On Methods of Election

I.—Take the well-known identity (w + zy = u?+2wz + z> = (w-zy + 4u>z (1). Now if we can transform 4wz into a square we shall have two square numbers whose sum is a square. This will be effected by taking w*=j», z = q\ for then 4wz = 4p*q = (2pqY and we have (2?+q>y={?-qy+(ipqy (2). See Mathematical Magazine, Vol. II., No. 5, p. 69. In (2) the values of p and q may be chosen at pleasure, but to have numbers that are prime to each other p and q must also be prime to each other and one odd and the other even. Examples.—1. Take p = 2, q = 1 ; then we find 3 + 4 = 5. 2, Take p = 3, q = 2 ; then we shall have