

THE LOGIC OF UELATIVES. 3C 



is, being denoted by an algebraic expression which denotes a Bcalar, a vet or, etc., in 

 geometry), are 



(157.) Form of a scalar : z(i -\- I) . 



(158.) Form of a vector : xi + y £ + z ^ — x ^ • 



(159.) Form of a versor : 



xfx S\-~\. , y(x A-lj , *f* ,\-i* , tt i\~* 



y \* 



<+I(M "V+Xl- 1 ) '**+ 



2 \# 



l v 



(160.) Form of zero : s* + **/ + "* + **' 



(161.) Scalar of?: S? = \{x + w)(» -f /) . 



(162.) Vector of q: Vq = i( x — »)* + *> + zk + i( w — *)' 



(163.) Tensor of q: Tq = \l~~xw — yz(i + I) . 



(164.) Conjugate of q: Kq = v»— J7 — *i + xl 



In order to exhibit the logical interpretations of these functions, let us consider I 

 universe of married monogamists, in which husband and wife always have country. 

 race, wealth, and virtue, in common. Let i denote "man that is -,» j "hush id 

 of — » k "wife of — " and /"woman that is — "; x negro that if -. // rich pei -n 



. e American that is -, and w thief that is -. Then, q being defined as 



that is 



above, the q's of any 



of so many individuals of that 



men or 



women-thieves together with all persons who are rich husbands or 



American wives of persons of this class. Then, 2S? denot by (100) all the 

 groes and besides all the thieves, while S, is the indefinite term which deao*. «. 

 the negroes and thieve, Now, those persons who are self-q's of any class (that is 

 the q's of themselves among that class) are rf + 1, add to these then spouses and 

 we have 2S* In general, let us term (/+*) the <• correspondent of - 



the double scalar of any quaternion relative, q, 



hich denotes 



„ a +o of«plfn'«=i of—." (To) 2 denotes all persons 



elf-q's, and, besides, all correspondents of self-< P ot . I q) i 



-i__l__ L ^.1 ~* ^^^Jin, self-o's mm all persons belonging to pairs of 



belonging to pairs of corresponding self-q 



ponding q's of each other 



iieapuiiuma 4 * « — ' ; o-eometrv to the logic of relatives, 



may take the following 



From this it folio 



• • i + w thP souare of a vector is a scalar, rrom uu* » — . 

 quaternion principle that the square ot a rf ^ 



««~> K. n k-7\ «*M*iia* vector, that the rich husbands ana 



by (157) yz + zh 



IX. 



50 





