468 



In regard to this theory, several capital discoveries belong to Pro- 

 fessor Sylvester, the Law of Reciprocity considered as a law relating 

 to the number of Invariants, Contravariants, which, although now 

 seen to be included in the notion of an invariant, were a conception to 

 which is due much of the progress of the theory, the theory of the 

 Canonical forms of binary functions of an odd order, and (less com- 

 pletely developed) the more difficult theory for those of an even 

 order, and Combinants, a theory, the resources of which are still to 

 be developed, but a first fruits of which was the determination, in a 

 manageable form, of the resultant of three ternary quadratic functions. 



Only a sketch of a singularly elegant geometrical theory of the 

 derivative points of a cubic curve has as yet been published, in a 

 paper in the l Philosophical Magazine ' (1858). 



The very original investigations forming the subject of the Lectures 

 on Partitions are also as yet published in an incomplete form. 



There are many other papers which might with propriety be 

 specially noticed, but it is obviously impossible on the present occasion 

 to give anything like a complete account of the labours of Professor 

 Sylvester ; among the latest of them are the researches on the Invo- 

 lution of six lines. The nature of the relation can be easily explained. 

 Six lines may be such that, considered as belonging to a rigid body, 

 there exists forces acting along these lines which keep the body in 

 equilibrium ; or, what is the same relation between them, they may 

 be such that the equilibrium of a system of forces about these lines 

 as axes, does not imply the complete equilibrium of the system of 

 forces. But the consideration of such a system of lines leads to a 

 long series of geometrical theorems relating to curves in space, and 

 ruled surfaces of the third and fourth orders, and opens a wide field 

 for future researches. 



PROFESSOR SYLVESTER, 



Passing over the metaphysical question as to the origin of those 

 simple conceptions from which as a starting-point all mathematical 

 inquiries must set out, it is plain that whatever is done afterwards is 

 the result of the exercise of the pure intellect ; and there is perhaps 

 nothing more remarkable in the history of human nature, or which 

 tends to give us so exalted a notion of the powers of the human mind, 

 as that out of such simple materials so marvellous a fabric should 



