PEOFESSOE C. J. JOLY ON QUATEENIOYS AND PEOJECTIVE GEOMETEY. 257 
vector function into a function representing a pure strain following or followed by 
a rotation. 
Multiply any function into its conjugate, and write 
ff = 
(191), 
where F is the self-conjugate function whose double operation is equivalent to the 
operation of the self-conjugate function jj'' (Art. 36). 
Introducing a new linear function g and its conjugate g’ defined by the relations 
/=% / = or p=:F-y,p'=/F-i 
it appears that this function is the inverse of its conjugate, for 
(192), 
g'g^\-gg> .(I 93 ) 
is a consequence of tlie equations of definition. 
The geometrical property of this new function is, that j^oints conjugate to the unit 
sphere remain conjugate after transformation. 
For if 
=: 0, then ^gpgq — ^pg'gq =1 0 .(194). 
In particular the unit sphere is converted into itself by the transformation. 
This transformation is orthogonal, points and planes being transformed by the 
same function (Art. 4). 
48. On counting the constants, it appears that an arbitrary function f cannot be 
reduced to the product of a self-conjugate function and a conical rotator 
R=r( )?'-!, Pd = = r"i( ) r.(195), 
there being sixteen constants in /, ten in F, and three in R, 
In order to determine tlie conditions, observe that by the last aidlcle 
V'^=ff if /= FR, and RR'= 1.(196). 
Now I say that if a scalar remains a scalar after the operation of R, the function 
is a conical rotator. For then 
SR'/) = SpR (1) = 0 .(197), 
and therefore Rp or Rp remains a vector whatever vector p may be; and, moreover, 
the angle between any two vectors is unaltered by the transformation.* 
Thus the condition required is simply 
/(I) = F (1), where T- = ff . (198); 
and when the reduction is possible it is generally determinate. 
* Compare the Appendix to the New Edition of Hamilton’s ‘ Elements,’ vol. ii., p. 366. 
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