[ 293 ]" 



XL. Note on a Theorem of the Integral Calculus. 

 By Professor Sylvester, F.R.S* 



PROPOSE briefly to lay before the mathematical readers of 

 the Magazine a wide generalization, and at the same time 

 a more precise statement, of the theorem contained at the close 

 of my paper in the last Number. The theorem, as therein enun- 

 ciated, was drawn from geometrical considerations, it having first 

 manifested itself dimly to the author by a sort of indirect reflec- 

 tion from a metrical theory of perspective. I have since obtained 

 a very easy proof of it in its extended form, which in spirit amounts 

 to a free algebraical paraphrase of the method indicated in the final 

 foot-note of the paper in question. The ultimate form of the 

 perfected theorem is particularly interesting from its simplicity of 

 application, and from its connexion with the grand and growing 

 theory of invariants. The proof of it will appear in its proper 

 place in the continuation of the paper in which, in its incipient 

 state, it first came to lightf. 



Theorem. — Let a figure, whether plane, solid, or hyperspatial, 

 be supposed to be limited by a locus or loci defined by one or 

 more algebraical equations, not necessarily the most general of 

 their respective degrees, but each at least the most general of its 

 degree and kind J, and let the density at any point of the figure 

 be any homogeneous function of the coordinates, and let the 

 mass of such figure be supposed to be known in terms of the 

 constants which enter into the defining equations ; next let the 

 density at each point of the mass be multiplied by a new factor, 

 which may be any rational integral homogeneous function of the 

 coordinates. Then the theorem affirms that the expression for 

 the new mass may be obtained by operating upon the expression 

 for the original one with differential operators precisely identical 

 with combinations of certain of those which serve to define an 

 invariant of the given system of equations, and which will be 

 found set forth in my paper " On the Calculus of Forms," in the 

 1 Cambridge and Dublin Mathematical Journal' §. Thus, for 



* Communicated by the Author. 



t Strange cradle this for the inception of a quasi-invariantive theory of 

 integration, " A geometrical construction of the centre of gravity of a trun- 

 cated pyramid" ! Ou la verite va-t-elle se nicher? 



X By kind I mean descriptive character, i. e. such character as is not 

 affected by perspective or homographical deformation. Thus, ex. gr., the 

 case of a cone may be treated apart from the more general case of a surface 

 of the second degree. So, again, a curve of the third degree with a multiple 

 point, or having one or both of its fundamental invariants zero, may be 

 treated apart from the case of a general cubic curve. 



§ The partial differential equations for invariants, covariants, and con- 

 travariants will be found therein stated with absolute generality for any 



