January 24, 1895J 



NA TURB 



309 



point images. This method is inapplicable when the (dielectric) 

 angle is not a su*imuliiple of two ritjht angles, as has been 

 shown by W. D. Niven, L.M S. vol. xii. p. 27. The only 

 other case which has been hitherto solved is, the author thinks, 

 that of the spherical bowl (Lord Kelvin, " Papers on Electro- 

 statics and Mignetism." p. 17S). In 'he piper by W. U. Niven, 

 mentioned above, an attempt is made to deduce the capacity of 

 such a conductor from the solution of a functional equation f >r a 

 particular value of one of the variables, but the result obtained 

 does not seem in the case of the spherical bowl to agree with 

 Lord Kelvin's. The results obtained ny the author also differ from 

 those given by Niven. The objfct of the paperis to obtain the 

 solution in the general case. To effect this, the functional 

 image of a point placed between two planes intersecting at any 

 angle is obtained in the form of a definite integral. In the 

 next few paragraphs the refiuction of thts integral to known 

 forms is effected in certain cases, and it is shoivn that the inte- 

 gration can be performed when the angle of intersection is any 

 submultiple of four right angles; the case in which it is re- 

 ducible to elli|)tic functions is also discussed. In § 5, the func- 

 tional image of a line of uniform density parallel to the inter- 

 section of the planes is deduced. In § 7, the potential due to a 

 freely-charged conductor bounded l)y two spherical surfaces 

 cutting at any angle is obtained an I some particular cases dis- 

 cussed. The capacity of such a conductor is obtained in §8, in 

 finite terms, and some particular cases are discussed in §9 ; one 

 of the most interesting of these is the capacity of a hemi- 

 sphere, which is found to be nearly one ant' a quarter 

 times that of the complete sphere, showing that the sharp 

 edge acts somewhat like a c 'n ienser. Some ca-cs are 

 mentioned in the last piragraph which could be deduced from 

 the resul's of the preceding ones.^The Dynamics of a Top, 

 by Prof. A. G. Greenhill, F. R. S. To construct a molelof the 

 articulated deformable hyperb iloid described by .M. Darboux 

 in Note xix. to Despeyrous' " Coars de Mecanique," t. ii., 

 which shall realise the motion of the axis of a Top, the 

 ratio of the axes of the focal ellipse mu~t be taken equal to the 

 modulus k of the associated elliptic funcrir)ns. The parameters 

 a and /' of the 'wo elliptic integrals of the third kind corre- 

 sponding respectively to the lowest and highest vertical positions 

 of the Top, which give the azimuth ^, will be of the form 



a = pVJi and i = y K'j + K, 



where p and q are real proper fractions. Then two points P and 

 Q must betaken on the focil ellipse wh >se excentric angles, 

 measured from the minor axis, a^e given by 



am{(l - p -k- ?)IC, k'\ and am!(l - / - ?)K', k'\ ; 



and ifthe tangents in and tif are drawn at Q and P, inter- 

 secting in H, these tangents mike angles 



am|(/ -1- i/)K.', k'\ and am!(/> — ?)K', k'\ 



with the minor axis. The parallel tangents OC and OG being 

 drawn, intersecting in O, the design of the m )del is completed, 

 in llenrici's m inner, by drawing any other two piirs of tangents 

 to the focal ellipse ; the tangents OG an 1 OC being replaced by 

 a pair of rods representing the generators through O, III and 

 HJ representing the parallel generators through II, the oiher 

 pairs of tangents representing the connect in g generators, all freely 

 jointed at the points of crossing. If OG is held fixed in a 

 vertical position, the point II opposite to O is constrained to 

 move in a fixe I horizontal plane ; and now if H is moved along 

 a herp ilh >de of parameter a + b, the generator OC will imitate 

 the m >' ion of 1 he axis of a Top. S ' art ing with the hyperboloiil 

 flattene I in ihe o'a le of its focil ellipse, and \\ at a maximum 

 distance from OG, the axis OC is in its lowest position ; and as 

 H moves along the herp^ilhode to its minimum distance from 

 OG, the axis rises to its highest position, when the hyperboloid 

 liecomes llitiened in the plane of its focal hyperbola. If the 

 herpolho le his points of inflexion, the pi'h of a point C in the 

 axis OC will he looped ; since the m >tion in aznnuth o( OC 

 vanishes as II pass-s through a point of inflexion. If 

 /--(/= I, the p )ini Q lies at the end of the minor axis of the 

 focal ellipse ; the pith of C now has cusps. In the sph -rical 

 pendulum the points IlanlO lie on the pe lal >f the focal 

 ellipse wi h respect to the centre ; this gives a geometrical inter- 

 pretation of the equation 



p'a = ±pY;, 



discussed in Ilalphen's "Konctions elliptiquc^," i. p. 1 10 

 The vector Oil represents the resultant angular momentum of 



NO. 1317, VOL. 51] 



