34T 



by weight, of 1 of manganese, 100 of sulphur, 1000 of sand ; 

 and on exposing ten grains of the mixed substances to a red 

 heat on a slip of platina, the mixture afforded, with pure 

 water, a solution of sulphate of manganese, which was ren- 

 dered turbid by prussiate of potash. The method is applicable 

 to organic substances, both vegetable and animal; but the 

 Professor's experiments on such bodies are still in progress. 



The Rev. Charles Graves communicated a general theo- 

 rem in the Calculus of Quaternions : 



Let Q be a variable quaternion, of which /"(Q) is a homo- 

 geneous function of the »"* degree; and let 



then we shall have 



S.NQ = nSf{Q). (1) 



And, more generally, if Q, Q', Q", &c., be any number of 

 variable quaternions, of which y(Q, Q', Q", . . . .) is a homo- 

 geneous function of the w"' degree ; and if 



S.dfi Q, Q', Q", ...)=S.NdQ+ S.N'dQ; + S.N"dQ" + &c., 



we shall have 



S.Na+ S.N'Q:+ S.N"q' + ... = nS.f{Q,Q:, Q",...) (2) 



Let us first establish the theorem in a particular case, and 

 it will afterwards be easy to show that the proof admits of ex- 

 tension to the most general one. Suppose, therefore, that 



f(Q) = RQRaR" 



where R, R', R" are any constant quaternions ; then 



S.dfiQ) = S.RdQ_R'QR"+ S.RQR'dQR"; 



or, in virtue of the rule which permits us to execute a cyclic 

 permutation on factors under the scalar sign, 



VOL. IV. 2 D 



k 



