I.—ON THE THEORY OF DOUBLE PRODUCTS AND STRAINS IN HYPERSPACE. 
By Epwin BipweL_L WILSON. 
Part I.—On A MULTIPLE ALGEBRA AS SET FORTH BY GIBBS. 
Introduction. 
1. History and apologies.—During the academic year 1899-1900, 
I followed a course of lectures on vector analysis by the late pro- 
fessor J. Willard Gibbs. These lectures, with some alterations as 
to additions and retrenchments, I published with his permission in 
the year 1901.1 Previously to this, during the academic year 
1900-1901, I had the opportunity of following a short course of 
twenty-five or thirty lectures on multiple algebra under the same 
master. These lectures have never been published, and very likely 
never can be. My own notes were very meager and most of them 
have unfortunately been lost. There remains, however, a set of 
notes taken by the late professor G. P. Starkweather of Yale Uni- 
versity. These notes were of a similar course given during the 
academic year 1895-1896 ; and as nearly as my memory and the 
fragments of my own notes will serve me, it appears that Gibbs 
had not materially changed his course during the intervening five 
years. It is therefore from Starkweather’s notes that the follow- 
ing articles on multiple algebra are drawn, with the practical cer- 
tainty that the presentation is essentially that followed by Gibbs 
in his later years. 
Gibbs’s course on multiple algebra, following immediately upon 
his lectures on vector analysis, began with a discussion of quaternions 
defined as the sum of the vector product and the negative of the 
scalar product of two vectors, and then turned to the geometric 

1 Vector Analysis, a textbook for the use of students of mathematics 
and physics, founded upon the lectures of J. Willard Gibbs. Yale Bicen- 
tennial Publications, Charles Scribner’s Sons, New York, and Edwin 
Arnold, London. xviii +436 pp. A second edition, merely corrected, 
has recently appeared. References in the text to Vector Analysis are 
to either edition. Gibbs’s original pamphlet on vector analysis, printed 
privately in 1881—1884 but never published, may now be found reprinted 
in the collection of The Scientific Papers of J. Willard Gibbs, volume 2, 
pp. 17-90. 
Trans. Conn. Acap., Vol. XIV. 1 SEPTEMBER, 1908. 
