78 Proceedings of the Royal Irish Academy. 



in general, and, remembering that the a are constant, aj, ag, &c., ar& 

 expressible in terms of aj, thus : — 



= — D^tto -I ai = (-L'/ + «r) tti, 



1 _ ch 1 



as - - D,a4 + - a3 = (D/ + (^i^ + «/ + ^^2) p;. + ^^2^^2) „ 



&C. 



Hence, it is easy to infer the general types of derivation 



^2ma2m+l = fm{D?) . Ci, and ff2,„+ia2m+2 = Fm{I>s") • Ds^l, 



where /„, and F„, are rational and integral functions of the order m. 

 Integration is now possible. For spaces of even order {2m), 



a^mo-zmn = 0> and f,n{,D,^)a^ = ; 



for those of odd order (2m +1), 



f-'2m+ia2m+2 = 0, aud i^,„ (D/) D,ai = 0. 



ITow, the general solution of f,n{D^~)ai = is 



ai = ^ (/? cos cs + ^' sin cs), 



where c is a root of fm{-c^) = O5 and the vectors ^ are constants of 

 integration. These vector constants are generally arbitrary ; but the 

 condition that ai should be a unit vector, or that its square should be 

 independent of s and equal to negative unity, requires generally the 

 mutual rectangularity of the vectors y8, and also the equality of the 

 tensors of /3 and /S'. Thus, the particular form 



o-i = ii{h cos CiS + u sin c^s) + lo{ij, cos CoS + i^ sin c^s) + . . . 

 + Ki {:i-i,n^\ COS c„,s + iVm sin c,„s) 



is obtained in which «i, ^2, . . . Hm are any set of mutually rectangular 

 unit vectors, and in which the scalars h are obliged to satisfy the 

 relation 



Ji- + J2'' + . . . bj = 1. 



