Sept, 22, 1887] 



NATURE 



483 



of digits will fluctuate about the mean 450 according to a pro- 

 ability-curve whose " probable error " is about 19. 



(i) One explanation of the failure of the law is that the 

 requisite plurality of items is wanting. Suppose we had taken 

 sums of two (instead of a hundred) digits, the grouping of these 

 sums would be best represented by a right line, or rather two 

 right lines. If we took three digits at a time, the resulting form 

 would be parabolic. A variant of this class of exception is when 

 the larger items are few or unique, while items of an inferior 

 order congregate in great numbers. Suppose each observation 

 to consist either of the digits 3 or 6, plus ten items taken at 

 random from the series "i, '2, . . . . '9. There would then be 

 generated a curve like those in Dr. Venn's Fig. 2. If, instead of 

 3 and 6, we had two digits, 4 and 5, differing by very little from 

 each other, the abnormal uniqueness of the larger items would 

 be disguised. It is upon this principle, doubtless, that the 

 population of a kingdom appears to conform (in respect of height 

 or other attribute) to the law of error, while at the same time each 

 province may present a distinct type. Suppose that the majority 

 of our returns were, as the last-mentioned case, either 4 or 5 

 plus an aggregate of smaller items ; but that a small proportion 

 of the returns were governed by a widely disparate "large 

 item," e.g. 8 or 9 ; in this case we might have the appearance 

 presented by Dr. Venn's Fig. i. The body of the curve would 

 seem to be of the probability family ; but there would be tacked 

 on a tail appertaining to a different type. Dr. Charles Roberts 

 has adduced some statistics of this species in a paper published 

 in the Medical Times, February 7, 1885. 



(2) We have hitherto supposed that the constituent items have 

 no bias in one direction. Suppose, however, that instead of 



the digits i, 2 8, 9 being each equally eligible, 8 and 9 



became inadmissible ; and, whenever one of those digits was pre- 

 sented, we had to substitute 6 and 7 respectively. There would 

 thus be two chances in favour of 6 and also of 7. An aggregate 

 of 100 digits each selected according to this unsynimetrical scheme 

 would be grouped about the mean value 10 x(i-f2 + 3-1-4-1- 

 5 -I- 2 X 6 -)- 2 X 7), or 410, in a form which as to the body of 

 the curve would be a probability-curve, but which would be 

 unsymmetrical at the extremities. The most familiar example 

 of this case is afforded by games of chance. If black and white 

 balls, in an unequal proportion and immense numbers, are 

 mixed up, then if you take at random batches of 100 (or 1000) 

 balls the percentage of white or black balls will fluctuate in the 

 manner described. It is quite possible that this principle 

 should govern what Dr. Venn calls a "one-ended phenomenon," 

 i.e. one in which unlimited variation is conceivable in one 

 direction but not in the other. Dr. Venn's Fig. i seems fairly 

 well to represent a biased probability-curve. 



(3) We have hitherto supposed that the individual observation 

 or return is the sum of the variable elements. But it may be 

 a more complicated function. Thus it may be a product. The 

 logarithm of the observations may fluctuate according to a 

 probability-curve, while the observations themselves obey a law 

 which has been investigated by Dr. Macalister in the Proceedings 

 of the Philosophical Society (1879) ; related to the geometrical 

 mean just as the probability curve is to the arithmetic mean. 

 This grouping is to be expected wherever the analogies of 

 Fechrier's law prevail. This may be the rationale of the fact 

 which I have elsewhere pointed out, that fluctuations of price rise 

 much higher above, than they fall below, the mean. But, where 

 the principle of estimation does not come in, it is not quite clear 

 why the geometrical curve should be more appropriate to a "one- 

 ended phenomenon" than the biased probability-curve which 

 has been described under our heading (2). At any rate, in 

 the case before us. Dr. Venn's Fig. i, the numerical statistics 

 which he has allowed me to inspect show much too close a 

 correspondence between the body of the figure and the proba- 

 bility-curve to admit of the geometrical explanation. There is 

 also this peculiar difficulty, that the longer limb of the given 

 curve is the lower one. F. Y. Edgeworth. 



King's College, London. 



A Null Method in Electro-calorimetry. 

 By reference to the last number of the Electrical Review 

 (vol. xxi. p. 262), wherein is printed a short abstract of our 

 paper on " A Null Method of Electro-calorimetry" read before 

 the British Association on September i, Mr. Iluntly will find 

 that the method of measuring specific heats suggested by him is 

 in principle similar to that described by Mr. Gee and myself. 

