January 31, 1895J 



NA TURE 



319 



LETTERS TO THE EDITOR. 



( The Editor doc not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond ivitk the writers of rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. ] 



A New Step in Statistical Science. 



Contribution's of an abstract kind to statistical science are so 

 liltle read \iy the hull; of statistician-, that the scope of the re- 

 markable memoir by Prof. Karl Pearson may be unappreciated 

 by them unless attention is pointedly direcled to it.' 



Slaiisticians are conversant with the use of curves to epitomise 

 masses of data. The forms of the majority of these curves are 

 skewed or humped, and have hilherlo been nondescript, except 

 as Prof. K. Pearson's previous memoir showed, some of them 

 may be dissected inlotwoor more normal curves, having different 

 constants. It is only a few curves that are symmetrical and con- 

 form closely to the normal law of facility of error. These few 

 have been much studied, the numerous and valuable properties of 

 the normal curve beinij extremely helpful in dealing with them. 

 When the conformiiy between the observations and the normal 

 law ceases to be close, the latter must be applied warily. When 

 the discrepancy is serious, it becomes unsafe to trust the theory 

 at all. Not a few statisticians have chafed under these limita- 

 tions, and felt that the normal law is too limited in its 

 };rasp to satisfy their needs. Now at length, it lurns out, 

 thanks to Prof. K. Pearson, that ihe normal law admits of 

 being regarded as nothing more thin a very special case of a 

 highly generalised theor\', by whose aid curves may be drawn 

 that shall fit every one of the observed curves he has tried as 

 yet. The shapes of these arc curiously varied. Their list 

 includes the dimensions of shrimps and prawns, of American 

 recruits and of school-girN, of IJavarian skulls, of frequency of 

 enteric fever, and of divorces, of variation in house value, in 

 butter-cups and in clover, in pauper percentages and in a 

 mortality curve. Only those who have studied the delicate an t 

 oddly-shaped drawings of these observed curves on a large 

 scale, upon which the equally delicate/iV/ty/ curves have been 

 superimposed, can adequately appreciaie the wonderful close- 

 ness between the pairs of outlines. There can be no doubt 

 that the descriptive efficiency of Prof. Karl Pearson's method 

 is of the highest order. 



The question of the utility of the method to ordinary 

 statisticians has now to be considered. First, as to its 

 descriptive [inwers. We are already able to describe the whole 

 of any normal series by means of only two numbers ; say, by 

 the average of all the members of the series, and by the mean of 

 the several departures of its individual members from that 

 average. Henceforth, by the use of more constants, we shall 

 be able to do the same to any series. A few figures will 

 always serve as the equivalents of a vast amount of tabular 

 matter. 



The second and higher use is to afford a clue to the cause or 

 causes of variety. It has long been a dream to me to select a 

 peculiar and often recurring form of curve, and to study with all 

 possible care the conditions under which it has been produced, so 

 that whenever a new curve of that same form was me' with, 

 there should exist some guidance towards discovering the cause 

 of iis iiroduction. I have made not a few attempts from lime 

 to time, but was discouraged by the then impossibility of sufii- 

 ciently defining the curves that were dealt with. Thatdifticuliy 

 seems now removed To explain myself further, let us sup- 

 pose that a man finds the mark of a more or less incomplete 

 circle on the ground, and wishes to discover how it was 

 made. Various possibiliiies exist, which might have been 

 recorded lo help him: (i) The mark may have been 

 made by a basin, &c., turned in a lathe, or, what 

 come to the same thing in principle, by something revolving 

 round a fixed centre. f2) It may have been stamped by a 

 hoop, formed by bending an el.istic rod until its ends met, 

 the circular form depending on the equal distribution of stress. 

 (3) II may have been the mark of a withy that had bound a 

 f*gg"l. which, after selling into shape, had broken away, the 

 circular form arising from the compressi m of the sticks of the 

 faggot within the smallest possible girlh. (4) It may have been 



' "M.iihcmatic.il Contributions to the Theury of Evolution " (Pan ii.), by 

 Karl I'carion. Read at the Ro>'al Society, January 24, 1895. 



