of Quaternions. 491 



are well known, viz. 



( + )"=( + ), (-r=( + ), (-r-> = (-), 



where n is an integer. But it is to be remembered, that, unless 

 w=l, these equations do not express identity, but only equi- 

 valence. ( — )2=:(-|-) is an identical equation; but (-f)2=i{-f ) 

 does not mean that ( + )^ is the &ame operation as ( + ), but 

 only that it produces the same result v/hen performed on the 

 operand. Thus ( + )^ = ( + ) is only true in the same sense in 

 which ( + ) = l is true. And this brings us to a point of con- 

 siderable importance, namely, the twofold use of the symbol 1 

 in algebra, as representing an operation on the one hand, or 

 a concrete quantity or operand on the other. Considered in 

 the former light, 1 represents the operation of taking the ope- 

 rand as it is ; considered in the latter, it represents a concrete 

 unit taken as a subject of operation. In the former case ( 4- ) 1 

 and ( — )1 may be used as synonymous with ( + ) and ( — ), and 

 '/( + )!, '/(—)! as synonymous with ( + )^ (— )*. In the 

 latter case 



((+)!)'' ((-)O'' ^(+11' ^Fn 



are all equally unmeaning, just as 



((+)£!/, ^/Fo^Ki; &c. 



are equally unmeaning. It appears to me that a great deal of 

 confusion has arisen from neglecting this distinction. At all 

 events 1 intend to preserve it strictly in the present paper, in 

 which all the symbols will represent operations, and never 

 concrete quantities, unless that oe expressly stated. Thus, if 

 x be one of the coordinates of a point in space, x represents, 

 strictly speaking, in any equation in which it may occur, not a 

 line, nor a number, but the operation of multiplying the ope- 

 rand (which is understood throughout) by the same number 

 by which the linear unit must be multiplied in order to pro- 

 duce a line equal to the distance of the point in question from 

 the plane of yz. If this be well understood, there is no ob- 

 jection to speaking of x, for convenience, as if it really repre- 

 sented such a line. 



To return, however, to the symbols + and — , the equations 



a -^ {■{■)}} — a — { — )h — a-\-b, a — { + )b = a + { — )b — a—d, 



which rest on grounds that need not be here examined, justify 

 us in omitting the use of brackets to save trouble, though it 

 would perhaps be theoretically desirable to have some equi- 

 valent mark of distinction. 



The principles of the usual interpretation of symbolical 



2 K2 



