August 2, 1888] 



NATURE 



3*9 



Three examples have been given by him, all very neat. Writing 

 for shortness the differential equations thus — 



Circle, R = o ; Parabola, S = o ; Conic, T = o, 



he has proved (in Journ. As. Soc. Bengal, vol. Ivi. p. 144, and 

 Nature, vol. xxxviii. p. 173) that in general in any curve 

 whatever, 



(1) Tan. z of aberrancy = q x . R; 



(2) Index of aberrancy = q t . S ; 



(3) Radius of curvature of aberrancy curve = t/ 3 . T ; 



where q x , q. 2 , q 3 are certain functions in general finite. Hence 

 the geometric meaning of the differential equations of the three 

 curves is at once 



(1) Circle. — Angle of aberrancy = o \ right round 



(2) Parabola. — Index of aberrancy = o r all curves 



(3) Conic. — Radius of curvature of aber- I of each 



rancy curve = o ; family. 



The verbal neatness of these interpretations can hardly be 

 excelled. 



A writer (R. B. II.) in Nature, vol. xxxviii. p. 197, objects 

 to the last that it really only means that a conic is a conic (be- 

 cause its aberrancy curve shrinks into the centre) ! Now, this 

 is precisely what was to be expected : the differential equation 

 of a curve expresses exactly that the curve of some family which 

 osculates it in the highest degree is the curve itself. Rut the 

 new interpretation puts this in a neat form, viz. in assigning a 

 meaning to the magnitude F, which differs from zero in general, 

 and whose vanishing at all points of every curve of a certain 

 family (say conic) indicates a property of high generality of 

 those curves. 



But the Professor makes, what I conceive to be, the mis- 

 taken claim (Proc. As. Soc. Bengal, 1888, p. 75, et sea.), that this 

 mode of interpretation is the only true one ; and further that, 

 accepting this mode of interpretation, only one meaning can be 

 attached to it (p. 76, 1. 29, op. cit.). 



Now it must be observed that the equal ion F = o implies 

 directly, not only that some one geometric magnitude F vanishes, 

 but abo that every geometric magnitude vanishing with F (such 

 as «F, a¥"\ sinF, cvc.) vanishes right round every curve of the 

 family. All of these are equally good geometric interpretations 

 of the same kind as proposed. 



But the equation F = o also implies, more or less directly, 

 countless theorems of position, osculation, &c. All of these 

 may be fairly considered geometric meanings of that equation. 

 Thus, attending to the meaning of "aberrancy," the results 

 quoted involve directly — 



(1) Circle. — Normal coincides with diameters. 



(2) Parabola. — Diameters are axes of aberrancy, and meet at 

 infinity. 



(3) Conic. — Diameters are axes of aberrancy, and are con- 

 current (in the centre). 



Surely these are also true geometric interpretations. 



Lastly, let the equation F =0 be multiplied by any of its 



integrating factors fi, and write for shortness / fiFdx = <p. It 



follows that <p = constant. Hence, since the number of in- 

 tegrating factors is infinite, another (indirect) geometric inter- 

 pretation arises, viz. that all the geometric magnitudes <j> are 

 constant right round every curve of the family. 



These latter general modes of interpretation, viz. theorems of 

 position, o?culation, and of first integrals (<p = c), I had given 

 eleven years ago (in Quart. Journ, Math., vol. xiv. p. 226). 



To the last of these the Professor has objected (p. 76 of his 

 paper quoted), that it is not an interpretation of the equation 

 F = o at all, but only of its fir.-t integrals <p = c. This is, of 

 course, admitted. But it is worth noting that the connection 

 between the two, F = o, <p = c, is so very close, that many 

 will accept an interpretation of the latter as a fair (indirect) 

 interpretation of the former also. 



In fact, since F = o is equivalent to D.^ = o, the former is 

 now seen to mean directly that there is no variation of any of 

 the magnitudes <p right round every curve of the family ; and 

 this is a strict direct interpretation of the equation F = o itself. 

 But many will probably prefer the shorter phrase <f> = constant, 

 even though it interprets F = o only indirectly. 



