328 [Dec. 18, 



III. &c. to XXII., viz. II. Cubic surface with a proper node, and 

 therefore of the tenth class, &c., down to XXII. Ruled surfaces 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 XIII. 1, 2. 



XIV. Cubic surface of the fifth class with a biplanar node and a 



proper node. Species XIV. 1 . 



