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§ 1. In Grunert’s “Archiv der Mathematik und Physik,” vol. xxxvi (1861), pp. 125-176, Baehr has given the formulæ for snnu , cnnu , dnnu , in terms of snu . for the casesn = 2, 3, 4, 5, 6, 7. These expressions are reproduced by Cayley in a tabular form in his “Treatise on Elliptic Functions,” Art. 109 (pp. 80-85). The object of the present paper is to give the corresponding formulæ in the case ofn = 8. These were deduced from the formulæ for the casen = 4 in the following manner.

Formulæ for sn 8 u , cn 8 u , dn 8 u , in terms of sn u