AND IMAGINARY ROOTS OF EQUATIONS. 
599 
Certain preliminary properties of circulation introducing some new notions of polarity 
must be first established, by way of Lemmas to the proof in question. 
By a type let us understand a succession of symbols of any subject matter whatever 
susceptible of receiving the signs H — , or any suchlike indications of opposite polarity. 
Let a,b,c , ... I , k, l be any such type, where the elements a,b,c , ... may be regarded 
either as points in a line or rays in a pencil affected respectively with the signs of + 
and — . 
Then by a per-rotatory circulation of such type, I mean the act of passing from the 
first element to the second, from the second to the third, &c., from the last but one to 
the last, and from the last to the first. 
By a trans-rotatory circulation of the same, I mean the act of passing from the first 
to the second, the second to the third, &c., from the last but one to the last, and from 
the last to the first, with its sign reversed. 
A type considered subject to per-rotatory circulation may be termed a Per-rotatory 
Type ; one subject to the other sort of circulation, a Trans-rotatory Type. 
If #, b, c, d, e be a per-rotatory type, its direct phases are 
a , b, c, d, e, 
b , c , d, e, a , 
c, d, e, a , b , 
d, e, a, b, c, 
e, a, b, c , d, 
and its retrograde phases 
a, e, d, c, b , 
e, d , c, b, a , 
d, c, b, a, e, 
c, b, a , e, d , 
b, a , e , d, c. 
If, on the other hand, a , b, c, d, e be a trans-rotatory type, its direct phases will be 
a, b , c, d, e, 
b , c , d , e , «, 
d, e, a, b , 
d, e, a, b , c, 
e, a , b, c, d, 
and its retrograde phases 
a, e , d , c, b, 
e, d, e , b, a, 
d , c , £>, a , 0, 
c, «, 0, d, 
e . d c. 
