490 keport— 1878. 



length of the free path of the molecules enclosed between the surfaces, and on the 

 difference of temperature of the latter. By duly observing the conditions of this 

 theory it was found possible to support a spheroid of ether on a surface of the 

 same liquid for upwards of an hour and a half. The author has shown * that the 

 supporting layer of gas need not necessarily consist of the vapour of the spheroid, 

 or of the liquid upon which it floats; since melted paraffin, which showed no 

 diminution in weight even when heated for an hour in vacuo at the temperature of 

 boiling water, readily yields spheroids at ordinary atmospheric tensions when its 

 temperature is 80-90° C. If the drops are kept cool by means of a gentle current 

 of air, they continue to float for a considerable time. 



6. On the Spherical Glass-Cubic with Three Single Foci. 

 By Henry M. Jeffery, M.A.* 



1. Let the three foci A, B, C be a quadrant apart, so that the triangle of re- 

 ference (ABC) is tri-rectangular. 



A group of class-cubics thus constituted may be thus denoted by line co- 

 ordinates : — 



2 dpqr +(ap + fiq + yr) (p 2 + q 2 + r 2 ) = o, 



where d is the parameter of the group, (a, /3, y) the satellite-point. 



2. When there are inflexional cubics in the group, the locus of the satellite- 

 point may be thus found, by equating the invariants to zero. 



S = { ^-(a 2 + /3 2 + y 2 ) y~-\2d a $y = o, 



- ~ I { d 2 - (a 2 + j3 2 + y 2 ) Y + 90d 3 aPy - 54<P (/3 2 y 2 + y 2 a 2 + a 2 /3 2 ) 



+ 18tfa|3y(a 2 + 2 +y s ) = o. 

 The eliminant of d is found to be of the eighteenth degree : — 



27 (a 2 + /3 2 + y 2 ) (/3 2 y 2 + y 2 a» + a 2 /3 2 ) 4 - 12 x 81a 2 2 y 2 (/3 2 y 2 + y V + a s /3 2 ) s 



- 8 x (27) 2 (a 2 + /3 2 + y 2 ) 2 (/3 2 y 2 + y 2 a 2 + a 2 /3 2 ) 2 a 2 /3 2 y 2 



- 112 x 17 x 27(a 2 + /3 2 + y 2 )(/3V + y 2 a 2 + a 2 /3 2 )a'»/3 4 y 4 + 10 x (27) 2 (a 2 + /3 2 + y 2 ) 3 a"/3V 



+ 10x(17) 3 a 6 /3y = o. 



If this equation be arranged according to the powers of a, its highest term is 



(27) 2 (/3 2 -y 2 ) 4 a ,0 + .. . 



The curve has four loops at each of its three foci and their antipodes, at which 

 the tangents intersect at right angles. It is petal-shaped, like the Rhodonese of 

 Abbate Guido Grandi. (Gregory's ' Examples of the Ditf. and Int. Calc.,' fig. 49.) 



3. There are seven critic lines at the most. 



For a critic value, S 3 -r I — J = o. This function will be found to be of the 



seventh degree in the parameter (d). 



The theorem may be also thus established. 



By partial differentiation with respect to the variables (p, q, r), 



a (p 2 + J 2 + r 2 ) + 2p(ap + @q + yr) + 2dqr = o. 

 /3 (p 2 + '$* + r 2 ) + %q(ap + $q + yr) + 2dpr = o. 

 7 (P* + Q 2 + r ) + 2>'( a P + /3? + yr) + 2dpq = o. 



Hence the critic lines are determined symmetrically by cubics with collinear 

 foci (2>,q,ap-Pq). 



op ft? yr 



—p 2 + q 2 + r 2 ~p 2 — q 2 + r 2 ~p 2 + q 2 — r* 



* ' Proceedings of Roj^al Dublin Society,' vol. i. (new series), p. 87. 



