366 Notices respecting New Books, 



A second object which the author had in view was to produce 

 " a really elementary treatise " on the subject. And this he has 

 in one sense accomplished ; for the exposition of the subject is 

 pot unduly long or hard, except so far as the subject itself is 

 hard; and a student who is already familiar with the matter 

 to which the method is applied in the last two hundred pages 

 of the volume, would probably master its contents without very 

 serious difficulty. Still we think something more might be done 

 in the way of simplification. The student is in danger of being 

 bewildered at the outset by a number of symbols which look like 

 what he has already had to do with, but which must be interpreted 

 in wholly new and unexpected w^ays. This difficulty yields to 

 labour ; - but it might be sensibly diminished by being brought more 

 gradually before the learner. There is the less need to dwell on 

 this, as a still more elementary treatise on Quaternions* has been 

 recently published (written in part by Professor Tait), in which 

 the treatment is perhaps as simple as the subject admits, and which 

 could be made out by any one to whom the study of the subject 

 w T ould be advantageous. 



The Expression of a Quadratic Surd as a continued fraction. By 

 Thomas Mtjik, M.A., Assistant to the Professor of Mathematics in 

 the University of Glasgow. Grlasgow : J. Maclehose. 1874. (8vo, 

 pp. 32.) 



If Gta? 2 =H, where Gr and H are integers, it is well known that .r 

 can be expressed as a recurring continued fraction in the form 



a+J- 1 * 



and that the first term of the first recurring period or cycle is a v 

 assuming H to be greater than Gr. In the calculation of the num- 

 bers A, a v a 2 , a s , . . . two other sets of numbers are used — viz. a set 

 of divisors Gt, d v cl 2 , d 3 , . . . and a set of rational dividends 0, AGr, 

 r 2 , r 3 , . . . The pamphlet before us is a systematic investigation of 

 the properties of these sets of numbers and of the successive conver- 

 gents of the continued fraction. It differs from the accounts of 

 the subject to be found in ordinary text-books by the greater gene- 

 rality and completeness of the treatment of the subject E. g. it is 

 usual to notice no more than the case in which a?= H ; but according 

 to Mr. Muir's treatment this is only a particular case having some 

 special properties, which he discusses in the last six pages of the 

 pamphlet. 



The cycle formed by the successive quotients is well known to 

 take the form a v a 2 , . . . a 2 , a v 2 A. Now there is, of course, a 

 means of deriving each convergent successively from that which 

 precedes it ; but the form of the above cycle suggests that there 

 ought to be some simple method of determining certain convergent^ 

 without calculating all the intermediate ones ; and this is in fact the 



* Introduction to Quaternions; by Professors Kelland an d Tait. 



