July 20, 1893J 



NATURE 



285 



only carries to an extreme characters which are present, but in 

 a much lesser deijree, and in varying gradations, in the difTerent 

 members of the Lcmiiroidea, both recent Lenmrid,c, and extinct 

 Adapida. Tn the very simple pattern of the molars, the super- 

 ior of which are of the pure tritubercular type, Megaladapis ap- 

 proaches closely to the Malagasy Lemurids Lipidolemttr &nA.%\\\\ 

 more to Chirogaleus. 



The diminutive size of the 'brain-case (comparable only with 

 what we tind amongst the Marsupialia and the Insectivora) is 

 viewed by the author, in this instance, as a degeneracy, other 

 characters being equally indicative of a retrogressive evolution 

 undergone by this Lemuroid. 



It is strongly insisted upon, generally, that "low " organisa- 

 tion in Mammalia is by no means always synonymous with 

 "primitive" organisation, and that retrogressive evolution is 

 more frequently to be met with amongst Mammalia than is 

 ^'ener.illy admitted. 



As regards the geological age of Megaladapis and its associated 

 fauna, one of whose members, the Crocodilus robustus, is still 

 living in the lakes of the interior, evidence of various kinds 

 lioes far to prove that these sub-fossil remains represent a fauna 

 which was living at a comparatively very recent period, and 

 that man himself was also contemporary with it, and in part 

 responsible for its destruction. 



The author adduced evidence in support of the proposition that 

 an older Tertiary vertebrate fauna will ere long be forthcoming 

 in Madagascar. 



"On Operators in Physical Mathematics. Part II." By 

 Oliver Heaviside, F.R.S. 



It is first shown how the ascending and descending series 

 for the first cylinder function may be algebraically harmonised. 

 If A is the ordinary ascending series in even powers, B the 

 equivalent series in odd powers, and C the equivalent descend- 

 ing series which is most useful for numerical calculation, then 

 C = \ (A-fB). This contains the explanation of a former 

 anomaly. A and C were known to the author to be equivalent 

 as operators, also algebraically and numerically equivalent 

 with positive argument. But when the argument is a pure 

 imaginary A remains real, whilst C becomes complex. A 

 becomes the original oscillating first cylinder function. C 

 contains it and the second oscillating function as well. But 

 the identity C =i (A-fB) explains it. Both sides remain 

 identical when the argument is imaginary. The second oscillat- 

 ing function is brought in by B. 



The generalised formula of which C is an extreme case is 

 then investigated. 



The extreme forms of the binomial theorem are then 

 examined. It is shown that when the index is a negative 

 integer the generalised formula becomes indeterminate, con- 

 sisting of the two extreme forms combined in any ratio. 



A more general operator and the equivalences to which it 

 le.ads are then examined. There is sometimes satisfactory 

 equivalence for numerical purposes by employing only the 

 initial convergent portion of the divergent series, but this fails 

 when the index of the operator is in the neighbourhood of a 

 negative integer. 



The general question of the meaning of equivalent and of 

 divergent series is then discussed. The difference stated by 

 Boole to exist between divergent series of the alternating and 

 of the continuous type, in that the former may, whilst the 

 latter cannot, be employed for calculation, appears to be 

 groundless. They are alike in the respect alluded to. But 

 the true meaning of numerical equivalence is unsettled, for 

 there are many cases in which formulae believed to be analytic- 

 ally and algebraically equivalent show no visible numerical 

 equivalence. 



Some generalised formulae involving the logarithm are then 

 iliscussed, and some independent verifications found. A 

 formula for Euler's constant is obtained and examined. 



One of these formulas brings in the second cylinder 

 function, which is discussed by means of an operator leading 

 to it ; also its connexion with the first solution, and of both 

 with the corresponding two oscillating functions. Various 

 analogies are pointed out, especially through an operator 

 containing two differentiators, leading to elastic wave solutions. 

 Some applications and extensions will follow in Part III. 



