528 Mr. A. Cayley on the Delineation of a Cubic Scrole. 



definitely settling the atomic weight of dicresole ; while its ex- 

 cessively high boiling-point, and its partial decomposition when 

 distilled, prevent the determination of its vapour-density. 



Dicresole may be distinguished from benzoine, which it re- 

 sembles in some particulars, by its different fusing-point (10° C. 

 higher), by the green colour which it produces with sulphuric 

 acid, and by its indifference to a strong solution of caustic pot- 

 ash. Benzoine itself, indeed, by the action of sodium -amalgam 

 and water, is partially hydrogenated, with the production of a 

 substance apparently identical with dicresole. 



I have obtained in the cumyle series a body seemingly homo- 

 logous with dicresole, but the quantity was quite insufficient for 

 quantitative analysis. It appears that the method of converting 

 an aldehyde into its alcohol by the action of sodium-amalgam 

 upon a solution of the aldehyde in an inert liquid such as ben- 

 zole, followed by treatment of the sodium compounds thus formed 

 with water, is of very general application. At the same time, 

 the alteration of the aldehyde consequent upon the presence of a 

 caustic alkali, as in FriedeFs process, is eliminated by thus divi- 

 ding the reaction into two stages. 



LXIX. On the Delineation of a Cubic Scrole, 

 By A. Cayley, Esq.* 



IMAGINE a cubic scrole (skew surface of the third order) ge- 

 nerated by lines each of which meets two given directrix lines. 

 One of these is a nodal (double) line on the surface, and I call it 

 the nodal directrix; the other is a single line on the surface, 

 and I call it the single directrix. The section by any plane is a 

 cubic passing through the points in which the plane meets the 

 directrix lines ; i. e. the point on the nodal directrix is a node 

 (double point) of the curve, the point on the single directrix a 

 single point on the curve ; the two directrix lines, and the cubic 

 curve, the section by any plane, determine the scrole. Consider 

 the sections by a series of parallel planes. Let one of these 

 planes be called the basic plane, and the section by this plane 

 the basic section or basic cubic ; and imagine any other section 

 projected on the basic plane by lines parallel to the nodal direc- 

 trix : such section may be spoken of simply as ' the section/ and 

 its projection .as ' the cubic/ The cubic has a node at the node 

 of the basic cubic ; that is, the two curves have at this point/owr 

 points in common. The two curves have, moreover, in common 

 the three points at infinity (or, in other words, their asymptotes 



* Communicated by the Author. 



