This is a supplement to the paper *All triples for orthogonal designs of order 40*,
Discrete Math, 308 (2008), pages 2796 - 2801, which appears in a special volume of Discrete Math honouring Jennifer Seberry on her 60th birthday.

All OD(40;*s*,*t*,*u*) where *s*, *t*, *u* are positive and *s*+*t*+*u* is at most 40
are presented.
By definition this matrix is an order 40 matrix, OD, with entries from the set
{**a**,**-b**,**b**,**-b**,**c**,**-c**,**0**},
such that OD times its transpose is
*s***a**^{2}+*t***b**^{2}+*u***c**^{2} times the identity matrix.

- The first variable, which occurs
*s*times per row, is represented as**a**or**A = -a**. These are displayed in the image as light or dark yellow respectively. - The second variable, which occurs
*t*times per row, is represented as**b**or**B = -b**. These are displayed in the image as light or dark red respectively. - The third variable, which occurs
*u*times per row, is represented as**c**or**C = -c**. These are displayed in the image as light or dark blue respectively. **0**is displayed as white in the images.

The orthogonal designs are constructed from either 8 or 4 matrices. The first rows of these component matrices are displayed as follows:

- They are for an amicable set of 8 cyclic matrices given in order
A
_{1}, A_{2}, ..., A_{8}where A_{1}is matched with A_{2}, etc. which are used in the Kharaghani array, K, of the paper. - They are for a set of 4 negacyclic [right negative shift] matrices A
_{1}, A_{2}, A_{3}, A_{4}which are used in a Goethals-Seidel array, G, of the paper.

In the first case, if the matching is special, then infinite classes of designs are constructible from these.
See Theorem 4 in the paper of H. Kharaghani: *Arrays for orthogonal designs*, J. Combin. Des., 8 (2000), 127-130.
Select *special matchings* to have them displayed.

*W.H. Holzmann, H. Kharaghani & B. Tayfeh-Rezaie, June 2003*