2 Proceedings of the Royal Irish Academy. 



<p and 8 in terms of those vectors whose directions are similarly altered by 

 the two functions. Sometimes, as in his paper* <>n " Scalar Invariants of 

 Two Linear Vector Functions," he finds it more convenient to express <f> 

 in terms of its own axes, and then to formulate in terms of (p. Two of bis 

 invariants will be useful in the present investigation. 



On the purely analytical side, the problem of the present paper is 

 analogous to that presented by the classification of pairs of bilinear forms, or 

 of pairs of collineations.t To the worker in quaternions, however, a purely 

 scalar treatment is unsatisfactory, first because the physical and geometrical 

 significance of the results is pretty thoroughly concealed by the method of 

 presentation, also because relations of singular and non-singular forms to 

 each other and to the invariants of the system cannot be, or at least has not 

 been, clearly brought out by ordinary algebra. A singular bilinear form (or 

 a singular collineation) corresponds to a linear vector function one of whose 

 latent ions is zero. It will appear below that either or both of the given 

 forms may be singular without altering the typical properties of the system : 

 in fact, it is only when the occurrence of simultaneous vanishing root- - 

 accompanied by the vanishing of two of Jolv's invariants that the system 

 falls into a more special type. 



mention a problem of another son in which Jolv's formulas for <f> and 

 (t prove to be of a oce : the general vector function of p 



can be writ ten , ^ p9p - ,, SSp, 1 2 , 



where 8 is a constant vector — a fact bearing on the theory of certain 

 functions defined by differential equations.* In factorizing a quadratic vector 

 in this way. simple methods for the simultaneous formulation of <f> and are 

 neces- 



Ag -.in, to take a problem from geometry, if a curve of the fourth order in 

 space be given by the intersection of two quadric surfaces, the equations of 

 the curve may conveniently be written 



Sp<p P + Sap +a = 0, , 



Sptif, - S/3p - 7 < = I ; °' 



where <£ and 8 a- te linear vector functions, a and /3 are constant 



vectors, and a and h are constant scalara Many of the properties of the 

 curve appear most clearly when <f> and 6 are expressed by formulas analogous 



•Trans R.I. A. 30 July, 1895,, p. 709. 



r a short account in English of this aspect of the matter, see Bocher and Duval's 

 '• Introduction to Higher Algebra," chap. xxi. 



: For a sketch of this theory see " A Classification of Quadratic Vector Functi 

 Proc. Nat. Acad, of Sciences (Washington), vol. i, No. 3 (March, 1915), p. 177. 



