October 12, 191 1] 



NATURE 



485 



a crow-shrike [fiarita destructor) pecked it only once, the 

 latter vigorously shaking his head and wiping his beak 

 after the taste. A Cuban mocking-bird (Mimus orpheus) 

 and a Brazilian hangnest (Ostinops viridis) attempted it, 

 but after a few pecks gave it up. Finally, the mangled 

 remains were eaten with much hesitation by a rufous 

 tinamou (Rhynchotus rujescens). Whether the latter would 

 have eaten it, if given the first refusal, it is, of course, 

 impossible to say : but there is no doubt that the other 

 birds found the butterfly highly distasteful. I was par- 

 ticularly impressed by its rejection by the two bustards, 

 which on previous occasions have eaten some of the most 

 unpalatable of British insects (see Proc. Zool. Soc, 191 1, 

 pp. .809-68). 



The birds used for these experiments belong to tropical 

 American, Asiatic, Australian, and African species, and 

 were purposely selected from a variety of families. Anosia 

 plexippus lias, I understand, comparatively recently invaded 

 the Old World from the New ; and the result of the above- 

 recorded experiments suggests that no serious barrier to its 

 dispersal will be offered by insectivorous birds. If it 

 Sue© eds in widely distributing itself it may, as a useful 

 model, bring about marked mimetic changes in the Lepido- 

 ptera of the districts in which it settles. 



The Zoological Society. R. I. Pocock. 



The Arithmetic of Hyperbolic Functions. 



Throughout the books treating of hyperbolic functions, 

 although elaborate series for their determination are given, 

 the possibility of calculating them directly from their 

 definitions, by means of common logarithms, is never sug- 

 gested, and it would appear, therefore, that the merits of 

 the direct method are insufficientlv recognised. 



If the hyperbolic functions of a quantity U are required, 

 it is convenient, for purposes of writing and printing, to 

 get rid of the exponential and to write A = e . Then 

 log, „ A = 0-43429448 L", and A is thus found at once from a 

 book of common logarithms. The functions can then be 

 calculated by a slide-rule, or by logarithms, in the simple 

 form 



cosh U = §(A-|-i/A) 

 sinhU = MA-i/A); 

 and similarly for tanh U, coth U, sech U, cosech U, and 

 versh U — all in terms of A. 



For example, calculate cosh 2 and sinh 2. Here 

 log, „ A = 04342944S X 2 = 0-86858896 ; 

 A is therefore 7-389060, and i/A is 0-135335. Hence 

 cosh 2 = 3-76220 and sinh 2 =3-62686 



In the more general case the functions of a complex 

 quantity (V + id) are required, and they have consequently 

 to be expanded in terms of cosh U, sinh U, cos 8, and sin 8. 

 So far as cosh U and sinh U are concerned, the direct 

 method by common logarithms is still available, and the 

 result is best dealt with in the form, for example, 



sinh (U + *'9) = i;(A- i/A) cos 8 + «(A + i/A) sin 8\. 

 The only real difficulty then left for the student is in 

 ensuring that cos 8 and sin 8 are given their proper signs. 

 That is to say, he must be clear regarding how many 

 quadrants are contained in 8, how many degrees there are 

 in 8 bevond that number of quadrants, and the proper sign 

 of cos 8 and sin fl, respectively, in each quadrant. 



In general, as is well known, if the functions are not 

 required to a greater degree of accuracy than 1 in 10,000, 

 it is permissible for all real values of U greater than 5 to 

 write 



cosh U = sinh U = £A ; 

 and the direct method has obvious advantages. For values 

 of U less than 8, Ligowski's excellent tables give sinh U 

 and cosh U, proceeding by increments of 001 of U ; but for 

 practical purposes these 001 steps are too great, and 

 " difference " columns have to be used. Consequently, to 

 find the functions for values of U less than 5, where U is 

 given to three or more places of decimals, it will usually 

 be as quick and as accurate to adopt the direct method as 

 to worry through the irksome arithmetic involved in 

 estimating " differences." 



In cable problems it is, as a rule, desirable to retain at 

 least four significant figures for U. 



October 3. Rollo AprLEYARD. 



XO. 2189, VOL. 87] 



Hot Days in 1911. 



By a "hot day" will here be meant one with 70 or 

 more. There are about seventy-seven of these at Green- 

 wich, on an average, in the year. I propose to show how 

 a method of forecasting recently described would apply to 

 those days in 191 1. 



The series of annual numbers (1841— 19 10) is first 

 smoothed with sums of five ; then we compare, in a dot- 

 diagram, each sum with the difference between it and the 

 fifth after. 



The last comparison before this summer was that 

 between 324 (for 1903) and its difference with 335 (for 



§<yt> '"io 'A-o '6 'flq tfCna'fUi 'ifQ 'Co 'fro- 



Comparison of hot day numbers, Greenwich. 



1908), i.e. +11. Next we come to 343 (for 1904) and the 

 position of the new dot (for 1909, representing the sum of 

 1907-11). 



Placing an arrow-head at 343 in the horizontal scale, we 

 might fairly expect the new dot to be above the zero line. 

 Suppose, however, to be on the safe side, we say not below 

 — 10. Then 343 — 10 = 333. Now the four years 1907-10 

 yield 243, and 333 — 243 = 90. So that we might say the 

 year 19 11 was likely to have at least 90 hot days. 



The actual number is 101. 



The method may be commended, perhaps, for application 

 to various weather items. Alex. B. MacDowall. 



Frequency of Lightning Flashes. 



A letter on rainless thunderstorms in Nature of 

 August 31 leads me to ask if any accurate counts have 

 ever been taken of the frequency of lightning flashes. 



Watching a severe storm from my bungalow about a 

 year ago, I made an attempt to separate and count the 

 flashes — to the unassisted eye the lightning was as con- 

 tinuous as a flickering arc-lamp. 



The only thing I could find to help me was a gramo- 

 phone ; I took its top works off, and on the horizontal disc 

 I put one radial white chalk line. The speed of the disc 

 was adjusted, by trial, exactly to 100 revolutions per 

 minute, and the instrument was placed where the storm- 

 light fell directly on it. 



The appearance of the revolving disc was as if 

 irregularly spaced phosphorescent spokes were being shown 

 instantaneously in sections of various sizes in continually 

 changing positions. It was difficult to estimate the number 

 of separate streaks in one revolution, but I finally settled 

 on eight as a fair average during the whole storm — suffici- 

 ently exact to show the order of figures being dealt with. 



This works out at 800 flashes per minute, or, say, 50,000 

 an hour. H. O. Weller. 



Jamalpore (Dist. Mymensingh), E. Bengal, 

 September 19. 



