which occur in the Calculus of Invariants, 467 



Mr. Cayley verifies this theorem when for E p the leading pro- 

 tectant, we substitute V [f a pertractant, as follows. Take only 



a single element # and its symbolical reciprocal -7-, so that 



P 1 =4;thenP 2 =|P,andP;.P{=(^© , )*(,4)- 

 is easily seen to be 



x i i * 1.2 x ~"' 



= Pi + ' + y Pl+i- 2 (fflPj + Jk=^pI)- P «+./-« (2 p s) . + . . . . 



as before. 



But I find that the theory admits of a still further and 

 most important extension. Thus far we have been dealing 

 with operants and operators derived from a single one of the 

 former. But we may easily form a set of two or more, 

 say k pertractants, i. e. functions lineo-linear in a,b,c, ... 



T' Ik' !~ ' ' ' commu ^ aD ^ e i n t er se "f > these being energized into 



t This imports into the subject a beautiful theory of commutable matrices. 

 In the case of two letters we have two types of commutable pertractors, 

 from which all the rest may be derived by the laws of pertraction stated in 

 the text. These two fundamental systems are, 



(1) xb x ; yh r 



(2) ("J)(*,y)(a«,8 y )5 *a,+yV 



In the case of three letters, the four following types of commutable sys- 

 tems present themselves : — 



(1) x8 x ; yh y ; z$ z . 



(3) axhy+byb x +czb K ; -y§x+p$y+-x*z. 



(4) (d, elf) (x, y, «)(8,, d y , 8*); *»,+y* y +«»*. 

 V K k/ 



Whether the above four systems are independent, and whether they con- 

 stitute an exhaustive enumeration in the case of three letters, I have not 

 yet had time to ascertain. 



The reader will please to bear in mind that any linear function of the terms 

 in each system, or of them and their derivatives, is commutable with those 

 terms themselves ; thus, ex. gr., the last system but one is quite as exten- 

 sive as if we included in it 



\axh y +\byb e +\czb x + \ y8 x + £s8y+ ~ xb z + s/xh x + s/ydy+ s/zb z , 



2H2 



