lib* Dr. L. Silberstein : Farther Contributions 



publishing o£ that book, no reasonably simple multiplicative 

 operation, independent of congruence and of parallels, pre- 

 sented itself to the writer. During the reading of the proof- 

 sheets, however, a not too-complicated non-metrical generali- 

 zation of the scalar product, and then also of the vector product, 

 of two vectors spontaneously suggested itself. And since, for 

 technical reasons, this theory of vector multiplication came too 

 late to be included in the said volume, it seemed worth while 

 to publish it separately in the present paper which, to facilitate 

 its perusal, will be kept uniform with the treatment (in- 

 cluding nomenclature and symbolism) adopted in ; Projective 

 Vector Algebra/ In order to avoid covering unnecessarily 

 many pages of this Journal, the definitions and theorems 

 concerning vector-equality and addition with their immediate 

 consequences will not be repeated here. As often as the 

 need arises the reader will be referred to the said book, 

 which he will be assumed to have read or at least perused. 

 It will be shortly referred to as P. V. A., with the section 

 number whenever necessary. 



Here but one more remark to justify this paper. As was said a 

 moment ago vector addition is sufficient by itself to cover the field of 

 projective geometry, i. e. to treat its problems algebraically. Thus, 

 strictly speaking, vector " multiplication " would be a superfluous 

 operation. So, in fact, it is in a certain sense. But " multiplication " 

 enables us to treat many of those problems in a considerably simpler 

 manner. The multiplication of vectors by vectors, especially if it is so 

 defined as to be distributive, is a powerful instrument of algebraical 

 investigation, and therefore a very desirable supplement of addition. 



2. Standard or Unit Conic. — It will be well to treat first 

 the relations in a plane. The passage to three-dimensional 

 space (section 6) will offer no difficulties. 



In P.V.A. the concept of equality was applied to such 

 pairs of vectors only as have different origins, while vectors 

 emanating from a common origin were * complete strangers 

 to one another, no relation between them having been de- 

 fined, unless one counts their distinctness. This distinctness 

 consisted in their being on distinct lines (straights), and 

 therefore in having different " termini ,J or 2-points, their 

 crosses with the conventionally fixed straight, the T-line. 

 In more familiar language, our co-initial vectors differ in 

 " direction, " and since their "lengths''' or " sizes " as tested 

 by "rigid transferers'''' are entirely foreign to our circle of 

 ideas, all these vectors have among themselves no feature in 

 common. 



* With the only exception of a vector OA and its negative OA' =. 

 -(OA). 



