j96 



iV.-i TURE 



[October 17, 1895 



About a certain Class of Curved Lines in Space of 

 » Manifoldness. 



The cla5> of curves to be considereJ is defined by ihe follow- 



: A curve of that class situated in plane space of ii 



- is cut by a S„ , in // (different or coinciding) 



in the plane it is therefore a conic, and in space a 



■ i cubic. 



li liirough ;; - I of its points a jiencil of S,,.; is drawn, then 

 each element of that pencil cuts out of the curve iwt additional 

 point, and has with a straight line one (xjint in common. The 

 coordinates of the cun'e must therefore be expressiMe as 

 rational functions of one parameter. If any fixed pyramid A^, 

 Aj, . . . A, VI is accepted as pyramid of reference, theii -any 

 point P of the curve : i"" • • 



• (2x.) . P = XiA, + . . . +X..+.A,,.,,, ■;■ .,' 



where the xi are the homo;;eneoH3 coordinates of P ; and it 

 follows 



Xi = Ri (A. m) . • . X. = R. {K /«), 

 where the Ii; are homogeneous and integer functions of the 

 A, )i~ To ensure that a S,,., has ii points exactly with the 

 cur\'e in common, necessitates that the degree of the R, is = ;;. 



It follows from the (lefinition that no Si can have more than 

 /• -i- I points in common with the curve (unless the curve is 

 wholly contained in the Si), as otherwise through this St and 

 II - k additional puints belonging to the curve a S,,., might be 

 constructed, having more than n points in common with the 

 cur%e. 



The curve is uniquely determined by any ;/ + 3 of its points ; 

 and between any " + 4 of its points a certain condition is (ul- 

 tilled (from which for « = 2 the well-known Chasles and 

 Pascal theorems for cunlcs are easily deducible). To construct 

 this condition and verify this proposition, let us return to the 

 article entitled "' Metrical Relations," A:c., of NATfRE, August 

 8. There it was pointed out that a point and a S«_, may have 

 ;i peculiar situation in regard to a pyramid of n manifoldness, by 

 virtue of which to each point of the S„ corres|>onds one S,,.,, 

 and liiei'ersii. It is not ditiicult to verify that when the co- 

 ortlinates of the point in regard to the pyramid are 



<lj . . . <7„+j, 



then the coordinates xi of the points of the S,,., satisfy the 

 condition 



■L + - 



+ ■■""' = o. 



If pjint and S„_i hive that relation to a pyramid, then they 

 may be called |iole and ix)lar to it. It will be remembered that 

 ;he construction of pole to |»lar, and vice versiu\% a purely 

 jirojective one, by means of cuts of plane spaces, iVc. The 

 iclation of // + 4 points of the curve to each other is now, 

 'hat Ihe polars of any three with regard to the pyramid of the 

 iiher // - I have a S,,.... in common. 



Indeed, let A, . . . A„+i be « + I points of the curve, and 

 I' any of its other points, also 



(2x4 • '^ = X|A, -(• . . . +X"-nA,,+i and xi - K; (A, /«)• 

 Then, .\, l)cing a p<:)int of the curve, R.^ . . . K„+i must have a 

 .Mirnon zero point ; and the same is true for R, K, . . . R„+i ; 

 ' . K4 . . Kn.fl, ^:c. It is therefore easily seen that the 

 . mates of P may be put into the form 



X' = , . where Oi and k, are constants. 



a,\ + bill 



The pv)lars to P form, therefore, a pencil ; that is, they h,ave a 

 ■■ , in common. 



!• he points of the curve are projected from any one of its 



into a .S,, . 1, thfv fiirm a curve of the class considered in 



:''■' I from Ihe representation of the co- 



I'Vjr // = 1 ihe curve becomes a 



rm a homographic range with that 



It', are the represcnialivcs of the 



• vvs, iherefore : four paints of the 



L;fmi|) of // - I .curve-points 4S„-i of 



I " • . , into straight 



'— almost iiii- 



■' irll 



PMC, that Im with It a p imt m common (hut is 

 ..J. 1355, VOL. 52] 



not situated in the same plane), or into three straight lines, of 

 which one has one point in common will) each of the other two. 



