SeptkiMBEr 27, 1894J 



NA JURE 



519 



theorem, which states that the multiplier is a rational 

 function of the squares of the moduli. The latter is 

 shown, in fact, to break down when the modular equation 

 has equal roots. 



The long memoir on the Theta and Omega Functions 

 was originally written as an introduction to the long- 

 e.xpected "Tables of the e Functions." It may be 

 regarded as an advanced work on elliptic functions, in 

 which the arithmetical treatment is given the prominent 

 place. The theory of the transformation, and in parti- 

 cular of the modular equations and the associated curves, 

 is exhibited with remarkable elegance. 



Everywhere the treatment is characterised by extreme 

 rigour. In fact, the subject matter, dealt with in these 

 volumes, leads to work of so recondite a nature that only 

 an investigator to whom any slurring over of difficulty, 

 or exceptional case is absolutely repulsive, can expect 

 to make a real advance. Those who look chiefly to 

 results, and do- not care to know the precise circum- 

 stances under which they exist, may be warned off the 

 monument to Prof. Smith's' genius which is given to the 

 world in these pages. 



On two principal occasions Prof. Smith found oppor- 

 tunity to place his views on mathematics in general before 

 the scientific world. We have the valedictory address to 

 the London .Mathematical Society, delivered in the year 

 1876, on his retiring from the office of president. He 

 took as his text some "comparatively neglected regions 

 of pure mathematics"; and now, after an interval of 

 eighteen years, it is a matter of great interest to re-survey 

 the ground and estimate the advances that have been 

 made. In the theory of numbers, then as always the 

 subject of his predilection, he called attention to the 

 state of knowledge with respect to (i) the theory of 

 homogeneous forms ; (2) the theory of congruences ; 

 (3) the determination of the mean or asymptotic values 

 of arithmetical functions. With respect ,to. quadratic 

 forms of four or more indeterminates, he referred to the 

 fundamental theorem of M. Hermite concerning the 

 finiteness of the number of non-equivalent classes of 

 forms having integral coefficients and a given dis- 

 criminant ; and to the researches of Zolokoreff and 

 Korkine on the minima of positive quadratic forms. 

 In a foot-note also he referred to his oivn great work 

 " On the Orders and Genera of (Quadratic Forms con- 

 taining more than Three Indeterminates." These 

 three papers mark the extent .to which the inquiry 

 had been pushed at that time. The latter is much 

 the most important, and, so far as I know, but 

 little further progress in the same direction has 

 since been made. In the theory of congruences an 

 important advance has been made by G. T. Bennett, in a 

 paper published in the Phil. Tra7ts. R S. vol. 1S4A. The 

 investigation is '" On the Residues of Powers of Numbers 

 for any Composite .Modulus, Real or Complex." Re- 

 marking that primitive roots exist only when the modulus 

 is a power of an uneven prime or the double of a power 

 of an uneven prime, and that a primitive root may be 

 said to "generate" by its powers the complete set of 

 residues, Bennett exhibits the mode of formation, and the 

 relations connecting, the most general set of numbers 

 capable of generating the '\>{m) numbers which are prime 

 NO. 1300, VOL. 50J 



to any composite modulus in, and extends his results to 

 complex numbers. 



Prof. Smith gave an historical 'account of our know- 

 ledge of the series of prime numbers. Prof. Sylvester 

 has made a considerable contraction of Tchebychel's 

 limits, and has established important general principles 

 in connection therewith. 



Passing on to the discussion of the transcendency off 

 and TT, it may be noted that since the address was de- 

 livered (in fact, six years subsequently) the question has 

 been triumphantly set at rest for ever by the labours of 

 Hermite and Lindemann. The former established the 

 transcendency of e, and the latter, standing on the 

 shoulders of the former, demonstrated the transcendency 

 of TT. Lindemann's proof shows that v cannot be the 

 root of any algebraic equation, and marks a distinct 

 epoch in the history of mathematical science. The 

 death-blow was thus given to the circle squarers in 18S2 

 [Mai/i. Ann. vol. xx.). O^ite recently extraordinarily 

 simple proofs of the transcendency of both numbers have 

 been given by Hilbert. Prof. Smith noted and lamented 

 the want of advanced treatises in English on various 

 branches of pure mathematics. Our position to-day in 

 this respect exhibits a marked improvement. On 

 dil'ferential equations, theory of functions, integral cal- 

 culus, theory of numbers, important works by English 

 and American authors have been published, and certain 

 eminent mathematicians are known to be engaged in 

 the preparation ot advanced works, which will shortly 

 appear and further fill in the gaps. 



Prior to the above, in 1S73, was delivered the address 

 to the Mathematical and Physical .Section of the British 

 Association. Remarking on the recent appearance of 

 .MaxwelTs " Electricity," he observes: " It must be con- 

 sidered fortunate for the mathematicians that such a vast 

 field of research in the application of mathematics to 

 physical inquiries should be thrown open to them at the 

 very time when the scientific interest in the old 

 mathematical astronomy has for the moment flagged, 

 and when the very name of physical astronomy, so long 

 appropriated to the mathematical development of 

 the theory of gravitation, appears likely to be 

 handed over to that wonderful series of discoveries 

 which have already taught us so much concern- 

 ing the physical constitution of the heavenly bodies 

 themselves. ' Mathematical astronomy to-day, it may be 

 said, no longer flags. Thanks to the work of Hill, 

 Poincard, and Gylden, the subject has received a new 

 impulse, and the world of science watches with intense 

 interest the process of its evolution under the powerful 

 hands of these mathematicians. 



Prof. Smith had much at heart the organisation of 

 scientific education as influencing the supply of scientific 

 men. He asserts the importance of assigning to physics 

 a very prominent place in education. He gives as his 

 opinion that from the sciences of observation the student 

 "gets that education of the senses which is after all so 

 important, and which a purely grammatical and literary 

 education so wholly fails to give." These are weighty 

 words when we consider the all-round attainments of 

 their author, and that he was, in particular, a classical 

 s:holar of the first rank. The effect of these volumes on 



