ON MR COTTERILLS GONIOMETRICAL PROBLEM. 
Thus, as regards the two forms of S Q’ the identity to be verified may be 
written 
c (d 2 Ab + e 2 Ba — cFde) = f(a 2 De + If Ed — fGab). 
Proceeding to reduce the factor a 2 De+ b 2 Ed—fCab, if we first write herein f— eD + dE, 
it becomes 
which is 
a 2 Be + b 2 Ed — (eD + dE) Cab, 
= aBe (a — bC) + bEd(b — aC), 
and then writing C=ab-AB, we have a - bC = a(l - b 2 ) + bAB, =B(aB + bA), = Be; 
and, similarly, b - aC = Ac; whence the term is = c (aeBD + bdAE); or, in the equation 
to be verified, the right-hand side is =cf (aeBD + bdAE), and by a similar reduction, 
the left-hand side is found to have the same value. 
The paper contains various other interesting results.