 The method has been employed for determining specific heats 



during the last two years, but we hive delayed publication till 

 the best working details of the method have been elaborated. 



In certain practical details our method differs from Mr. 

 Iluntly 's suggestion. The mass of liquid in each calorimeter 

 is not the same. It is much preferable to have the masses in- 

 versely proportional to the specific heats, so that the thermal 

 capacities of the liquids are equal. In this way it will be readily 

 understood that the correction for radiation can be made to 

 disappear altogether. For since the calorimeters are precisely 

 equal, and their temperatures equal, the loss of heat by radiation 

 must be the same from each ; further, since the thermal capacities 

 of the liquids are the same, as well as that of vessels and stirrers, 

 it follows from the equality of the resistances that the same 

 current will produce the same rise in temperature in each case, 

 and conversely, since the heat radiated from each calorimeter is 

 the same, and since the thermal capacities of the calorimeters 

 and stirrers are equal, it follows that, if the same current travers- 

 ing the equal resistances produces the same rise in temperature 

 in each liquid, the thermal capacities of the two liquids are the 

 same, whence the specific heat can at once be determined by 

 determining the masses of the liquids. Virtually, then, the null 

 method of obtaining the same rise of temperature in each 

 calorimeter is attained by varying the mass of liquid in either or 

 both calorimeters. In practice we approximate as nearly as 

 possible to the condition by adding liquid to that calorimeter 

 which rises in temperature most quickly, and then make a final 

 adjustment by shunting a very small fraction of the current by 

 means of the high resistance in the box. This is, we believe, 

 the first time that a method for measuring specific heats 

 has been published in which the correction for radiation and 

 for the thermal capacity of calorimeters and stirrers has been 

 entirely eliminated. 



With the first apparatus we had made to embody these ideas, 

 viz. that described in t\iQ Electrical Ranew (loc. cit.), an accuracy 

 of at least one-tenth per cent, could be obtained from a single 

 experiment, thoroughly confirming Mr. Huntly's anticipations 

 as to the delicacy of the method. We have just introduced 

 some considerable improvements in the apparatus which we hope 

 will enable us to insure much greater accuracy than that hitherto 

 obtained. 



A few words are required in reply to some observations of 

 Mr. Iluntly. First, he suggests a bolometric method of deter- 

 mining the difference of temperature. We have so far preferred 

 a thermo-electric method, which, without a specially constructed 

 galvanometer, enables us to detect with certainty 1/2000 of a 

 degree ; the necessary corresponding variation in the resistance 

 of a Pt wire would only be I "6 parts in a million ; besides some 

 difficulties may arise in procuring two pieces of Pt wire which 

 shall have- the same temperature coefficient to i part in a 

 million, even if they be cut from the same piece originally. 

 Secondly, the time method described by Mr. Huntly at the end 

 of his paper seems to me to have a fatal objection : it would be 

 quite impDSsible to keep the current constant for a long time to 

 the 1/2000 part which would be requisite to secure such 

 accuracy as we can get with present arrangements. 



William Stroud. 



Mental Development in Children. 



I SHOULD like to hear the opinion of psychologists on the 

 following circumstance : — A female child, quick and intelligent, 

 when about fifteen months old, learned to repeat the alphabet, 

 shortly afterwards the numeral?, days of the week, month, &c., 

 and, subsequently, scraps of nursery rhymes, English and Ger- 

 man ; then to spell words of two and three letters. All this was 

 learned readily, eagerly indeed, and for a time she remembered 

 apparently every word acquired, indelibly. At about two years 

 old further teaching was for a time remitted, as she was observed 

 to be repeating audibly in her sleep what she had learned during 

 the day. Subsequently, tuition was resumed, unler a governess, 

 but she had not only forgotten much of what she had previously 

 known perfectly, but learns far less readily than formerly. She 

 is now about three and a half years old, in perfectly good health 

 and spirits, quick, and particularly observant, but the capacity 

 for learning by rote is materially diminished ; she is remarkably 

 imitative, but shows no faculty whatever for writing, and as 

 little for music. 



I should like to hear of any parallel cases, and what the ulti- 

 mate development has been ; with any opinions upon the cause 

 of their appearances. M A. 



September 18. 