" On a Failure of the Law in Photography that when the 

 Products of the Intensity of the Light acting and of the Time 



NO. 1238, VOL. 48] 



of Exposure are Equal, Equal Amounts of Chemical Action 

 will be produced." By Capt. W. de W. Abney, C.B., 

 F.R.S. 



The above law has been generally assumed. In some recen 

 experiments, however, I have disctvered that this law breaks 

 down under certain conditions. Quite lately I have described 

 the method of comparing the photofiraphic value of sunlight 

 with that of candle light {Photographir. yotinm.', June, 1893), 

 which was as follows : — Abeam light would be admitted through 

 a narrow slit to sensitive bromide paper stretched round a drum 

 of about 4 inches in diameter. The drum could be caused to 

 rotate round its axis at any speed up to about sixty revolutions 

 per second, by means of an electromotor. Part of the experi- 

 ment was to place an amyl acetate lamp in position at any con- 

 venient distance from the slit exposure being given for a fixed 

 time during rotation. The slit was next replaced by a small 

 square aperture, of some J inch side, and other portions of the 

 same paper were exposed to the amyl acetate light at the same 

 distance, for varying but known exposures, with the drum at 

 rest. On development the papershowed, amongst other things, 

 a narrow band of deposit of the width of the slit caused by the 

 light from the amyl acetate lamp, and a row of squares of vary- 

 ing blackness of deposit due to the different exposures given 

 with the drum at rest. 



By comparing the blackness of the band with the scale of 

 blackness, the width of the slit would evidently be calculated, 

 supposing the usually accepted law to hold good under all cir- 

 cumstances. On making such calculations in every case the 

 calculated width of the slit was always considerably less than 

 what it was in reality, the difference being far beyond that which 

 would be caused by any error in the measurement. This led 

 to an investigation into the cause of this difference. 



The following experiment was made. The circumference of 

 the drum with the paper stretched round it was I2'25 in. The 

 width of the slit was arranged to be o'oi2 in. The amyl acetate 

 lamp was placed 2 feet from the slit, and a rotation of 30 per 

 sec. was given to the drum for one exposure, and I per sec. for 

 a second exposure. In the first case the time of exposure during 



each revolution was x — sec, or about 1/30,000 sec. 



1225 30 



The sum of the exposures during 20 min. was thus I '176 sec. 

 In the other case the exposure was 



12-25 



or about i/iooo sec, 



and the sum of the exposures was, as before, I "176 sec. Thus 

 the first individual exposures had only ^\ of the duration of 

 the second exposures, though in the aggregate they were the 

 same. 



A scale of blackness was made on the same paper, through a 

 square aperture, without shifting the lamp, the exposures being 

 i, i, 4, I, 2, 4, and 8 sec. The scale and blackness of the bands 

 were measured accurately, and the times of exposure which had 

 been given to each band, on the assumption that the law enunci- 

 ated held good, was calculated and found to be for the first band 

 o'6 sec. , and for the second liand o'gi sec, instead of I'I76 

 sec. which was really given in all. Another example is «here 

 the slit was opened to o'll in., and the time of exposure reduced 

 from 20 to 10 min. It was found that in this case the exposures 

 given on the same assumption were 37 sec. and 5'28 sec, the 

 real exposure given being 5 '36. Other experiments are quoted. 



It is to be remarked that the more sensitive a surface is to 

 radiation the less marked are the differences observable for the 

 same speeds of rotation. This is what might be expected. 



As an outcome of the experiments so far made, it seems that 

 when exposures less than l/iooo sec. are made, and the source 

 of illumination is an amyl acetate lamp (Von Altneck's) placed 

 I foot from the sensitive surface, the law fails. 



The question of very low intensities of light acting, and of 

 the sensitiveness to different spectrum colours, is now under 

 consideration. 



[jVc/^ />y the author. It may be stated that it has subse- 

 quently been proved that the law equally fails where feeble 

 intensities of light act on the sensitive surface.] 



Geological Society, June 21— Dr. II. Woodward, F.R.S., 

 Vice-President, in the chair. — The following communications 

 were read : — On composite dykes in Arran, by Prof. J. W. 



