600 
PEOEESSOE STLYESTEE ON THE EEAL 
where the sign ( — ) is, for greater convenience of writing, placed over instead of before 
the elements which it affects ; and so on in general a type of n elements, whether per- 
rotatory or trans-rotatory, will admit of n direct and n retrograde phases. 
If we count the number of variations of sign in the circulations of any phase of a 
per-rotatory type, this number will be the same for all the phases, and will be an even 
number ; this even number may be termed the variation-index of the type. 
So, again, if whatever be the original signs of the element in a trans-rotatory type, we 
count the number of variations in the circulation of any of its phases, this number also 
will be constant and will be odd, and this odd number may then be termed the variation- 
index of the type. 
(16) Let any phase be taken of a per-rotatory type, and out of such phase let any 
element be suppressed ; then we obtain a type one degree lower in the elements, which, 
if we please, we may consider as a trans-rotatory type, and such trans-rotatory type 
may be termed a derivative of the original per-rotatory one. 
In like manner any phase being taken of a trans-rotatory type, one element may be 
suppressed, and the reduced type treated as a per-rotatory one, and termed a derivative 
of the original trans-rotatory one. 
We may now enunciate the following important general proposition, viz. 
Any trans-rotatory type or any per-rotatory type whose variation-index is different 
from zero being given, a per-rotatory derivative of the one and a trans-rotatory deri- 
vative of the other may be found such that the variation-index of the derived types in 
either case shall be less by a unit than the variation-index of the types from which they 
are derived. 
Case (1). Let the given type be per-rotatory. Then by hypothesis, since it has some 
variations, we may find a phase of it beginning with + and ending with — , by which 
I mean beginning with an element that is positive and ending with one that is negative. 
This gives rise to two sub-cases. 
T, the phase, in question, will be +..... 4 — 
0, the phase in question, will be 4- • 
In either sub-case let the last sign be suppressed, and the result treated as a trans-rotatory 
type ; then T, 0 become respectively T', 0', where 
T' is + + 
and 
0' is 4- - 
and evidently the variation-index of T — variation-index of T'= number of changes of 
sign in H 1- less changes of sign in 4 =2 — 1=1; and again variation-index of 
0— variation-index of 0'= number of changes of sign in — (- less changes of sign in 
=1 — 0=1. Hence the theorem is proved for the case where the given type is 
per-rotatory. 
Case (2). Let the given type be trans-rotatory. 
Then, again, there must either be a phase of the form P, or one of the form O, where 