the Top ; and the tangent to the path of H is thus perpendicular 

 to the vertical plane GOC. When the niomental ellipsoid at 

 O of ihe Top is a sphere, then OH repre-ents also the resultant 

 angular velocity. Hut in the general case, when the momental 

 ellipsoid at O is a spheroid, with axis OC, the resultant angular 

 velocity is represented by the vector OI to a point I fixed in 

 the generator HQ ; also H and I describe equal curves with 

 respect to OC in parallel planes perpendicular to OC, which are 

 herpolhodes of parameter a - //. Since I can be joined to a 

 fixed point in the opposite generator OG by a rod of fixed 

 length, we have Darboux's theorem that the motion of the 

 Top can be realised by rolling a herpolhode of parameter a-b, 

 on a fixed sphere. The connection of the motion of a Top with 

 herpolhodes has also been discussed by Dr. Kouth, Q.f.M. 

 xxiii. p. 34. As an application, consider Ilalphen's alge- 

 braical herpolhode 



(I- + t"-) (V- + ^)=a* 

 or 



lr^siii=2» .r b-r- + 6* ~ a' = o, 



produced by rolling the hyperboloid of two sheets 



+ -<■ — . -(--. = I 

 - a-! - *^ a- 



on a fixed pl.ane at a distance b from the centre. 



In the associated motion of the Top, p + 1 = 

 ellipse of the articulated hyperboloid is given by 



1 ; I he focal 



the coordinates of II and O in its 

 „ ., b* la- V " 



\- 



plane are given by 



£(';-T{s 



The motion of the axis of the Top is given by 



ab° / f a* -!- 2*' 



'1- 



■ -o. . / a*- / f a« -!- 2*» „ ) 



sin- 9 cos 2ii = 4 x' 2 - — — ;— / \ — — ;— . - cos 9 > 



sin'-'fl sin 2J< 



^{a* -!- U*) 



cos 9 



/ ; a^ -I- 4/^1 

 V '. V(a'-)-8*' 



8**) 



-t- cos 9 



a- + ±i>- , „ "I 



—. — - „ - + cos 9 - 

 v/la* -I- 8/-') I 



Thus for instance, if a- = ilr, k = \; the point Q is at an 

 end of the minor axis of the focal ellipse, and the curve de- 

 scribed by C has cusps. If a- = 31^-, k — \\ If alphen's herpol- 

 hode has points of inflexion where >~ = ^^'b'-, and >- varies 

 between 4/'- and S/'- ; the equation of the focal ellipse is 



and the coordinates of H and O are ±J^3(>, ±h^6i. — Some 

 properties of generalised Brocard circles, by Mr. J. Griffiths. 

 — On fundamental systems for algebraic functions, by Mr. 

 H. F. Baker. 



Physical Society, January 11. — Extra meeting, in the phy- 

 sical laboratories of University College (by invitation of Prof. 

 Carey Foster). — Prof. Kii-ker, President, in the chair. — Prof. 

 R imsay read a paper, by himself and Miss Dorothy Marshall, on 

 the measurement of latent heats of vaporisation of various 

 organic liquids. The liquid to be examined is placed in a small 

 flask with a narrow neck, and within this is a platinum wire 

 which has its two ends fused through the bottom, so as to be 

 capable of conveying an electric current. The flask is com- 

 pletely enclosed in a jacket, which is filled with the vapour of 

 liquid of the same kind. Before the current is turned on. the 

 vapour jacket is ke|it going for some time, so that the liquid in 

 the fl-isk is raised juvt to its boiling point, but no appreciable 

 evaporation takes place. As soon as the current is turned on, 

 boiling commences, and all the heat developed in the wire is 

 expended in producing evaporation. By weighing the flask 

 before and after, the mass oI'liquiH vaporised is determined. So 

 far the authors have only used the method for comparative 



