;92 



NATURE 



[April 20, 1893 



RoMian letteri used for qMaternions an! scalars. A glance tells 

 us what kind of quantity we have to deal with before we are 

 cilled upon to inquire into its comp >sition. There is no such 

 eye-catching virtue in Gibbs's notation ; and Heaviside largely 

 destroys the contrast between the quantities and selective symbols 

 by using capital letters for all. In print the vectors are made 

 heavy and stand out prominently enough. But a vector analysis 

 is a thing to he used; and with pencil or pen or chalk on a 

 blackboard it is hopeless to prevent confusion between A and A- 

 In suggesting a suffix notation for manuscript, Heaviside un- 

 consciously condemns his own system. Two conditions for a 

 good notation are (i) an unt)iistak<able difference between 

 ^aj?/y ?fr2V/^« symbols for scalar and vector quantities; (2) the 

 scalar and vector parts of products and quotients thrown out in 

 clear relief. This second is quite as important as the first con- 

 dition. So far, Hamilton's notation easily hold- its own. 



A very important symbol of operation is the Nahla, v, "hich 

 may be defined in the form o9] + /SS^ + 783 where dfijb-:^ are space- 

 differentiations along the mutually rectangular directions of the 

 unit vectors oySy. Since Heaviside and Madarlane makeo'3^7^ 

 each equal to + i, they find that V"«, where n is any scalar, is 

 d-ujdx- + d'^uldy" + d~uldz-. The real V"" is minus this 

 quantity. When v^ acts on a vector, Heaviside boldly defines 

 v'^ai as having the same significance ; but Macfarlane, rejoicing 

 in his non associative algebra, finds that vCvcd) is quite a 

 different quantity from (vv)cd. The net result attained by this 

 tinkering of the signs is to get a pseudo-nabla'non-associative 

 with itself! 



Gibb? moves more cannily. He defines separately the 

 quantities v«, vxw vw, and V.Vw, which mean the same 

 things as the quaternion quantities V«, Vvw, - Svco, and 

 - V"w. [In quaternions there is one definition of v, and every- 

 thing else follows.] But even with these four definitions (all of 

 which are properties somewhat distorted of the real Nabla) Gibbs 

 finds his system lacking in flexibility. He has, so to speak, to 

 lubricate its joints by pouring in the definitions of four other 

 functions with as many new symbols. One of these is the 

 Potential ; the others are cilled the Newtonian. Laplacian, aad 

 Maxwellian. They are symbolised thus — Pot, New, Lip, Max. 

 Their meanings will be evident when they are exhibited in 

 quaternion form. Thus, as is well known, 



V dx'- 

 from which at once 



92^ 



Pot u = 



47r«, 



V^ Pot u 



Siinilarly, if. 



Then we have 



Pot u = dirv'-u. 

 is a vector quantity, 



47rV--o 



Pot 



New ?< =: V I'ot 71 = 47rV"'.v 

 Lap w = Vv Pot w = 4TrVv"'a' 

 - Max u) = Sv Pot oi = 47rSv-'ctf. 

 Now, Prof. Gibbs gives a good many equations connecting 

 these functions and their various derivatives, equations which 

 in quaternions are identities involving the very simplest trans- 

 forma'ions. But there is no such simplicity and flexibility in 

 Gibbs's analysis. For example, he lakes eight distinct steps to 

 prove two equations, which are special cases of 



V-V'« = ti ! 

 Another of his theorems, nanely, 



4ir Pot a> = Lap Lap w - New Max w 

 is simply the quaternion identity 

 47rV"-£0 = 4TrV"'V""'« 



= 4TV-Wv-'w + 4irv-^Sv-^«. 

 Similarly the equation 



47r P.,t u -— - Max New ti 

 is a travesty of 



4TrV~-?/ = 4irV""^V""^ «! 

 These extremely simple quaternion transformations cannot be 

 obtained with the operator used by Gibbs. Hence the necessity 

 he is under to introduce his Pot, New, Lap, Max, which are 

 merely inverie quaternion operators. . . . 