Ihe mark of a warped sheet of bark, hide, &c., the circular 

 form of which depended on the unequal coniraclion of its 

 outer and inner layers. (5) It could have been made by a 

 projecting nail, near the angle formed by two straight rods 

 securely joined at one end ; ihe apparatus being caused to 

 slraddle and press upon two pegs in the ground, and moved in 

 Ihe only way possible under those conditions. Here the frag- 

 ment of a circle would be due 10 Ihe conslancy of the angle 

 subtended by the same chord. .A. catalogue of these and other 

 possibilities, which are numerous and include circular forms of 

 animal and vegetable growth, would certainly enlarge the 

 speculations of the observer as to the cause of the circular mark. 

 So It would be wiih ihc curves of which I spoke. Each form 

 of curve would be a serious study in itself: still the results 

 would gradually accumula'e, audit is reasonable to look forward 

 to a time when a set of such curves, each defined by Prof. K. 

 Pearson's, method, shall have been studied. 



I venture upon one criticism as to the completeness of 

 his generalisaiion. Variation of ihe normal kind is sup- 

 pose 1 to be due to the combined eftect of (1) an infinite 

 number of causes, that are (2) equally likely to err in 

 excess or deficiency, and {3) that are independent of one 

 another. These three restrictions are removed in the 

 generalised curve, which bears a ceriain relation to the 

 uinomial point curve formed by the expansion of (/ -H q)" 

 when (I) « need not be infinite, (2)/ need not be equal 10 ?, 

 while (3) the binomial form which implies independence of the 

 conlriljuioiy causes, is modified. Now though the condition (3) 

 is removed, it does not, as yet seem to me that the supposition 

 which replaces it. and which is based on such considerations as 

 \ ihe effect of withdrawing r cards from a pack of ns cards 

 > containing s suits, is analogous to what commonly occurs in 

 rerum natiinu Namely, the intermingling of contributory 

 cause; of various degrees of cfliciency, some of which are very 

 lew in number and have large efTects. Thus the number of 

 persons who walk day by day down St. James' Street, is 

 occasionally vastly augmented by some national spectacle, and 

 it is considerably and irregularly affected by changes in the 

 weather. It does not wholly depend on a multitude of equipotent 

 causes. So again, the time of ripening of the fruit on a tree 

 generally, is much afi'ec ed by the aspects of the pariicular 

 branches on which it grows. I thereiore conclude that the 

 effects of an aggiegate of large and small causes, ought to 

 be distinctly included in a thoroughly generalised formula of 

 variation. 



Francis Galton. 



NO. 1318, VOL. 5 l] 



The Kinetic Theory of Gases. 



I THINK sufficient stress has not been laid on the distinction 

 between the purely mathematical proof of the fundamental 

 determinantal relation connecting the differentials of the co- 

 ordinates and momenta of dynamical systems and the purely 

 statistical applications of that relation which form the subject 

 of the Kinetic Theory. 



The well-known determinantal equation is perfectly general 

 and applicable to any dynamical system whatever. It merely 

 asserts that if the initial coordinates and momenta receive any 

 small independent variations whatever, the resulting variations 

 in the fiiml coordinates and momenta after any fixed interval of 

 time are so related to them that the dift'erential product of the 

 former variations is equal to that of the latter, conformably to 

 the well-known laws of Jacobians, by which the_variables are 

 changed in a multiple differential or integral. 



In general the variations in question are purely hypothetical, 

 just as, in the principle of Least Action, the .actual motion is com- 

 pared with varied motions which have no real existence. If, 

 for example, we consider the system to be the Karlh, the 

 variations could only be made real by making the Karth move 

 dilTerently to what it actually moves, and doing this in every 

 possible w.ay. 



]'>ut when it is required to assign any physical meaning to the 

 determinantal relation, these hypothetically varied molious must 

 be represented by actual motiims, and this can only be done by 

 taking an indefinitely large number of independent systems, all 

 simikar to the one we started with, and setting them moving in 

 all possible ways. The determinantal relation then tells us 

 tluit if the coordinates and momenta of these systems are 

 initially distributed according to what in my Report I have called 



