4o6 



NA TURE 



IFeb. 28, 1884 



Proportionally they suffer from the ravages of the birds equally ; 

 the percentage of losses is the same ; they are on equal terms. 

 No matter how long they continue the association, neither gains 

 nor loses on the other ; though through one being more 

 numerous it loses more individuals, yet equally in proportion 

 with the other. So that, if one is twice as numerous as the other 

 at the time of assimilation, it must always — other conditions 

 being equal — remain twice as numerous. 



We now give the mathematical reduction : — 



Designation of species A IJ 



(1) Original number a > b 



(2) No. lost without imitation , 



(3) Remains without imitation . 



(a - e) 



(4) No. lost with imitati( 



(b-c) 

 _b_ 



a + h' I a + b' 



(5) Remains with imitation a ( i - -^ — V/'li - — ^ I 



V a + bl \ a + bf 



(6) Excess of remains due to 



imitation, or absolute ad- 

 vantage (z)-(S) -^ 



(S) Ratio of excess to remains 

 without imitation (6) : (3), 



or proportional advantage 



(9) Ratio of proportional ad- 

 vantage of B to propor- 



tional advantage of A 



a + b 



a (a - e) 



From (8) we see that, if e < b < a, there is a proportional 

 advantage to both, the mimicry "is twice blessed," but the pro- 

 portional advantage to B is greater. If e is zero, there is no 

 advantage to either. It e = b < a, the prop, advantage to B is 

 infinite, while that to A is still finite ; this is as it ought to be, 

 seeing that to 1! it is a case of "to be or not to be," of existence 

 with mimicry or extinction without. And in this extreme case 

 it must l5e evident to every one that the ratio of a''' : b°, both 

 terms finite, caimot be the ratio of the infinite advantage of B 

 to the finite advantage of A. The greater e the greater are both 

 advantages. 



From (9) we .see that, if e is small compared to b and a, the 

 ratio is nearly a": b- (Midler's law), but the larger e is the 

 further it deviates from that law, the ratio becoming rapidly 

 greater than a- : b", and approaching infinity as f approaches b. 



To conclude, we may point out that MuUer's law, as given in 

 his own words and quoted above, is incompletely enunciated, 

 and but for the numerical examples, it might lead any one astray 

 as to what the law is. It ought to have the ratio of interpolated 

 between "and" and "ihe proportional" ; then " advantage" 

 and " sijuare" ought both to be plural ; "relative" ought to be 

 respective ; and, lastly, the fact that the ratio is inverse should 

 be explicitly stated. 



Finally we enunciate our law. Let there be two species of 

 insects equally distasteful to young birds, and let it be supposed 

 that the birds would destroy the same number of individuals of 

 each before they were educated to avoid them. Then if thesi 

 insects are thoroughly mixed and become undistinguishable to 

 the birds, a proportionate advantage accrues to each over it-- 

 former state of existence. 'Vhfie proportional e advantages are in- 

 versely in the duphcate ra'.ioof their respective original nuQ3,beis 

 compounded with the ratio of the respective percentages that 

 would have survived without the mimicry. 



This last "ratio compounded" corrects Midler's law, but we 

 still think with Mr. Wallace that the law, even when corrected, 

 has not much bearing on the question that the individual 

 aljsolute advantages (6) above, together with the (jrobable value 

 of ^ and the ratio <; : b indicated by relative frequency of capture, 

 solve the whole question. In our first paper above mentioned 

 we established formula; for calculating these last-named items, 

 although in a different manner from and quite independent of 

 MitUer's law, which we iiad not then seen. 



