of Edinburgh, Session 1862-63. 115 
kinetic energy than belongs to any other motion fulfilling the given 
velocity conditions. 
6. Note on a Quaternion Transformation. By Prof. Tait. 
The following paper gives an idea of the nature of the physical 
applications of quaternions to which I referred in a previous note 
(Proceedings, April 1862), but which other avocations have, as yet, 
prevented me from developing into a form and bulk suitable for 
publication in the Society’s Transactions. The equations I now 
give form the basis of the investigations in question, which I hope 
to present to the Society in detail on some future occasion. 
I. If the vector of any point be denoted by 
p^ix + jiy + hz, (1) 
there are many interesting and important transformations depend- 
ing upon the effects of the quaternion operator 
^ .d .d d 
+ 
upon various functions of p. When the function of p is a scalar, 
the effect of <i is to give the vector of most rapid increase. Its 
effect on a vector function is indicated briefly in my former note. 
II. I shall commence with one or two very simple examples, 
which are not only interesting, but, as we shall see, very useful in 
subsequent transformations. 
and, of course. 
np 
(Tpy 
(Tp)”+2 ■■■ 
.(5) 
(5)' 
'^Tp~ V V 
whence, 
( 6 ) 
