Oct. 25, 1888] 



NATURE 



619 



The Geometric Interpretation of Monge's Differential 

 Equation to all Conies. 



Neither the note of Prof. Asutosh Mukhopadhyay in 

 Nature of the nth inst. (p. 564), nor that of Lieut. -Colonel 

 Allan Cunningham in the number of August 2 (p. 318), has 

 satisfied me that the criticism implied in my short note (June 28, 

 p. 197) on the Professor's first note (June 21, p. 173) is unfounded. 

 Permit me, therefore, to develop that criticism a little more at 

 large. 



I have not yet had an opportunity of referring to the papers 

 of the Professor in the Proceedings of the Asiatic Society, but 

 from what I can gather as to their contents from his notes in 

 Nature, I am in no way disposed to underestimate the accuracy 

 or the value of his results. It is only to his claim to find in 

 them "the true interpretation of Monge's differential equation 

 to any conic" that I demur. 



To my apprehension the interpretation in question is a truism, 

 not a truth. What has been put into the question as a defini- 

 tion emerges afterwards, as might have been anticipated, as 

 an interpretation. If the Professor has given a definition of 

 aberrancy, independent of a conic and its known proper: ies, of 

 cour.-e I am wrong ; but I gather from his note that by aberrancy 

 he merely means (if I may thus express it) deviation from 

 cdnicity. Whatever measure of aberrancy, then, he adopts for 

 curves generally, must necessarily become zero for a conic, 

 which has, from the very meaning of the words, no "deviation 

 from conicity." 



The difference, as I conceive it, between an interpretation 

 properly so called and an interpretation that is a mere truism, 

 may be clearly illustrated by the case of the circle. The Pro- 

 fessor tells us that "the differential equation of all circles 

 I V f-)r - 7>p<r — o, means that the angle of aberrancy 

 vanishes at every point of every circle." If thus read, what 

 I have said above applies, and the interpretation is but a truism. 

 It admits, however, of a different reading. For it is easy to 



show that (1 +f)r - 3^/ = (1 + f-f *J* t where s, <p are the 



as 2 

 usual intrinsic co-ordinates of the curve, so that the differential 

 equation is equivalent to d-^/ds' 1 = o. Now d<p/ds is the measure 

 of the curvature of a curve, defined as the rate of change, per 

 unit of arc, of the inclination of the tangent to a fixed direc- 

 tion, a definition which is quite independent of the circle; and 

 d-<plds- is the rate of change, per unit of arc, of the curvature. 

 Hence the equa'ion d' 2 <p/ds 2 = o, being true at every point of 

 every circle, expresses the truth that in a circle there is no 

 change of curvature from point to roint— or, in other words, the 

 property that the curvature of a circle is the same at every point. 

 I submit that this, lather than the Professor's, involving the notion 

 of aberrancy, has a right to be regarded as the true interpretation 

 of the equation. 



In like manner, the true interpretation of the differential 

 equation to a conic, if it ever is discovered, will express that 

 some magnitude or concept connected with a curve, and defined 

 independently of the particular curves, the conic sections, vanishes 

 at every point of every conic. 



Even admitting the Professor's interpretation, I agree with 

 Colonel Allan Cunningham that it has no prerogative right over 

 others of the same character to be called the interpretation of 

 the equation. To go no farther, any number of " aberrancy 

 curves" may be imagined; as, for instance, the locus of the 

 focus, instead of the centre, of the osculating conic, for which 

 it will be true that " the radius of curvature of the aberrancy 

 curve vanishes at every point of every conic " ; for in fact, in 

 this case the aberrancy curve degenerates into a single point, 

 and to say that the radius of curvature vanishes, or that the 

 curvature is infinite, at every point of a curve, is, to my appre- 

 hension, only a roundabout, and not very instructive, way of 

 saying that the curve becomes reduced to a single point. 



Harrow, October 13. R. B. H. 



A Shadow and a Halo. 

 The following notices of anthelia may be interesting to the 

 readers of Nature. Frances Kidley Havergal thus described 

 a sunset on the Faulhorn : " At one juncture a cloud stood still, 

 apparently about two hundred yards off, and we each saw our 

 own shadow gigantically reflected on it, surrounded by a com- 

 plete rainbow arch, a full circle of bright prismatic colours, a 

 transfiguration of our own shadows almost startling; each, more- 

 over, seeing only their own glorification" ("Swiss Letters and 

 Alpine Poems "). 