Gibbs's system of dyadics, which Heaviside regards with such 

 high admiration, differs from Hamilton's treatment of the linear 

 and vector function simply by virtue of its notation. In his 

 letter to Nature he gives reasons why this notation is prefer- 

 able to the recognised quaternion notation. As develoi>ed in 

 the pamphlet, the theory of the dyadic goes over much the same 

 ground as is traversed in the last chapter of Kelland and Tail's 

 " Introduction to Quaternions." With the exception of a few of 

 those lexicon products, for which Prof. Gibbs has such an affec- 

 tion, ^ there is nothing of real value added to our knowledge of 

 the linear and vector function. As usual, the path is littered 

 with definition after definition, 'i hus the direct prodttct- of two 

 dyads (indicated by a dot) is defined by the equation 

 ]o/3j. {75} =;8.7o5. Quaternions gives at once 



<p^p = oS;8(7SS/)) + &c. = aS5/)S;37 -f .'vc. 



Then there follow the -definitions of the .f/^c-w products of <^ 

 and p, thus— 



(p\p— aXxp+^fx-xp + yvxp 

 pX(p=:pXa\ + p x P/u. + p X yy. 

 These are not quantities but operators. To see what they mean 

 let them operate on some vect )r <r. Then we find 

 <p X p . (T — aS\p<T + ... — <p\pff 

 p X (p . a = \'paS\(r + ... = Yp(f><T. 

 The first is simply <pw, the old thing ! The second is a well- 

 known and important quantity in the theory of the linear and 

 vector function. It is interesting to note, as bearing upon the 

 intelligibility of the notation, that Heaviside, who dotes so on 

 the dyadic, writes <|> x p in the form W(pp, so that he makes 

 (piW pa = — Yaipp ! ! 

 As one example of our gain in following Gibbs's notation, 

 take his dyadic identity — 



ii ■ {p X <p) = {^ X p] <(>, 

 on which the comment is that "the braces cannot be omitted 

 without ambiguity." The quaternion expression is (//Vp^xr, 

 where there is no chance of ambiguity, where everything is 

 perfectly straightforward, and where there is much greater com- 

 pactness in form. It seems to me that this last equation given 

 by Gibbs condemns his whole principle of notation. It shows 

 that one use of connecting symbols is to obscure the significance 

 of a transformation. . . . 



A beautiful example of virtually giving back with the left hand 

 what he has taken away with the right is furnished on p. 42 of 

 Gibbs's pamphlet. He writes: "On this account we may 

 regard the dyad as the most general form of product of two 

 vectors. We shall call it the indeterminate product." And 

 then he shows how to obtain a vector and a scalar "from a 

 dyadic by insertion of the sign of skew or direct multiplic'ation." 

 This is exquisite. From the operator a\ + 0fi + yv, he forms 

 ■ — heedless of his high toned scorn for the quaternion product — 

 the conception of the sum of three similar though more general 

 products, but quiets his conscience by calling them indetermin- 

 ate ! This sum of products then becomes by simple insertion 

 of dots and crosses the vector 



(py. = axx + ^X[jL + yxv, 

 and the scalar 



<ps~a,.K + ^.[j, + y.v. 

 Why, we naturally ask, is this indeterminate product welcomed 

 where the quaternion product is spurned ? 



The truth is the quaternion, or something like it, has to come 

 in ; and in it most assuredly does come when Gibbs proceeds 

 to treat the versor in dyadic form. The expression 

 {2j3j3 - 1} . {2aa - 1} represents in Gibbs's notation the 

 quaternion operator 



j3a( )oj3, or more simply q{ )q~'^. 

 The I is called an idem/actor and is simply unity. . . . 



There is something almost naive in the way in which Heavi- 

 side introduces the dyadic as if nothing like it was to be found 



1 We are surprised that s-> much etymological erudition should accept such 

 a monstrosity as parallelOpiped, 



- Gibbs calls the quantity 0.0- (which is simply Hamilton's (^o-) the direct 

 product of the dyadic </> and the vector a. The direct product of two vectors 

 is u.ff, and this Heaviside calls the scalar product. Similarly translating 

 the Gibbsian dialect, he speaks of^xr as bemg the '' scalar product of the 

 dyadic and the vector" — and gets a scalar product equal to a vector! 

 This "is most tolerable and not to b; endured." 



225, VOL. 47] 



