Note on the Derivation of Elliptic Function Formulae from Confocal Conics
Proceedings of the London Mathematical Society
Bead Nov. 9th, 1882.] 1. The ordinary formulae for sn(w+«) and cn(u+v) may be derived from the following simple construction, viz., draw the tangent, as in the accompanying figure, at any point P u on the inner of two confocal ellipses so as to intersect the outer curve in two points Qu-e> Q u *9* say; through each of the latter draw a confocal hyperbola cutting the inner ellipse in P tt _ c and P M+ ,; then the elliptic injtegral of the first kind, say u-v, becomes u as we pass from the point
... ass from the point P u . p to P B , and similarly P o + , corresponds to the function u-\-v, the period in each case being v, which is a constant quantity when we take two fixed ellipses. 2. The equation of the outer ellipse in terms of the foregoing elliptic parameter v may thus be found:-Taking the eccentricity of the inner curve to be k and the semimajor axis as unity, the equation of this carve is z*+ %~ = 1, where tf+Jc* = 1; also, if we draw a tangent from B, an extremity of the minor axis of the outer ellipse, to the inner curve, and P , be the point of contact, then the period v = I . -, ft being the J o vl-A^sinV eccentric angle of P o , or fi = am v. Hence we may write for the equation of the outer ellipse -=-=-+ *£J = -?-• r dn'v k* cn l « The above construction enables us at once to form a quadratic whose roots sn(u+v), sn(it-v) are given in terms of snu and snv.