Notices respecting New Books. 251 



so that the right-hand member is 



O + q)f+ i j^ (p 2 q +pq*), or /+ \ n -^. 



The left-hand member may be represented by f+df/dp,, so 

 that ultimately 



df 1 ffif .,.#. 



d^=^d? (1&) 



Accordingly by the same argument as before the chance of 

 an excess r lying between r and r -{-dr is given by 



— i ^e-^P^dr (19) 



We have already considered the case of p = q= z i' Another 

 particular case of importance arises when p is very small, 

 and accordingly q is nearly equal to unity. The whole 

 number /j, is supposed to be so large that pp,, or m, repre- 

 senting the most probable number of occurrences, is also 

 large. The general formula now reduces to 



,<l \ e-* 2 ' 2m dr, (20) 



V (257T/H) 



which gives the probability that the number of occurrences 

 shall lie between m + r and m + r+dr. It is a function of m 

 and r only. 



The probability of the deviation from m lying between +r 



"■ '/ fl i [ r e-^dr= -|- f T '—**> ' ' ' ( 21 ) 

 y/ (Jtirm). Jo vttJo 



where T = r/ N /(2m). This is equal to "84: when t=1*0, or 



r — s/{2m) ; so that the chance is comparatively small of a 



deviation from m exceeding + s /(2m). For example, if m 



is 50, there is a rather strong probability that the actual 



number of occurrences will lie between 40 and 60. 



The formula (20) has a direct application to many kinds of 



statistics. 



XVII. Notices respecting New Books. 

 Textbook of Algebra with exercises for Secondary Schools and Colleges. 



By G. E. Fisher, M.A., Ph.D., and I. J. Schwatt, Ph.D. Part I. 



(pp, xiv-f-683: Philadelphia, Fisher & Schwatt, 1898). 

 r PHlS is a big book for the comparatively small extent of ground 

 **- it covers. The usual elementary parts are discussed up to 

 and including simultaneous Quadratic equations, and then, in the 

 remaining 80 pages, we have an account of liatio, Proportion, 

 Variation, Exponents, aud Progressions. The Binomial Theorem 

 for a positive Integral Exponent occupies about a dozen pages, the 

 treatment by Combinations being reserved, we presume, for Part 11. 



