

 precise a meaning as those called positive." In vol. xvii. 

 (191 3) of the Proceedings of the Cambridge Philosophical Society, 

 H. C. Pocklington discusses some Diophantine impossibilities, 

 and Prof. A. C. Dixon gives a short proof that a determinant 

 cannot exceed its leading term. In 1914 Prof. L. E. Dickson 

 published an elementary introduction to the general theory of 

 Linear Algebras (Cambridge University Press, 35. net), including 

 non-associative algebras. Among papers on algebra, we may 

 notice Dr. J. Brownlee's {Proc. Roy. Soc. Edinb., vol. xxxii. 191 3) 

 analysis of observations on inheritance of hair and eye colour. 

 He finds that Mendel's laws are satisfied to a remarkable extent. 

 In analysis and the theory of functions the most important 

 books recently published are Prof. A. R. Forsyth's Lectures 

 Introductory to the Theory of Functions of Tzvo Complex Variables 

 (Cambridge University Press, 19 14, 10s. net), delivered at 

 Calcutta University in 1913 ; the second edition of G. H. Hardy's 

 Course of Pure Mathematics (Cambridge University Press, 1914, 

 12s. net), which has several interesting additions to the first 

 edition published in 1908 ; an English translation by S. E. Rasor 

 of Dr. H. Burkhardt's admirable Theory of Functions of a Complex 

 Variable (London: D. C. Heath & Co., 1914, 12s. 6d.); an able 

 tract by G. N. Watson on Complex Integration and Cauchys 

 Theorem (Cambridge University Press, 1914, 3s. net); and a 

 fourth edition of Forsyth's well-known Treatise on Differential 

 Equations (Macmillan, 1914, 14s. net). 



Every linear differential equation of the second order leads 

 to a continued fraction, so that theorems involving the trans- 

 formation of one such differential equation into another would 

 be expected to lead to transformations of continued fractions 

 hitherto unknown. A. Lindsay Ince {Proc, Edinb. Math. Soc, 

 vol. xxxii. 1914) uses a method depending on these facts to get 

 certain theorems on continued fractions equivalent to Riemann's 

 and other transformations of the /'-function. A fairly large 

 proportion of the papers in this volume is devoted to the 

 subject of harmonic analysis. Of these, the most noteworthy 

 seem to be papers by Prof. E. T. Whittaker on the calculation 

 and investigation of the functions of Hermite and some other 

 functions by continued fractions, and the solution of Mathieu's 

 equation by introducing a new parameter and avoiding the use, 



