464 



NATURE 



{March 15, 1888 



or not. Prof. Capellini, of whom I asked the question at the 

 meeting of the British Association at Manchester, could not 

 answer me. Frimd facie we should certainly expect the Italian 

 evidence to support the French, but this is by no means the 

 conclusion to be drawn from text-books, in which it is generally 

 taken for granted that in Italy the elephant and mastodon have 

 been found at the same horizon. 



The question is one of very great interest and importance, 

 and an answer to it would be especially valuable to me. Perhaps 

 some of your readers may have the means of answering it. 



Henry H. Howorth. 



21 Earl's Court Square, February 28. 



True Average of Observations ? 



I HAVE long been dissatisfied with the method of taking the 

 arithmetic mean as the most probable value of a comparatively 

 few direct observations of a quantity. This is certainly the 

 legitimate result of the theory of probability, or "method of 

 least squares," when one knows nothing to guide one in giving 

 more weight to one than to another observation. 



But without knowing anything of the conditions under which 

 the observations were made, or, otherwise, no choice among them 

 being possible by considering these conditions, still, when one 

 comes to compare the results among themselves, this comparison 

 seems to me to afford means of judging between them. Thus, 

 if all the results are plotted on sectional paper, they are found to 

 be grouped closely together at one place and to be scattered 

 wide apart at others. Now the most probable result (whatever 

 be the right method of finding it) lies certainly somewhere about 

 the place of close grouping ; and it seems fair to consider those 

 results that come near this place as the liette?- ones, and to allow 

 to them more weight than to the others in calculating the mean. 



If the observations were extremely numerous, there can be no 

 objection to taking the arithmetic mean as the true probable 

 value. But one has usually to content one's self with a few only, 

 and in order to get a better approximation in this case I have 

 constructed the following formula. I would be glad if some of 

 your correspondents will express their opinions as to its legiti- 

 macy. In a case of this kind one ought not to trust entirely to 

 one's own judgment ; one should submit one's own judgment to 

 be checked by that of several others. 



The method I propose is as follows. 



First fix upper and lower limits outside which the true value 

 cannot possibly lie, and reject absolutely all measurements 

 outside these limits. The result will not be appreciably affected 

 by taking these limits a little higher or lower, and it is better to 

 err in taking them too wide apart than vice versa. One usually 

 has, or ought to have, a general notion of the quantity sought 

 for, sufficient to determine these limits ; but if this be not so, 

 they may be determined by adding to and substracting from the 

 arithmetic mean what is thought to be the maximum possible 

 error. 



Let x^, X2, x^, &c., be the excesses of the various measurements 

 above the lower of the above possible limits. Let x^ be the 

 excess above the same limit of the as yet unknown most probable 

 value as determined by the formula below. 



_ ^ \2 1 



a)td take as 



Attach to each x the weight \ i - 



Xf) the mean of the x's with these weights attached. 



Note that equal weights are given to measurements equally 

 above and below x^. Also to an x coinciding with the lower 

 possible limit, a weight zero is given. Zero weight is also 

 given to an x as much above Xq as the lower possible limit is 

 below it. 



The rule results in the following formula : — 



Weight ion X — 1 - 



X X weight = ^^^ ^ ^ . 



Xq 



Therefore, the mean equals — 



^ ^ 2XoS.y- - S-r* 



This is a quadratic for x^, the solution of which is — 



° 4 2;f I \ 9 {-ZxY) 



Of course the labour of finding this mean is greater than that 

 of finding the arithmetic mean ; it involves summing the first. 



second, and third powers. But the method is only intended to 

 be used when the number of values to be dealt with is not 

 large, and with the help of a table of squares, cubes, and square 

 roots, the work is not really very laborious. 



It is easy to prove that this result is identical with the arith- 

 metic mean in the following three cases : (i) all the ;r's equal ; 



(2) the x's all equidistant, i.e. forming an arithmetic progression ; 



(3) the x's infinitely numerous. 



The practical meaning of the rule may perhaps be made 

 clearer by the annexed table, giving the weights attachable to 

 various values of x where x,, is taken equal to unity. 



The following is a numerical example 



2x = 5-42 



2x" — 6794 



5-178 

 •705 

 •074 



17715 

 I -6016 



2X3 = 9-330 



/, 8 2x2x^ , , 



, / I -. — — - = -162 and Xft = I '0925. 



The arithmetic mean or — = i 

 5 

 Mason College, February 4. 



Robert H. Smith. 



Crepuscular Rays in China. 



Immediately after sunset enormous rays of light are fre- 

 quently seen spreading from the part of the horizon where the 

 sun has disappeared, and also — though somewhat fainter — from 

 the opposite part of the horizon. Sometimes the rays stiet ch 

 right across the sky, and when strongly developed they appear 

 first in the east, and then in the west, and resemble auroral rays, 

 glowing in a yellow or red colour, while the sky between the 

 rays is deep blue or greenish. They appear to be caused by 

 invisible cirro-stratus clouds high up in the air. This pheno- 

 menon is never seen in England, or at any rate it is by no means 

 so conspicuous as here. Ancient Greek mariners may have had 

 their imagination impressed by a similar phenomenon, po5o- 

 Sct/cTuAos T]<ijs being so frequently mentioned in Homer. 



Crepuscular rays at sunrise or sunset are seen at all seasons in 

 Southern China, but they are most frequent at the height of the 

 typhoon season, and most intense just before typhoons, which 

 latter are indicated beforehand by crepuscular rays as well as by 

 halos. 



The following table exhibits the number of evenings when 

 sti'ong crepuscular rays were registered in each month of the past 

 three years, and also the mean monthly frequency of the strongly 

 developed phenomenon : — 



May. June. July. Aug. Sept. Oct. Nov. Dec. 



1885 — — 32 43 — — 



1886 — III 37 — I 



1887 I — — 2 3 — - — 

 Mean 



0-3 



0-3 13 



17 



3-3 33 00 o- 



W. DOBERCK. 



Hong Kong Observatory, December 31, 1887. 



"An Unusual Rainbow." 



I READ with interest a letter with the above heading in 

 Nature (vol. xxxvi. p. 581) from Mr. S. A. Hill of Allahabad, 

 India, of date September 18, 18S7. He describes a brilliant rain- 

 bow which he saw after the sun had set, and states that such a phe- 

 nomenon " must be of rare occurrence," and that he had " never 

 before seen anything similar, nor read anywhere a description of 

 a rainbow after sunset." I had not read his letter when, on the 



