78 Mr Levy on ilte modes of ' Notation 
first, and w" that corresponding to the second, we must have 
+ 2 
+ ^ ^ In 
and 
hence 7i" = 
4 
, + l 
This expression shews, that when n! is positive, greater than 
1 and less than 4, the value of n" is positive, and less than 1 ; 
that is to say, that, to a rhomboid produced by a decrement in 
breadth of less than four rows on the superior angle of the pri- 
mitive, corresponds a rhomboid measuring exactly the same 
angle, produced by a decrement in height on the same superior 
angle. If n' is greater than 4, n” is negative ; that is to say, 
that, to the rhomboid produced by a decrement in breadth on 
the superior angle of more than four rows, corresponds another 
rhomboid measuring the same angle, produced by a decrement 
in height on the inferior angle of the primitive. All these results, 
as well as those referring to negative values of n\ might easily be 
verified by geometrical considerations. There is no difficulty, 
by means of the preceding formulae, to determine the index of 
a secondary rhomboid, with respect to another secondary rhom- 
boid assumed as the primitive. If the incidences of the two se- 
condary rhomboids are given, the formula (B) will immediately 
give the quantity required. If the indices n', n" of these rhom- 
boids, with respect to the primitive, are given, we shall have to 
determine the index n'" of the second. With respect to the 
first, the following proportion, 
(ji (n -f- in'" 4 - • in'" 
which gives for n"' the two following values : 
7X' 
n' n'' 4 - -2 
n'" :=. 
— {n' n" — 4nf 4- Bnf* — S) 
9.71' 7l'' 4- 71' 4- — 4 ' 
I shall now advert to the dodecahedrons derived from a rhom- 
boid. In the forms of this kind derived from a rhomboid, 
three of the pyramidal edges meeting at the same summit, are 
in the same vertical planes with the axis, as the three oblique 
diagonals of the primitive, and the three others in the same ver- 
tical planes, at the superior edges. Let ABCD, Fig. be the 
primitive rhomboid, and from one of the summits A draw AE, 
parallel to two of the pyramidal edges of a dodecahedron 
derived from it. These lines, in agreement to the preceding 
