The aim of the following articles is to meet certain difficulties and defects in the accepted book-work of the “Higher Algebra” of school work, especially in connection with infinite series.

The equation of the curve of order 14, on which lie the points of contact of the 28 bitangents of a quartic curve, was first given by Hesse (Crelle’s Journal, Vols. 36, 40, 41). A very concise form of the equation in the Clebsch-Aronhold symbols was obtained by Dersch (Math. Ann., Bd. 7). The method adopted in his Memoir was practically that given by Salmon (Phil. Mag., Oct. 1858), and followed by Cayley (Phil. Trans., 1859).

Huygens’ approximation to the length of a circular arc is , where l0 is the chord of the arc and l1 is the chord of half the arc. This result is most easily obtained from the series for sin and sin by neglecting powers of beyond the third, L being the true length of the arc and R the radius of the circle. For arcs less than 45° Huygens’ rule is very accurate, the error for an arc of 30° being less than 1 in 300,000.