MINDING’S THEOREM. 683 
Most of the preceding formulz may be looked upon as results of the elimi- 
nation of the function y from these equations. This forms probably the most 
important feature of such investigations, so far at least as the quaternion 
calculus is concerned. 
I employed the equation (5) as the basis of an investigation, one or two of 
whose results were communicated last session to the Society.* 1 will now give 
the main features of that investigation. 
vit 
12. It is evident from (5”’) that the vector-perpendicular from the origin on 
the central axis parallel to x8’ is expressed by 
T=ya'xX8" . 
But there is an infinite number of values of y for which U7 is a given versor. 
Hence the problem ;—to find the maximum and minimum value of Tr, when 
Uris given—z.e., to jind the surface bounding the region which ts filled with the 
Seet of perpendiculars on central axes. 
We have 
Tr=—S. xBox 8’, 
0=T78.x8'Ur. 
Hence 
0=S8.x6'ax6’, 
0=S.x8'Ur. 
But as Tf’ is constant 
0=S8.x8yB’. 
These three equations give at sight 
(w at ux’ =o Wr, 
where w, w’ are unknown scalars. Operate by 8.x@’ and we have 
—T’;—u=0, 
so that 
St(o +7) '7=0. 
This differs from the equation of FRESNEL’S wave-surface only in having 
. : LY : 
w+7° instead of a+7~° (ie, Tr for i) , and denotes therefore the reciprocal 
* Proc. Roy. Soc. Edin., 1879, p. 200. 
VOL. XXIX, PART II. 8A 
