McAULAY'S "UTILITY OF QUATERNIONS" 277 



its power, he snatches up the magnificent weapon which Hamilton tenders to all, and 

 at once dashes off to the jungle on the quest of big game. Others, more cautious or 

 perhaps more captious, meanwhile sit pondering gravely on the fancied imperfections 

 of the arm ; and endeavour to convince a bewildered public (if they cannot convince 

 themselves) that, like the Highlander's musket, it requires to be treated to a brand-new 

 stock, lock, and barrel, of their own devising, before it can be safely regarded as fit 

 for service. "Non his juventas orta parentibus...." What could be looked for from 

 the pupils of a school like that? 



Mr McAulay himself has introduced one or two rather startling innovations. But 

 he retains intact all the exquisitely designed Hamiltonian machinery, while sedulously 

 oiling it, and here and there substituting a rolling for a sliding contact, or introducing 

 a lignum vitae bearing.... 



The " startling innovations," however, as we called them above, are unquestionably 

 Mr McAulay's own and he has certainly gone far to justify their introduction. He 

 has employed the sure tests of ready applicability and extreme utility, and these have 

 been well borne. Objections based upon mere unwontedness or even awkwardness 

 of appearance must of course yield when such important advantages as these (if they 

 be otherwise unattainable) are secured ; but it certainly requires a considerable 

 mental wrench to accustom ourselves to the use of 



as an equivalent for the familiar expression 



dX 



dx' 



If this be conceded, however, it is virtually all that Mr McAulay demands of 

 us, and we are free to adopt his system ....... A single example, of a very simple 



character, must suffice. Thus in the strain of a homogeneous isotropic solid, due to 

 external potential u, we have for the strain-function <f> (when there are no molecular 

 couples) the equation 



= o 



which (in virtue of the property of a, already spoken of) is equivalent to three 

 independent scalar conditions. Suppose we wish to express these, without the a, in 

 the form of one vector condition. Mr McAulay boldly writes the first term as 



5.00'jVj or rather as S.a^V, 



for in so simple a case the suffixes are not required, and the strain-function is self- 

 conjugate under the restriction above. Then, at once, the property of a shows us that 



<V + V = o, 



which is the vector equation required. Here it is obvious that, in the usual order of 

 writing, 



This simple example shows the nature of the gain which Mr McAulay's method 



