296 
ON MR COTTERILLS GONIOMETRICAL PROBLEM. 
[685 
arbitrary, and each of the remaining angles is then determinate save as to an even 
multiple of 7T. And it may be remarked that these angles a, b, d, e may represent 
the inclinations of any four lines to a fifth line, and that the remaining angles are 
then at once obtained, as in the figure. The small roman letters are here used to 
denote as well angles as points, being so placed as to show what the angles are 
which they respectively denote ; the points *, * are constructed as the intersections of 
the lines ac, be by the circle circumscribed about fxy, and the angle z is the angle 
which the points *, * subtend at x or y. It will be observed that the sum of the 
three angles in a line or column is in each case = 7r. 
But this in passing : the analytical theorem is, first, we can form with the sines 
and cosines of the angles in any two lines or columns a function S presenting itself 
under two distinct forms, which are in fact equal in value, or say S is a symmetrical 
function of the two lines or columns, viz. for the first and second lines this is 
^(d e’ f )= d2Ahc + e2Bca +f iCab 
= a 2 Def + b‘ 2 Efd + c 2 Fde, 
where, as already mentioned, a, A denote sin a, cosa, and so for the other letters. 
Secondly, if to the 5 of any two lines or columns we add twice the product of 
the six sines, we obtain a sum M which has the same value from whichever two 
lines or columns we obtain it ; or, say M is a symmetrical function of the matrix of 
the nine angles. Thus 
which is one of a system of six forms each of which (on account of the two forms 
of the S contained in it) may be regarded as a double form, and the twelve values 
are all of them equal. There are, moreover, 15 other forms, of M, viz. 3 line-forms, 
such as 
bedx + caey + abfz (belongs to line a, b, c), 
3 column-forms, such as 
dxbc + xaef+ adyz (belongs to column a, d, x), 
and 9 term-forms, such as 
e 2 z 2 + / 2 y 2 -1- 2efyzA (belongs to term a), 
and the 12+15, =27 values are all equal. 
The several identities can of course be verified by means of the relations between 
the nine angles, or rather the derived sine- and cosine-relations 
G = ab — AB,