498 Proceedings of Royal Society of Edinburgh. [sess. 
plane : — when all three are equal, </> becomes a mere magnification, 
and x is, of course, wholly undetermined. 
[When the roots of <£ are all real, we have 
S.a f3y xp = /qaS./3yp + /i 2 /3S.yap + /? s yS.a/3p . 
When two are imaginary we may preserve this form ; or, if we 
wish to express it in terms of real quantities only, we may write 
it as 
8. a f3y xp — /qaS./^yp + (h 2 f3 — h^y)S.yap + (h . 2 y + h 3 fi)&.a/3p , 
where the meanings of h 2 , h 3 , f3, y, are entirely changed. 
It is well to notice that the squares of these functions preserve 
the form, so that in the first 
S.a/3y x 2 p = /q 2 aS./3yp + hfftS.yap + h{yS.yap ; 
and in the second we have the value 
/q 2 aS. /5yp + (( hf — hf)f3 - 2h 2 h 3 y)S.yap -f ( (h 2 2 — hf) y + 2h 2 h 3 f3)$.aftp . 
Thus the square roots of such expressions may be obtained by 
inspection.] 
2. Had tire known factors been different in the two members, 
i.e., had the equation been 
^X^X*/' • • • ( r )' 
the same process would still have been applicable, though the 
result would have been very different. For a being a root of \j/, we 
have 
4>X a = 9X a 
as before. But we can no longer conclude from this anything 
further than that the scalar roots of i f/ must be the same as those 
of <f>, in order that the given equation may not be self-contradic- 
tory. Thus, if i f/ have three real roots, so must <jf>, and conversely. 
If this necessary condition be fulfilled, x is an y function which 
changes the directional roots of if/ into those of (f > . Its own scalar 
roots remain indefinite. 
3. Let the equation be 
( t>X=X<t > '- 
( 2 )- 
