ON THE SOLUTION OF CERTAIN TRIGONOMETRICAL EQUATIONS.* THERE are several trigonometrical equations whose roots. can be readily obtained by taking into consideration their connection with the general binomial equation whose last term is unity. If for example we have the equation cose+cos20+ cos 30+...+ cos(n-1) 0=0, we can find its roots by means of the equation x2 – 1 = 0. We know that the roots of this last equation are of the form ±1, coso±√(−1) sino, cos20± √(− 1) sin 24..... cos (n − 1) $ ± √(— 1) sin (n − 1) 4. Now as the equation wants the second term, the sum of its roots must be 0: and as the possible and impossible parts do not affect each other, they must be separately equal to 0; also, as the roots +1 and 1 destroy each other, there remains cos+cos24+cos3p+...cos (n − 1) p = 0. Comparing this with the original equation, we see that the former will be satisfied by making 0 = 4. But to determine 4, we have the equation or {cos+√(−1) sinø}TM – 1 = cos 2nd+√√(-1) sin2no=1. * Cambridge Mathematical Journal, Vol. I., p. 44. Whence, as the possible and impossible parts are in where m has any integer value from 0 to n, making in all n+1 values of p. But two of these cannot be taken as values of 0, as we excluded the roots +1 and − 1, which correspond to the values 0 and n of m. So that will be found from the equation where m has any value from 1 to n-1, making in all n-1 values of which answer the given equation. In exactly the same way we might shew how to solve the equation cose + cos 30+ cos 50+...+ cos(2n − 1) 0=0. For its roots would be deduced from the equation where m has any value from 0 to n-1, giving on the whole n values of 0 which satisfy the equation. Similarly, the equations and may 1+ cos 0+ cos 20+ cos30+...+cosne = 0, cose+cos30+...+ cos (2n − 1) 0 = 1, be solved by means of the equations where m has n values from 0 to n-1. The same method may be extended to other equations. For the roots of x2" − 1 = 0 being of the form cos+√√(−1) sino, if we multiply each by cosa + √√(−1) sina, it becomes cos (a +$) + √(− 1) sin (a +¤). But the sum of the roots will still remain equal to 0 when multiplied by cosa +√(−1) sina, and taking away the terms which destroy each other, there will remain cos (a+4)+cos (a +24) + cos (a +34) +..... consequently the equation +cos {a+ (n − 1) 6} = 0, cos (a + 0) + cos(a + 20) +...+ cos {a + (n − 1) 0} = 0, will be satisfied by the same values of as the first of the given equations; and similarly we might proceed with the others. EVOLUTE TO THE ELLIPSE.* THE equation to the evolute of the ellipse may be found very readily by considering it as the locus of the ultimate intersection of consecutive normals. be the equation to the ellipse. Then the equation to a normal passing through a point x, y, will be where a and B are the coordinates of the normal itself. To find the locus of the ultimate intersection of the normals, we must differentiate considering a and B as constant, x and y as variable. We then have from equations (1) and (2) b2 = 0...... (3) a2 λ (3)+(4) gives, on equating to 0 the coefficients of each |