480 



Mr. E. H. Glaisher. 



the effect per degree of the former being greater than the latter. The 

 shortest latency for the heated frog muscle occurs at 29— 30' 5°, and 

 closely precedes rigor. 



Muscles maintained for a considerable time at a given elevation or 

 depression of temperature, preserve a constant latency. 



Up to the present the gastrocnemius of the frog has been the only 

 muscle upon which we have experimented, and our chief object in 

 making this communication is to call attention to the changes in the 

 duration of the latency brought about by varying conditions of 

 stimulation. We hope to continue our observation by making similar 

 experiments on different muscles of various animals, and in a future 

 paper to enter more fully upon the changes occurring in the other 

 phases of the contraction, for which purpose further use will be made 

 of the tables we have the honour of laying before the Royal Society. 



"Formulas for sn 8u, cn Su, dn 8u, in terms of sn w." By 

 Ernest H. Glaisher, B.A., Trinity College, Cambridge. 

 Communicated by J. W. L. Glaisher, M.A., F.R.S. Re- 

 ceived and Read June 16, 1881. 



§ 1. In Grunert's "Archiv der Mathematik und Physik," vol. xxxvi 

 (1861), pp. 125-176, Baehr has given the formulas for snnu, cn?ut, 

 dn nu, in terms of sn u for the cases n = 2, 3, 4, 5, 6, 7. These ex- 

 pressions 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 formulae 

 in the case of n=S. These were deduced from the formulae for the 

 case n = 4< in the following manner. 



We have 



2sn Asu cn 4u dn 4u 



sn 8%: 



cn mi- 



1— & 2 sn 2 4^ ' 



l-2sn 2 4^+Fsn 4 4^ 

 1— & 2 sn 2 4<u 



dn gf f - 1 ~ 2h * Sn2 4/11 + W Sn4 4u 

 1 — & 2 sn 2 4zt 



and therefore, denoting the numerators of sn4%, cn4^, dn 4iu, and 

 their common denominator by P, Q, R, S respectively, so that 



sn 4u = ^-, cn4w=— , dn4%=— , 

 S S S 



