14 Proceedings of the Royal Irish Academy. 



It is apparent if n exceeds the number of the units, that the group 

 F„ does not occur. In fact the highest order that can occur is given 

 by the equation 



n - 2m = N, or by n - 2m = JV- 1, 



■where iV is the number of the units, according as n and iV are both 

 odd or even, as in the first case, or one even and the other odd as in 

 the second. 



We shall now show that there is the closest analogy between a 

 Grassmann combinatorial product of n point symbols and the function 

 Vj^wi'Wz . . . •7ir„. We shall suppose, in the first place, n to be less than 

 iV, the number of the units. 



Comparing, then, a combinatorial "product" of point symbols 



and the group of highest order in the product of n vectors 



f^n'^l'^2'^3 . . . -zr„, 



we see at once the following points of similarity : — 



(1). Both vanish if two constituents are the same. For if 



-sTi = izr2 = ^i«i + ^iiz + . . . + a;^ijy, ■wi' = - Xi" ~ x<^ - . . . - Xj^, 



and the complete product 'sTi~'ST^-Wi . . . ■vTn has its highest group of 

 the order % - 2 at most. 



(2). No change is produced in either if we replace p^ and -ara by 

 ^2 + tpx a^iid -3r2 + ^•zTi, t being a scalar. 



(3). Interchange of contiguous symbols changes the sign of both. 

 We may replace •zr2 by ^-zet^ + -nr'a where -zzr', is at right angles to -zb-j, or 

 where ts^i'^'i = - -zr'a^i. Hence 



Vn'^YTiTi ... -zzr,, = F'^-zTi'TO-'a . . . -zir„ = - V^{m\'!JTi ... -zr„ = - /'"j-zToiri-zirs . . . -zr„. 



(4). Both obey the associative law expressed by 



PlP-i •••Pm-Pmn ■■'Pn 



and F",, Fj^-zs-i-zra . . . -zzr,,, F„_„j-zzr„j^.i . . . tsr^. 



For, in forming the group of highest order in the complete product 

 -zTi-zzTa . . . -zzr,,, . -ro-„,^i . . . -zzr,,, it is useless to retain any but the highest 



