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The theory of the 27 lines on a surface of the third order is due to Mr. Cayley and Dr. Salmon; and the effect, as regards the 27 lines, of a singular point or points on the surface was first considered by Dr. Salmon in the paper “On the triple tangent planes of a surface of the third order,” Camb. and Dub. Math. Journ. vol. iv. pp. 252—260 (1849). The theory as regards the reality or non-reality of the lines on a general surface of the third order, is discussed in Dr. Schläfli’s paper, "An attempt to determine the 27 lines &amp;c.,” Quart. Math. Journ. vol. ii. pp. 56-65, and 110-120. This theory is reproduced and developed in the present memoir under the heading, I. General cubic surface of the third order and twelfth class; but the greater part of the memoir relates to the singular forms which are here first completely enumerated, and are considered under the headings II., III. &amp;c. to XXII., viz. II. Cubic surface with a proper node, and therefore of the tenth class, &amp;c., down to XXII. Ruled surface of the third order. Each of these families is discussed generally (that is, without regard to reality or non-reality), by means of a properly selected canonical form of equation; and for the most part, or in many instances, the reciprocal equation (or equation of the surface in plane-coordinates) is given, as also the equation of the Hessian surface and those of the Spinode curve; and it is further discussed and divided into species according to the reality or non-reality of its lines and planes. The following synopsis may be convenient :— I. General cubic surface, or surface of the third order and twelfth class. Species I. 1, 2, 3, 4, 5. II. Cubic surface with a proper node, and therefore of the tenth class. Species II. 1, 2, 3, 4, 5. III. Cubic surface of the ninth class with a biplanar node. Species III. 1, 2, 3, 4. IV. Cubic surface of the eighth class with two proper nodes. Species IV. 1, 2, 3, 4, 5, 6. V. Cubic surface of the eighth class with a biplanar node. Species V. 1, 2, 3, 4. VI. Cubic surface of the seventh class with a biplanar and a proper node. Species VI. 1, 2. VII. Cubic surface of the seventh class with a biplanar node. Species VII. 1, 2. VIII. Cubic surface of the sixth class with three proper nodes. Species VIII. 1, 2, 3, 4. IX. Cubic surface of the sixth class with two biplanar nodes. Species IX. 1,2, 3,4. X. Cubic surface of the sixth class with a biplanar and a proper node. Species X. 1, 2. XI. Cubic surface of the sixth class with a biplanar node. Species XI. 1, 2. XII. Cubic surface of the sixth class with a uniplanar node. Species XII. 1, 2. XIII. Cubic surface of the fifth class with a biplanar and two proper nodes. Species X III. 1, 2. XIV. Cubic surface of the fifth class with a biplanar node and a proper node. Species XIV. 1. XV. Cubic surface of the fifth class with a uniplanar node. Species XV. 1. XVI. Cubic surface of the fourth class with four proper nodes. Species XVI. 1, 2, 3. XVII. Cubic surface of the fourth class with two biplanar and one proper node. Species XVII. 1, 2, 3. XVIII. Cubic surface of the fourth class with one biplanar and two proper nodes. Species XVIII. 1. XIX. Cubic surface of the fourth class with a biplanar and a proper node. Species XIX. 1. XX. Cubic surface of the fourth class with a uniplanar node. Species XX. 1. XXI. Cubic surface of the third class with three biplanar nodes. Species XXI. 1, 2. XXII. Ruled surface of the third order and the third class. Species XXII. 1, 2, 3.—A. C. I.General cubic surface, or surface of the third order and twelfth class. Art. 1. As the system of coordinates undergoes various transformations (sometimes imaginary ones), it becomes necessary to adhere to an invariable system of a real meaning, for instance the usual one of three rectangular coordinates. We shall call this the system offundamental cordinates , and define it by the condition that the coordinates of every real point (or the ratios of them, if they be four in number) shall be real. Consequently any system of rational and integral equations, expressed in variables of a real meaning, and where all the coefiicients are real, will be termed areal system (of equations), whether there be real solutions or none, provided that the number of equations do not exceed that of the variables, or of the quantities to be determined. The degree of the system will be the number of solutions of it when augmented by a sufficient number of arbitrary linear equations; and such degree will generally be the product of the degrees of the single equations. It is obvious that the system, whenever its degree isodd , represents areal continuum of as many dimensions are independent variables; for instance, every real quaternary cubic represents a real surface.

philosophical transactions of the royal society of london

X. On the distribution of surfaces of the third order into species, reference to the absence or presence of singular points, and the reality of their lines