October 24, 1907] 



NA TURE 



639 



the quail {Coturnix novae zealaiidiae), and it is reason- 

 able to suppose that it may still be represented on some 

 flats that settlement has not reached. My inquiries 

 extended only to the mainland ; I did not deal with the 

 islands included in the colony's boundaries. 



J.\s. Drummond. 

 Christchurch, New Zealand, September 8. 



Showers from near 3 and 7 Piscium. 



Ox October 12, at qh. 50m., I saw a second-magnitude 

 meteor at 346° + 3°, and it appeared to be nearly stationary 

 at that point, but I recorded the object imperfectly, as I 

 was looking toward the western sky at the time. 



On October 2, 1902, I noticed a small meteor almost 

 stationary at 345° -I- 3°, and several others directed from 

 the same point. This shower in Pisces is rather a 

 prominent one in the months of August and September, 

 and it has frequently been observed. The following are 

 some of the determinations of the radiant : — 



July 25 to Aug. 12, 1879 . 



Aug. 10, 1897 



Aug. 13-16, 1893 



Aug., 1893 



Aug. 16-20, 18S5 



Aug. 15-21, 1901 



Aug. 19, 1900 



Aug. 21, igor 



Aug. 21-23, '879 



Aug. 24-Sept. 7, 1SS6... . 



Sept. 1-4, 1885 



Sept. 3-14 



Sept. 8, 1899 



Seft. 14, 1901 



Sept. 14, 1875 



Sept. 15-20, 1876-1879 



Sept. 



Sept. 1858-63 



Sept. 17, 1S85 



Sept. 17, 1898 



Sept. 20-Oct. 4, 1886 ... . 

 Sept. -Oct. I, 1891 ... . 



Sept. 27, 1906 



Sept. 29-Oct. 2, 1877-1902. 



343 + 3 Weiss 6 meteors 

 345-!- 3 Libert 6 „ 

 347-1-0 W.F.D. 6 ,, 

 347 + Corder 6 ,, 

 345-1-0 W.F.D. 7 



345 -hO W.F.D. 7 

 346+1 W.F.D. fireball radiant 

 341 +5 W.F.D. meteor radiant 

 350 + W.F.D. 10 meteors 

 346+1 W.F.D. 5 „ 

 346+0 W.F.D. 9 ,, 



346 + 3 Schmidt 



347 + 3 W.F.D. fireliall radiant 

 345^1 W.F.D. „ 



348 + Tupman ,, ,, 

 346 + W.F.D. 10 meteors 



344 - 3 Schmidt 



346 - 3 Heis-Neumayer 



345 + W.F.D. 4 meteors 

 343 + W.F.D. meteor radiant 

 347+0 W.F.D. 5 meteors 

 345 + Milligan 



347 + 2 W.F.D. fireball radiant 

 347 + 3 W.F.D. 13 meteors 



Possibly several showers may be involved in producing 

 these radiants. As they nearly agree with the radiant 

 point computed for Daniel's comet on September 12, they 

 possess an interest of rather special character, and it is 

 to be hoped that observations will be augmented, par- 

 ticularly at the middle of September. 



Bristol, October 14. W. F. Denning. 



The "Quaternary" Period. 



In Dr. Wright's interesting review of " Les Grottes de 

 Grimaldi," by M. L. de Villeneuve (Nature, October 10, 

 p. 590), I find the following : — " M. Riviere attributed 

 them [the deposits] to the Quaternary period, M. Mortillet, 

 on the other hand, regarded them as Neolithic." Now 

 it is impossible to conceive any defensible use of the word 

 " Quaternary " that does not include the Neolithic. 

 Many authors have condemned the expression on the 

 ground that the Pleistocene and Recent are nothing more 

 than the latest and very subordinate portions of the 

 Tertiary period. For my own part I believe that the 

 great influence which man has already exerted on the 

 character and distribution of the forms of life upon the 

 earth, as well as on the purely physical conditions of its 

 surface and the still greater changes that his activity 

 must occasion even in the near future, are ample justifi- 

 cation for marking his effective appearance on the scene 

 by the commencement of a new period in the earth's 

 history, a period the threshold of which we have scarcely 

 passed. If, however, the Quaternary " period " is to be 

 considered to close at the end of the Pleistocene, it 

 becomes so insignificant in comparison with the long ages 

 of its predecessors that it would be better to dispense with 

 it altogether. John W. Evans. 



Imperial Institute, London, October 11. 



NO. T982, VOL. 76I 



The separation of the Quaternary period from the 

 Recent period, which begins with the Neolithic, is attribu- 

 table to the fact that an interruption was supposed to 

 have occurred in Man's occupation of Europe. .According 

 to this view, the Recent period begins with his re- 

 appearance. Of late years it has been shown that such a 

 view is untenable, and that no such interruption occurred. 

 There is therefore much reason in Dr. Evans's conten- 

 tion that the Quaternary period should be extended to 

 include the Recent period. The term " Quaternary " has, 

 however, a recognised meaning which could not be 

 changed without entering into a discussion of the reasons 

 for the step — a discussion which would be quite outside 

 the province of the writer of a short review. 



William Wright. 



To Deduce the Polar from the Intrinsic Equation. 



I SHALL be grateful if one of your mathematical readers 

 can give me the polar equation of the spiral which satisfies 

 the condition ps = c, i.e. the spiral the curvature of which 

 is a linear function of the arc. A. B. Porter. 



324 Dearborn Street, Chicago, September 19. 



The curve in which the radius of curvature is propor- 

 tional to the arc is easily seen to be an equiangular spiral. 

 If, as your correspondent assumes, the radius of curvature 

 is inversely proportional to the arc, the problem is more 

 complicated, and it is best in the first instance to express 

 the Cartesian coordinates in terms of a third variable 

 before attempting to form the polar equation. If instead 

 of ps = c we write ps = Jfe-, we get with the usual 

 notation 



i/'2 --? = .r, whence <() = -,, 

 ds i- 



(choosing axis so that i = o when (^=0). 

 Put 



/; 



and we have 



r , , f « J k /"cos (* dii 



\ ds cos *=/■ / cos irdi( = - I 7 — 

 ./ .' 2./ v> 



I rfj- sin a> = i I sin li-dii- 



'sin ip dip 



By a suitable choice of origin, the lower limit of integra- 

 tion can be made to be zero in each case. 



The integrals are known functions closely allied to the 

 well-known error function. In fact, we have 



: + n' = f: I ei"-dit = — ei;/ 11 , 

 I Ji 



To find the polar equation, we first transform the 

 coordinates to new axes of X and Y, making an angle a 

 with the old axes. Thus 



X=.t- cosa + vsin a = i- l cos (ir - a) dii 



Y = V cos a - X sin a = /• I ."^in (?r - o) dii 



If now r, 6 are the polar coordinates, we may adapt the 

 last results to polar coordinates by taking X=r, Y=o, 

 a = 9. The polar equation is thus the eliminant of the two 

 simultaneous equations 



r = /; I cos (ir - 9)di( 



J 



= 1' sin («- - e)ai/. 



In terms of 1^ we have 



.=^f*<="ii*^W, 0= f''"'»-% 



2.1 „ V<?' •' V> 



while the inclination + of the tangent to the radius vector 

 is given by \li = ^ — d. 



-This method can be applied to find the polar equation 

 of a curve the radius of curvature of which is any func- 

 tion of the arc, but, as in the present example, the integra- 

 tions cannot always be evaluated in terms of the functions 

 discussed in elementary text-books. G. H. B. 