There is, moreover, a slight disadvantage in the former mode 

 of interpretation, viz. that the meaning of the magnitude F 

 must necessarily be sought in curves other than, and usually 

 more complex than, the curves denoted by F = o ; whereas the 



interpretation of <p = c only requires the finding a meaning for 

 (p, which is explained in my paper quoted to be any fundamental 

 geometric magnitude of the curve itself. 



Allan Cunningham, Lt.-Col., K.E. 



British Earthworms. 



The occurrence of any new animal in England is a point of 

 some interest, however humble that animal may be ; and, in order 

 to work out the species of British earthworms, I sent a letter to 

 the Field some time back, requesting readers of that journal to 

 forward me specimens. In reply I received a large number of 

 worms from various people, amongst them being Mr. F. O. 

 Pickard Cambridge, of Hyde, who his very kindly sent me 

 several parcels of worms. One of these parcels contained some 

 very fine gravel taken from the bed of a stream, together with a 

 number of small worms about l| to 2 inches in length. These 

 turned out to be a species of Allurus, a genus formed by Eisen 

 for a worm in which the male pores are on the thirteenth segment 

 instead of on the fifteenth, as in the other genera of the family Lum- 

 bricidae. Only one species is at present known, viz. A. tetra'e trus ; 

 it is of a beautiful sienna colour, with a dull orange clitellum. 



I wish to record, for the first time, its occurrence in England, 

 and also to draw attention to the fact that it lives below water, 

 at any rate for some part of the year. Mr. Cambridge has been 

 most obliging in giving me the facts as to the place in which he 

 found the worms : they occur in the gravelly bed of a stream 

 which at certain times of the year runs down, so low as to leave 

 small gravelly islands 2 or 3 inches high. In these islands he 

 found Allurus ; but he finds none in the banks of the stream. 

 We already know of Criodrilus as being a thoroughly aquatic 

 earthworm, living in the muddy beds of rivers and lakes ; and 

 although this worm has not yet been recorded in Great Britain, 

 I see no reason to doubt that it exists here. 



I should add that Mr. Beddard has informed me that he re- 

 ceived a specimen of Allurus from Lea, Kent, some time after 

 I received these from Hyde. It has been recorded also from 

 Sweden, Italy, and Tenerife. Wm, B. Benham. 



University College. 



THE SUN MOTOR. 



INDIA, South America, and other countries interested 

 in the employment of sun power for mechanical 

 purposes, have watched with great attention the result of 

 recent experiments in France, conducted by M. Tellier, 

 whose plan of actuating motive engines by the direct 

 application of solar heat has been supposed to be more 

 advantageous than the plan adopted by the writer of 

 increasing the intensity of the solar rays by a series of 

 reflecting mirrors. The published statements that " the 

 heat-absorbing surface" of the French apparatus presents 

 an area of 215 square feet to the action of the sun's rays, 

 and that " the work done has been only 43,360 foot- 

 pounds per hour," funvsh data proving that Tellier's 

 invention possesses no practical value. 



The results of protracted experiments with my sun 

 motors, provided with reflecting mirrors as stated, have 

 established the fact that a surface of 100 square feet 

 presented at right angles to the sun, at noon, in the lati- 

 tude of New York, during summer, develops a mechanical 

 energy reaching 1,850,000 foot-pounds per hour. The 

 advocates of the French system of dispensing with the 

 " cumbrous mirrors " will do well to compare the said 

 amount with the insignificant mechanical energy repre- 

 sented by 43,360 foot-pounds per hour developed by 215 

 square feet of surface exposed to the sun by Tellier, 

 during his experiments in Paris referred to. 



The following brief description will give a clear idea of 

 the nature and arrangement of the reflecting mirrors 

 adopted by the writer for increasing the intensity of the 

 solar heat which imparts expansive force to the medium 

 propelling the working piston of the motive engine. Fig. 

 1 represents a perspective view of a cylindrical heater, 

 and a frame supporting a series of reflecting mirrors 

 composed of narrow strips of window-glass coated with 



