Mr. A. Cayley on the Double Tungent of a Plane Curve. 471 



Halley first pointed out some of the financial applications of the 

 Life-Table, and the new Table shows that the mean duration of life 

 among large classes of the population exceeds considerably the ex- 

 pectations of life deduced from the ordinary Tables. The science of 

 public health has been developed since Halley' s day ; and here a 

 new apphcation of the Life-Table is found. If we ascertain at what 

 rate a generation of men die away under the least unfarourable 

 existing circumstances, we obtain a standard by which the loss of 

 life under other circumstances is measured ; and this I have endea- 

 voured to accomplish in the New Life-Table. And recollecting that 

 the science of public health was almost inaugurated in England by a 

 former President of this Society, who crowned the sanitary dis- 

 coveries of Captain Cook, I feel assured that the Society will receive 

 with favour this imperfect attempt to supply sanitary inquirers with 

 a scientific instrument. 



April 14. — Sir Benjamin C. Brodie, Bart., President, in the Chair. 



The following communications were read : — 



"On the Means by which the Actiniae kill their Prey." By 

 Augustus Waller, M.D., F.R.S. 



" On the Double Tangent of a Plane Curve." By Arthur Cayley, 

 Esq., F.R.S. 



The author notices that the problem of finding the number of 

 double tangents was first solved by Pliicker in 1834 from geometrical 

 considerations, and he gives a sketch of the subsequent history of 

 the problem. The complete analytical determination of the double 

 tangents was only obtained very recently by Mr. Salmon, and is given 

 in a note by him in the Philosophical Magazine, October 1H58 : it is 

 there shown that the (« — 2) points in which the tangent at any point 

 of a curve of the order ?j again meets the curve, are given as the 

 points of intersection of the tangent with a certain curve of the order 

 («— 2) ; if this curve be touched by the tangent, then the point of 

 contact will be also a point of contact of the tangent and the curve 

 of the order n, or the tangent will be a double tangent. The pre- 

 sent memoir relates chiefly to the establishment of an identical 

 equation, which puts in evidence the property of the curve of the 

 order (h — 2), and which the author considers to be also import- 

 ant in reference to the general theory of binary quantities : viz. if 

 YU=lI(#][x, y, zy, DU=(Xd^,+Yd^ + Z9JU,andY,SY are what 

 U, DU become when {x, y, z) and (X, Y, Z) are interchanged ; then 

 the equation is of the form I . Y+II . i3Y + III . DU + IV. U=0. 

 Taking {x,y, z) as current coordinates and U=0 as the equation of 

 the curve, then if (X, Y, Z) are the coordinates of a jjoint on the 

 curve, Y=0, and we have for the equation of the tangent at the 

 point in question SY=0, The equation shows that the intersections 

 of the curve U = and the tangent 30Y=0, lie on one or other of 

 the curves 111=0, 1)U=0, and that they do not lie on tlie curve 

 DU=0; consequently they lie on the curve 111 = 0, which is iu 

 fact the before-mentioned curve of the order (?«— 2). 



" On the Action of Acids on Glycol." By Dr. Maxwell Simpson. 



The glycol employed iu the following research was prepared 



