576 



SCIENCE. 



[N. S. Vol. XIX. No. 484. 



which this form is due; (fc) it enables one to 

 reduce the consideration of any skew curve 

 to that of the normal curve; (c) the sim- 

 plicity of the application." 



A popular discussion of the origin of nor- 

 mal curves follows. The curve, as is well 

 known, is given by the expansion of (1/2 -H 

 1/2)". Professor Pearson derives his skew 

 curves by studying the expansion of (p + g) ". 

 where p-'f- q^l. Now Professor Kapteyn 

 considers the exponent n as giving the 

 number of causes which enter into the prob- 

 lem of gTowth, and shows that with a suffi- 

 ciently large value for ti, and natural 

 causes must be looked upon as almost infinite 

 in number, (p -|- q) " approximates closely to 

 a normal curve or, quoting Bessel : " What- 

 ever be the efiect of the various causes of 

 deviation, as long as they are: (a.) very nu- 

 merous; (b) independent of each other; (c) 

 such that the effect of any one cause is small 

 as compared with the effect of all such causes 

 together, we shall obtain a curve which ap- 

 proximates the nearer to the normal curve the 

 greater n is." 



But, though we may assume the effect of 

 certain causes in producing deviations in cer- 

 tain quantities x to be independent of the 

 value of X, this can not be the case with quan- 

 tities proportional to x', 1/x, or any non-linear 

 function of x. The resultant curves under 

 these conditions are the skew curves. To ob- 

 tain these the author supposes that ' on certain 

 quantities x, which at starting are equal, there 

 come to operate certain causes of deviation, 

 the effect of which depends in a given way on 

 the value of x.' Let us imagine certain other 

 quantities depending on the quantities x in 

 the way given by z ^ F{x). 



Then we have 



As 



Lz^=F'{x)^o^, or Aa:=; — ,, . , 

 ^ ' ' F'(x) 



where J 2 represents a series of deviations of 

 the quantity z independent of the value of 2. 

 Thus the effects of the causes of deviation 

 operating on x are proportional to 1/F'(x). 

 Now since, according to assumption, the quan- 

 tities z are distributed in a normal curve, say 



the quantities a; must be distributed along the 

 curve 



y- 



.F'{x)e 



This is the frequency curve generated under 

 the influence of causes, the effect of which is 

 proportional to 1/F'{x), no limits being placed 

 as to the form of this function. 

 The author next takes up the case 

 F{x) = {x + K)-> 

 the equation of the curve now being 



y = -^ {x + k)'~'c-" [{'+ «)'-• »]2, 



and derives complete formulae and tables for 

 the finding of the five constants A, h, M, q, n 

 for the five possible cases 



g ^ and g ^ ± 00. 



The solution is left in a rather unsatisfac- 

 tory state, as we can not find A directly, while 

 it is necessary to know A in order to find the 

 other constants. As A is in most cases unity, 

 he assumes this value for it, and computes 

 the other constants. These having been found, 

 A is readily computed. If A computed =j= A 

 assumed, try again with some other value for 

 A until a perfect agreement has been obtained. 

 Another weakness of the solution is that only 

 four of the observations of a set are used. 

 These are so chosen that their abscissse are 

 in arithmetical progression. The author, how- 

 ever, considers this very fact an element of 

 strength. 



It can not be denied that Professor Kapteyn 

 gets some very good results and his theory is 

 undoubtedly full of possibilities. 



C. C. Engberg. 



The University of Nebraska. 



The Mammals of Pennsylvania and New 

 Jersey. A Biographic, Historic, and De- 

 scriptive Account of the Furred Animals 

 of Land and Sea, both Living and Extinct, 

 Known to have Existed in these States. By 

 Samuel N. Ehoads. Illustrated with plates 

 and a faunal map. Philadelphia, privately 

 published. 1903. Pp. 252. 

 Mammalogists have been so busy in recent 



years describing, classifying and getting their 