In each point of the curve there is one straight line, that has 

 two coinciding points in conuuon with the curve, and one plane, 

 that has three |x>ints of intersection which all coincide, ^:c. 

 They may be called tangent lines, planes, Cv:c., of the curve. 

 Cut the curve byaS„.i. If the ;/ points of intersection are 

 distinct, draw the 11 tangent S„_i througli them : and if only 

 11-2 are distinct, and 2 coincide, draw the 11-2 tangent S„-i. 

 and the one tangent S„ . -j ; and so on. 



The point of intersection of these piano sp.ices may be called 

 \\\e pole of the original S„_i to the curvi; : and this one, the 

 polar of that point. The |)olar of any point of tlie ^wlar passes 

 the pole. Let the pyramid of reference be chosen so that the 

 equation of the ciirve is 



Xi=A-" Xo= A';-V. ■ • \■■^^ = " ■ 

 The S„-i may satisfy the equation 



AXi + . . . + /,-fix«-i-i = '3- 

 The n points of intersection arc then given by 

 pi\"-¥ . . . -l-A.+iM'' -o- 

 Their roots may be 



A/)u = ttj, a^ . . . a„. 

 Through xi — a" Xj =»"' ' • ■ • the tangent S,,., (whose co- 

 ordinates may be |,)fli{i + . . . + <r,n-i{„_, = o will be such 

 that 



(I, = I ,i_, = II . p a, = («)jfl- . . . rt„-n = B", 



where /8 is a parameter, whose value is found = - a. The point 

 of intersection of the « S„ . „ whose equations are 



ii - It . ai J.. + (//)a . or {j - . ±ai"|„+, = O 



is' obviously 



{..- 



(") 



Oiic. 



(on account of the equation satisfied by the a). 



If {i is any point, and Xi any point on its imlar, the equation 

 exists 



{» + iXi - "f-Va H-(")2l" 1X3 - . - . = O, 



which is symmetrical, and therefore proves the proposition. 



The polar to a line joining two points is the cut of their 

 polars ; and so generally. It is therefore jiossible in speak ot the- 

 l)olar, or pole, of any plane space, in regard to the curve. The 

 two are uniteil only when the two sets of coordinates are equal, 

 that is, when tlioy satisfy a condition of the second degree. 

 Pole and polar cut a straight line in involution, as immediately 

 follows from the .symmetry o^ the equation connecting tlieni. 

 The double jjoinls of the involution are the points in which the 

 straight line cuts that surface of the second nrder. 



Much more could be siiid concerning this class of curves, 

 the properties of which are so much like those of the conies ; 

 but I hojie that wh^l has already been nientioned will be found 

 sullicient to interest mathematicians in their existence. 



London, September 6. IvMANt Kt. Lasker. 



The Freezing Point of Silver. 



The subject of high temperature thermometry has recently 

 attracted considerable attention, and on account of the ease with 

 which silver can be obtained in a pure slate, coupled with its 

 grc.tt thermal conductivity, the freezing pnint ■)f this metal ha> 

 l)een suggested as a slandaril temperature. We therefore wish 

 to call attention to an error into which we believe M. le 

 Chatelier has fallen with regard to this cmstant. In the 

 Ziiliclirifl fur I'hysikcitiuhf Cliciiii,, Band viii. p. 186, he says 

 that the nieltinL' point of silver can be lowered by as much as jO 

 through Ihe absorption of hydrogen ; again, in the Coniplis 

 reiidiis for .\ugusl 12, 1895, he stales that Ihe melting point of 

 this metal is I iwered by a reilucing atmosphere. Me therefore 

 recommends that when Ihe melting point of silver is useil as a 

 fixed point in calibrating pyroinelers, the experiment should be 

 performed in an o.xiiltsiiij,' atiniisphere. This icmclusion is 

 contradicted liy I'rof. Callendar's experiments ami l>y our own, 

 fur in the /'lii'l. J/n.v'., vol. xxxiii. p. 220, Callemlar sliows that 

 the freezing point of silver is lowered and tendered irregular by 

 an oxidi.sing atmosphere; and our own results confirm this 



