347 



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 y(Q) is a homo- 

 geneous function of the ?»"■ degree ; and let 



then we shall have 



S.NQ^nS/iQ). (I) 



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.d/{Q, Q', Q", ...)=S.NdQ+ S.N'dQ'+ S.N"dq"+&c.y 



we shall have 



S.NQ+S.N'Q:+ S.N"Q" + ... = nS.fiq,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 



AQ) = ItQR'QR" 



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



S.d/{Q) = S.RdQR'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 



