QUATERNIONS AND VECTOR ANALYSIS. 181 



indeed reaches high-water mark in the paragraphs in which he men- 

 tions them. Concerning another notation, $x^ (defined in Nature, vol. 

 xliii, p. 513) [this vol., p. 160], he exclaims, "Thus burden after burden, 

 in the form of new notation, is added apparently for the sole purpose 

 of exercising the faculty of memory." He would vastly prefer, it 

 would appear, to write with Hamilton m^'' 1 , "where m represents 

 what the unit volume becomes under the influence of the linear 

 operator." But this notation is only apparently compact, since the 

 m requires explanation. Moreover, if a strain were given in what 

 Hamilton calls the standard trinomial form, to write out the formula 

 for the operator on surfaces in that standard form by the use of the 

 expression m^'- 1 would require, it seems to me, ten (if not fifty) times 

 the effort of memory and of ingenuity, which would be required for 

 the same purpose with the use of J^x^- 



I may here remark that Prof. Tait's letter of endorsement of Prof. 

 Knott's paper affords a striking illustration of the convenience and 

 flexibility of a notation entirely analogous to $*&, viz., $:$. He 

 gives the form SVV X So-o^ to illustrate the advantage of quaternionic 

 notations in point of brevity. If I understand his notation, this is 

 what I should write V<r:V<r. (I take for granted that the suffixes 

 indicate that V applies as differential operator to a; and V l to <r lt 

 a- and & l being really identical in meaning, as also V and V r ) It 

 will be observed that in my notation one dot unites in multiplication 

 the two V's, and the other the two cr's, and that I am able to leave 

 each V where it naturally belongs as differential operator. The 

 quaternionist cannot do this, because the V and or cannot be left 

 together without uniting to form a quaternion, which is not at all 

 wanted. Moreover, I can write <3? for Vcr, and <if?:3? for V0-.-V0-. The 

 quaternionist also uses a 0, which is practically identical with my < 

 (viz., the operator which expresses the relation between da- and dp), 

 but I do not see how Prof. Knott, who I suppose dislikes $ : $ as 

 much as ^x*^ would express SVV X Scro^ in terms of this 0. 



It is characteristic of Prof. Knott's view of the subject, that in 

 translating into quaternionic from a dyadic, or operator, as he calls it, 

 he adds in each case an operand. In many cases it would be difficult 

 to make the translation without this. But it is often a distinct 

 advantage to be able to give the operator without the operand. For 

 example, in translating into quaternionic my dyadic or operator $xp, 

 he adds an operand, and exclaims, " The old thing ! " Certainly, when 

 this expression is applied to an operand, there is no advantage (and 

 no disadvantage) in my notation as compared with the quaternionic. 

 But if the quaternionist wished to express what I would write in the 

 form (^x/o)- 1 , or |$x/o|, or (3>x/o) s , or (<3>x/3) x , he would, I think, 

 find the operand very much in the way. 



