444 Mr. J. W. L. Glaisher. [June 16, 



Iron, cobalt, nickel, copper, magnesium, and potassium have, up to 

 the present, given negative results, but will be examined again. 



XXI. "Formulae for mSu, cn8w, dn 8w, in terms of snw." By 

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

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

 ceived June 16, 1881. 



(Abstract.) 



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

 (1861), pp. 125-176, Baehr lias given the formulae for sn nu, cn nu, 

 dnnu in terms of snu for the cases n=2, 3, 4, 5, 6, 7. These expres- 

 sions are reproduced in a tabular form in Cayley's " Treatise on 

 Elliptic Functions," Art. 109. 



The present paper contains the corresponding formulae for the case 

 of w=8. Denoting the numerators and common denominator of 

 sn 4u, cn 4<u, dn 4<u by P, Q, R, S respectively, then the numerators 

 and common denominator of sn 8u, cn 8u, dn 8u are respectively 

 2PQRS, S 4 -2P 2 S 2 + 7<; 2 P 4 , S 4 -2k 2 P 2 S 2 + & 2 P 4 , S 4 -M> 4 ; and the 

 paper contains the values of these quantities and also of P 2 , S 2 , P 4 , 

 S 4 in terms of sn u, arranged in a tabular form. 



XXII. " On Riccati's Equation and its Transformations, and on 

 some Definite Integrals which satisfy them." By J. W. L. 

 Glaisher, M.A., F.R.S., Fellow of Trinity College, Cam- 

 bridge. Received June 16, 1881. 



(Abstract.) 



The memoir relates chiefly to the different forms of the particular 

 integrals of the differential equation 



g:-A = KH±)„ (1), 



and to the evaluation of certain definite integrals which are connected 

 with this equation. 



Transforming (1) by assuming u=x~Pv and putting 2p=n — 1, it 

 becomes 



d 2 v n — ldv 



dx 2 x dx 



a?v=0 (2), 



and this may be transformed into Riccati's equation, 



