Mr. R. Phillips on the Electricities of Steam, 503 



ternion expresses, in this system, what Sir W. Hamilton has 

 called a. geometrical quotient^ or the operation which must be 

 performed on one directed line to make it coincide in length 

 and direction with another directed line. If a, a' be two 

 such lines, /, w, w, /', m', «' their direction cosines, and jot the 

 arithmetical ratio of the length of a to that of a', then the geo- 

 metrical quotient — is expressed, as it is easy to see, by the 



quaternion 



qz=^u{ll' + mm' + 7m' + i{m'n — mn') +J{n'l — nl') + k{l'm — Im') }, 



a' 

 and — by another quaternion obtained from q by writing jw.~* 



instead of ju,, and changing the signs of the three terms affected 

 by /, J, k. According to this view, since e, J, k are symbols 

 of operation and not of quantity, there is nothing in the least 

 strange in their not being commutative symbols, for we are 

 familiar with non- commutative operatiojis in other parts of 

 analysis. 1 am not without a hope that this, with other con- 

 siderations suggested by the preceding investigations, may 

 tend to obviate any prejudice which may be felt against the 

 theory of quaternions, as containing something arbitrary, 

 mysterious, opposed to common notions, and incapable of in- 

 terpretation. I trust it has sufficiently appeared that the 

 theory may be completely divested of any such characteristics. 

 At the same time I am far from thinking that they really 

 apply, in any sense implying an objection, to Sir W. Ha- 

 milton's mode of considering the subject. On the contrary, 

 that mode possesses, in my estimation, besides its undoubted 

 practical utility, the attraction of suggesting a novel and very 

 interesting subject of inquiry; the properties, namely, of sets 

 or sums o\l heterogeneous quantities. This, however, is a topic 

 upon which I cannot here attempt to enter. 

 Oxford, May 18, 1850. 



LXIII. On the Electricities of Steam. 



By Reuben Phillips, Esq, 



[Continued from p. 108.] 



94-. A COMMON laboratory brass tripod stand with 



"^^ straight legs was insulated, by putting each leg into 



a glass tube previously closed at one end by the blowpipe; 



the glass tubes were securely jammed on by means of worsted, 



and a small piece of cork was placed in the bottom of 



each tube, to prevent the metal bearing on the glass; the 



