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IX.—On the Establishment of the Elementary Principles of Quaternions on an 
Analytical Basis. 
by Prof. Tarr. 
By Gustav Parr, Docteur és-sciences. 
Communicated 
(Read March 2, 1874.) 
INDEX TO PARAGRAPHS. 
PAGE 
PAGE 
§ 1. Definitions as to addition, &c., c AG § 5. Generalisation of the rule of the distributive 
§ 2. Addition, Subtraction of vectors and qua- law in its application to vector and qua- 
ternions ; Expression of vectors, . a JUrAS} ternion products, . : ; ; oye) 
§ 3. Multiplication of two vectors, one by § 6. Associative property in multiplication, . 198 
another, : : : ; : so § 7. Division, and some other results, . . 200 
§ 4. Determination of values of constants; Dis- 
cussion of results, . f : : ss 
The extension which is required to be given to the meaning of algebraic 
addition and subtraction, for the purpose of representing a vector by a poly- 
nomial expression (by the algebraic sum, namely, of components differing in 
direction from one another), renders it necessary to investigate into the question 
of what the geometrical signification of the result of the other elementary 
operations of algebra will become when these operations are performed on those 
polynomial expressions ? 
- The question, taken in its widest generality, is indeterminate; but we 
render it determinate by the introduction, step by step, of conditions and of 
limitations, and of rules, the leading condition being—that whatever be the 
result of the multiplication by the distributive law of two polynomial expres- 
sions of vectors into one another, that result shall be independent of the par- 
ticular mode of composition of the vector factors. Those conditions and those 
propositions agreed to by definition, and the further condition, that the multipli- 
cation of two conjugate quaternions shall be made according to the distributive 
law applied to the two heterogeneous elements of the quaternions, give us as 
consequences the known relations between the unit vectors of a triple rec- 
tangular system, and the value of the square of a unit vector. 
It may be remarked that the method followed in this paper is proceeding 
in a direction inverse to that which is commonly followed in the demonstration 
of the distributive property of vector multiplication. If there be an advantage 
in the present method, it may be the one that, by interverting the roles of the 
antecedent and the consequent of the proposition, it provides itself with a 
broader basis for deduction. 
The associative property in multiplication is then easily established, and 
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