Tennant, in his book on Ceylon, states that this curious 

 phenomenon, whi h may probably have suggested to the early 

 painters the idea of the glory surrounding the heads of beatified 

 saints, is to be seen in singular beauty at early morning in 

 Ceylon. When the light is intense, and the shadows propor- 

 tionally dark, when the sun is near the horizon, and the 

 shadow of a person is thrown on the dewy grass, each drop of 

 dew furnishes a double reflection from its convex and concave 

 surfaces ; and to the spectator the shadow of his own figure, but 

 more particularly the head, appears surrounded by a halo as vivid 

 as if radiated from diamonds. 



•S. T. Coleridge described the phenomenon thus : — 



"Such thou art, as when 

 The woodman winding westward up the glen 

 At wintry dawn, where o'er the sheep track's maze 

 The viewless snow-mist weaves a glisi'ning haze, 

 Sees full before him, gliding without tread, 

 An image with a glory round its head : 

 The enamoured rustic worships its fair hues, 

 Nor knows he makes the shadow he pursues." 

 Benvenuto Cellini saw, probably, this phenomenon, and sup- 

 posed it peculiar to himself. F. Robertson ci'es it as a proof of 

 inordinate vanity. Hesajs: "Conceive a man gravely telling 

 you that a vision of glory encircled his head through life, visible 

 on his shadow, especially on the dewy grass at morning, and 

 which he possessed the power of showing to a chosen few " 

 (" Life and Letters of F. Robertson," vol. ii. p. 192). 



Bardsea, October 22. Edward Gcoghegan. 



I have frequently, on the South Downs, seen a halo round 

 the shadow of my head, as described in your last number by Mr. 

 A. S. Eve. I have noticed that the further off the shadow, the 

 brighter is the halo. I have also observed, wher. looking at my 

 shadow in the sea, that rays of light appear to surround the 

 shadow of my head. Charles Cave. 



Ditcham Park, Petersfield, October 22. 



On the Grass Minimum Thermometer. 



The average readings of the self-recording grass minimum 

 thermometer for every month during the past three years have 

 been compared with the average minimum damp bulb tempera- 

 tures, obtained from the means of hourly readings, and the 

 following figures show the corrections to be applied to the latter 

 in order to obtain the former : — January -o 0- 3, February +o 0- 3, 

 March -o° - 3, April -o°8, May -o°'2, June - f'l, July - i°r, 

 August — o°*9, September +o°'2, October +i 0, 4, November 

 -+- l c, 9, December + o°'4. 



The grass minimum is nearly a degree below the damp 

 bulb minimum in the wet season, and nearly 2° above it in 

 the driest month. The comparison between the minimum air 

 temperature and the minimum on grass does not measure the 

 terrestrial radiation, although the difference is to some extent 

 influenced by radiation. Moreover, the epochs of the two 

 minima need not coincide — e.g. in Hong Kong the early morning 

 hours are more cloudy than the evening hours. 



During the daytime in summer the thermometer, exposed an 

 inch above the short grass, shows as a rule temperatures rising 

 to 120 or 130 , especially in calm weather ; but even when it is 

 not perfectly calm, the force of the wind is not felt so near the 

 ground, from which the air rises laden with minute particles of 

 dust, which are observed adhering to the cloth of damp bulbs 

 and other objects cooled by evaporation, and which may occa- 

 sionally be smelt in the air. At night such minute particles 

 would of course tend to return to the ground, and the unhealthy 

 character of the ground-fog during early morning hours in 

 tropical countries may be intensified by this circumstance. 



Hong Kong Observatory, W. Doberck. 



September 10. 



ON THE ELECTROMOTIVE VARIATIONS 



WHICH ACCOMPANY THE BEAT OF THE 



HUMAN HEART. 



THE observation of these variations is extremely easy, 

 the only requisite being a sufficiently sensitive 

 capillary electrometer. 1 



' The electrometers I used were made by Mr. Dean, glass-blower, 8 Cross 

 Street, Haf.on Garden. 



