Ross — Verb-Functions. 



37 



those indicated. Moreover, we are scarcely justified in giving arbi- 

 trarily different powers to two elements of the same expression : 



a + h ~ 



cx 



must be eitlier a factor or an operation — it cannot be both. The 

 equation can be rendered accurately by writing 



\_aB' + hD'] u = au + [hB] tc, where = 



(See § 23.) 



9. To sum up, there are two proposals contained in the preceding 

 pages : one a proposal to recognise the unit of operation by a special 

 symbol, and the other to adopt a special operative bracket. Con- 

 sideration will show that it is scarcely possible to represent operations 

 explicitly and accurately without these two conventions. We may 

 employ another symbol in the place of /?, and other brackets than the 

 square brackets ; but the fundamental conventions appear to be inevit- 

 able. Nor should the suggested notation be mistaken for a symbolic 

 one. A symbolic notation may, perhaps, be described as one which is 

 used for convenience, although it is not strictly in accordance with 

 algebraic usages. But these proposals do not interfere with algebraic 

 usages ; they merely suggest additions which are as rigid in their own 

 way as those of algebra. 



At one point a symbolic notation has been used above, namely, in 

 the expression [c^]". Strictly speaking, the index has already been 

 allocated for the use of algebraic involution ; but as an algebraic 

 power of ^ can be rendered by (^)", there is little chance of ambiguity 

 if [</)]" be taken to represent operative involution. It is, however, 

 advisable, and even at times necessary, to use a symbol of operation in 

 order to express operative involution correctly. We may suggest 

 that 



so that 



[y..][y»J* = [<^]""' 



and 



[a + by + cy2 + dys . . .] </> = a + bcf) + clcfj^ + <?[</)]^ . . . 



10. In order to illustrate the practical advantages of these pro- 

 posals, two courses are open. One to draw isolated examples from 

 many diverse branches of mathematics; the other to deal more 

 thoroughly with a single field. The latter course is adopted ; and 

 the field selected is that of the common algebra of the subject — 



