18 Proceedings of the Royal Irish Academy. 



the laws of operating in this way, or as we call it of generalised multiplication. 

 are summarised by stating that like signs give + and unlike -. 



As (- a) a = nj3 the symbol - enables us to dispense with the symbol 



/3, and any quantity may be expressed by the unit a and a qualifying 



adjective + a or - a. This qualifying adjective is our generalised number, 



and consists of an ordinary number a with the symbol + or - prefixed. 



We denote it usually by the letters x, y, a\, y„ &c. Xow my object is to 



establish the ordinary use of symbols. Were I to follow what would 



appear to be a more natural course, I should not introduce the symbols 



+ and - at all, but should introduce one new symbol j, which would 



have the same meaning as -, in analogy to the symbol i, which will be 



introduced presently, ami so our generalised number would be a or aj. 



If any student wishes to introduce algebraic operations hi a sound 



logical manner, I advise him to use this symbol j, as it is rather hard to 



limit the well-known symbols + and - to their precise usage as defined 



above. 



The meaning now of xy:a is clear, and as we see generally that 

 xyza = yxza, because the numerical part of xyza is independent of the 

 order of .ry, and also the final sign + or - in ryza does not depend on the 

 order of the signs + or - hi x and ;/ ; hence it follows that in the process of 

 generalised multiplication generalised numbers obey both the commutative 

 and the associative law, by a method of proof precisely analogous to that 

 given before in proving the corresponding results for addition of quantities. 



As we may prefix the sign + = + 1 to any quantity, we may dispense 

 with the comma, and according to my view mere juxtaposition of the 

 quantities will signify that they are to be added. Thus xa + ya means that 

 xa is to be added to ya, xa - ya means that xa is to be added to y(5. 

 The same quantities may also be denoted by (x + y) a and (x - y) a, 

 whereas xi/a means that a is to be multiplied first by y and then by x. 

 We call + aa = o.a a positive quantity, - aa = a/3 a negative quantity, 

 + " or a a positive number, - a a negative number, and in future in con- 

 formity witli usage we shall call a generalised number a real number. By 

 adding any small positive quantity, say ea, repeatedly to a large negative 

 quantity we alter it by steps as small as we please to a large positive 

 quantity. We usually talk of this process as increasing the quantity, but 

 we might equally well talk of the reverse process as increasing the quantity, 

 for by continually adding c/3 to a large positive quantity we proceed by 

 steps as small as we please to a large negative quantity. Subtraction is a 

 term which may be dismissed in a line: to subtract any quantity ya from 

 xa means that we look for a quantity such that when ya is added to it we 