Tokio, Japan, Novemlier, 1883 



Thomas Blakiston 

 Thomas Alexandei; 



Christian Conrad Sprengel 



I became acquainted with Christian Conrad Sprengel's 

 work, " Das entdeckte Geheimniss der Natur im Bau und 

 ill der Befruchtung der Blumeu" (Berlin, 1793) in 1850 at 

 the University of Berlin through Brof. C. II. Schultz-Schul- 

 zenstein, who brought it forward in one of his lectures on botany, 

 praiving Sprengel's good observations and illustrations, but 

 making his teleological views appear so irksome as to dispose 

 his hearers rather to depreciate and reject the book than be 

 attracted to it by respect. The value of Sprengel's treatise in its 

 bearing on the theory of selection was first recognised by Charles 

 Darwin, whose writings recalled the remarkable book to my 

 mind, and induced me to buy it, which I did at a very cheap 

 rate at an old book -shop. K. MoBlus 



Kiel, February 18 



Circular Cloud Bows 



I fancy that the phenomenon described by Mr. Fleming in 

 your issue of January 31 (p. 310) is not a very uncommon one. 

 It has twice fallen to my lot, when in Switzerland, to be a wit- 

 ness of these spectral shadows. 



On the first occasion I was with a party of three on the 

 mountains to the north-east of Montreux, almost opposite the 

 Cape de Moine. It was midwinter, and the day was very cloudy, 

 even in the valleys, while the high ground on which we stood, 

 and all the surrounding peaks, were completely swathed in mist. 

 Suddenly, and under the impulse apparently of a blast of wind 

 from below, the mists around us were almost entirely dissipated, 

 and a few sickly gleams of sunshine filtered through the fog. At 

 t' at moment we saw gigantic images of ourselves ijrojected on to 

 the wall of vapour enshrouding the Cape de Moine, immediately 

 opposite the point where the sunbeams had permeated. The 

 effect was very transitory, and, so far as I remember, there were 

 no prismatic colours. 



The circumstances under which I saw the second appearance 

 were as follows : — 



In August last I was standing, just before sunset, on the sum- 

 mit of the Niesen, in company with a friend. The day had 

 been very hot, and we were just remarking on the extraordinary 

 difference in temperature between our elevated position there 

 and our situation a few hours before on the Lake of Thun, when 

 we saw some scattered wisps of cloud rising out of the depths 

 below. These increased rapidly, both in size and number, 

 uniting as they rose, till the uhole abyss presented the appear- 

 ance of a seething cauldron, from which was escaping a dense 

 cloud of steam. The prospect towards the east was quickly 

 blotted out, while the sky in the opposite quarter remained as 

 clear as before. 



We then saw dim and fragmeirlary signs of prismatic colours 

 in the curtain of cloud, and these became more defined and vivid 

 as the thickness of the cloud increased. Finally there appeared 

 a very distinct circle of rainbow hues, with our own figures 

 looming, weird and awful, in its centre. Both images were 

 visible to myself and my companion, though each could see the 

 other's reflection more distinctly than his own. 



Mr. Whymper, iu his "Ascent of the Matterhorn," mentions 

 an instance in which the prismatic colours assumed the shape of 

 crosses. This effect, occurring as it did soon after the fatal acci- 

 dent v\'hich marked the conquest of the m luntain, filled the 

 minds of the guides with superstitious horror. From my own 

 experience on the Niesen I can well imagine that, as Mr. 

 Whymper suggests, this form could be accounted for by the sup- 

 position that there were several circles interlaced, and that only 

 segments of them were visible from the point at which he and 

 his companions stood. 



Perhaps some of your readers may be able to explain the 

 exact atmospheric conditions under which these appearances 

 become possible. E. H. L. FiRMSTONE 



B^wdley, Febiuary 21 



On the Absence of Earthworms from the Prairies of the 

 Canadian North-West 



In Nature of Jan. 3 (p. 213) Mr. Robert M. Christy writes 

 on the absence of earthworms from the prairies of the North- 

 West. I can confirm his statements, and extend them to cover 

 the prairies of Kansas, the Indian Territory, Idaho, and Wash- 

 ington Territory. In all the above-mentioned territory of the 

 United States the soil is more or less alkaline, and it seems to 



