June 21, 1888] 



NATURE 



173 



experiments before the year 185 1 with different coals suitable for 

 the Navy. These trials were conducted near London, under a 

 small marine boiler at atmospheric pressure. 



(3) At the English Government dockyards, various interesting 

 experiments have been made under small marine boilers, and the 

 results published in Blue-books. 



(4) Messrs. Armstrong, Longridge, and Richardson published 

 in 1858 an account of some valuable experiments they had made 

 with the steam-coals of the Hartley district of Northumberland, 

 under a small marine boiler, for the Local Steam Colliery 

 Association. 



(5) At Wigan many excellent experiments were made by 

 Messrs. Richardson and Fletcher about 1867, to test the value of 

 Lancashire and Cheshire steam-coals for use in marine boilers. 

 The water was evaporated under atmospheric pressure from a 

 small marine boiler. This station was afterwards abolished. 



In none of the above do the gases of combustion appear to 

 have been analyzed. 



(6) A fuel-testing station was worked at Dantzig in 1863. 



(7) An important station was opened at Brieg, on the Oder, 

 by the colliery-owners of Lower Silesia in April 1878, with the 

 primary object of testing the value as fuel of the important coal- 

 seams of that province. After working with the most satisfactory 

 results for two years, and establishing the superiority of the Lower 

 Silesian coal, the experiments terminated in 1880. The testing 

 boilers had each 40 square metres of heating surface. Gases 

 and coals were analyzed. 



Existing Continental Stations. — (8) The Imperial Naval 

 Administration Coal-testing Station at Wilhelmshaven, Germany, 

 was established in 1877. 



(9) Dr. Bunte's coal-testing station, erected at Munich about 1878, 

 particulars of which have been published in the Proceedings of the 

 Institution of Civil Engineers, vol. lxxiii. Here some hundreds 

 of trials have been reported on and published ; much valuable 

 work has been done, and many fuels tested, including coals of the 

 Ruhr valley, Saar basin, Saxon and Bohemian coal-fields, and 

 those of Silesia and Upper Bavaria. The boiler of the station has 

 about 450 square feet of heating surface. The gases and coals 

 are analyzed, and all particulars carefully noted. It is one of 

 the most complete stations I have seen. 



(10) In Belgium, near Brussels, there is a Government station 

 for testing fuels, under the administration of the Belgium State 

 railways ; locomotive boilers are used. The establishment has 

 been at work for the last two years, but no results are published, 

 as they are considered the property of the Government. Private 

 firms can, however, have their coals tested and reported upon. 



(11) The Imperial Marine Station, Uantzig. 



(12) Boiler Insurance Company at Magdeburg. 



The above is a slight outline of the work already done in this 

 direction. 



With the view of obtaining the opinions of those interested in 

 starting a fuel-testing station, I ask you kindly to give this letter 

 publicity. If the necessary sum can be raised, we may hope to 

 have before long a practical and useful establishment in London, 

 and to gain from it many interesting practical results respecting 

 the combustion of fuels. Bryan Donkin, Jun. 



Bermondsey, S.E., June n. 



The Geometric Interpretation of Monge's Differential 

 Equation to all Conies — the Sought Found. 



The question of the true geometric interpretation of the 

 Mongian equation has been often considered by mathematicians. 

 In the first place, we have the late Dr. Boole's statement that 

 "here our powers of geometrical interpretation fail, and results 

 such as this can scarcely be otherwise useful than as a registry of 

 integrable forms " ("Diflf. Equ.," pp. 19-20). We have next 

 two attempts to interpret the equation geometrically. The first 

 of these propositions, by Lieut-Colonel Cunningham, is that 

 "the eccentricity of the osculating conic of a given conic is 

 constant all round the latter" {Quarterly Journal, vol. xiv. 

 229) ; the second, by Prof. Sylvester, is that "the differential 

 equation of a conic is satisfied at the sextactic points of any 

 curve" (Atner. Journ. Math., vol. ix. p. 19). I have elsewhere 

 considered both these interpretations in detail, and I have 

 pointed out that both of them are irrelevant ; the first of them 

 is, in fact, the geometric interpretation, not of the Mongian 

 equation, but of one of its five first integrals .which I have 

 actually calculated (Proc. Asiatic Soc. Bengal, 1888, pp. 74- 

 86) ; the second is out of mark as failing to furnish such a 



property of the conic as would lead to a geometrical quantity 

 which vanishes at every point of every conic (Journal Asiatic 

 Soc. Bengal, 1887, Part 2, p. 143). In this note I will briefly 

 mention the true geometric interpretation which I have recently 

 discovered. 



Consider the osculating conic at any point, P, of a given 

 curve ; the centre, O, of the conic is the centre of aberrancy at 

 P, and as P travels along the given curve, the locus of O will 

 be another curve, which we may conveniently call the aberrancy 

 curve. Take rectangular axes through any origin ; let (x, y) be 

 the given point P, and a, £ the co-ordinates of the centre of 

 aberrancy. Then it can be shown without much difficulty that 



a = x - Zqr 



W - y 



fi=y - 37(/ r - 3 ? 2 ) 



whence 



where 



da. 



ll.X 



= \T, 



3qs - 5r- 

 d$ 



dx 



= h T, 



KW ~ 5"*)* (3'/* - Sr ! f 



T = 9ft - i&qrs + 40;- 3 , 

 p, q, r, s, t being, as usual, the successive differential coefficients 

 of y with respect to x. 



If dip be the angle between two consecutive axes of aberrancy, 

 ds the element of arc, and p the radius of curvature of the 

 aberrancy curve, we have 



p = *L t ds* = </a 2 + d$ 2 , 

 dty 

 whence 



, dx 



p = <x» + m 2 )* . T . % 



But it is easy to show that 



fty = y(3^ - S r - ) 



dx r 1 + (rp - 3<7 2 ) 2 ' 

 so that 



f = T t { r 2 + (rp - 3? 2 ) 2 } 1 , 



q(w - 5^) 3 



Now, T = o is Monge's differential equation to all conies, 

 and when T = o we have p = o. Hence, clearly, the true 

 geometric interpretation of the Mongian equation is : 



The radius of curvature of the aberrancy curve vanishes at 

 every point of every conic. x 



This geometrical interpretation will be found to satisfy all the 

 tests which every true geometrical interpretation ought to satisfy, 

 and I believe that this is the interpretation which, during the 

 last thirty years, has been sought for by mathematicians, ever 

 since Dr. Boole wrote his now famous lines. I will not take 

 up the valuable space of these columns with the details of calcu- 

 lation : they will be found fully set forth in two of my papers 

 which will be read next month before the Asiatic Society of 

 Bengal, and will in due course be published in the Journal. 



Calcutta, May 18. Asutosh Mukhopadhyay. 



PERSONAL IDENTIFICA TION AND 



DESCRIPTION? 



I. 



T is strange that we should not have acquired more 



power of describing form and personal features 



than we actually possess. For my own part I have 



I 



1 The differential equation of all parabolas, 

 2,qs - 5^ = O, 

 is also easily interpreted, viz. calling the distance OP between the given 

 point and the centre of aberrancy the radius of aberrancy, and the 

 reciprocal of this (= I) the index of aberrancy, we have, easily, 



39 s - 5** 



I = 



iq {r 2 + (rp - 3? 8 ) 8 } * 



so that the interpretation is that the index of aberrancy vanishes at every 

 point of every parabola. . 



2 'the substance of a Lecture given by Francis Galton, F R.S., at the Royal 

 Institution on Friday evening, May 25, 1888. j 



