447 



pointic contact with any plane curve; but his method does not 

 appear to admit of direct extension to curves of higher orders than 

 conies having contact with any plane curve. The method here 

 suggested would admit of such extension ; and it may therefore be 

 said to contain potentially the solution of the general problem of 

 contact. But the labour of completely working out even the case 

 of a cubic of nine-pointic contact would be very great. The case of 

 the conic of five-pointic contact has been worked out, and the result 

 agrees with Mr. Cay ley's. 



XIII. "On the Calculus of Functions." By WILLIAM SPOT- 

 TISWOODE, Esq., M.A., F.R.S. Received October 9, 1861. 



(Abstract.) 



In a paper published in the ' Philosophical Transactions ' for 186 1, 

 p. 69, Mr. W. H. L. Russell has constructed systems of multiplica- 

 tion and division for functions of certain non-commutative symbols, 



viz. p=x, and ir=x , and has given the quotient and remainder 



after both "internal" and "external division" of the symbolical 

 function 



by the factor p^i(ir) + \^ (v). But in the case of 



he has given only the quotient and remainder after internal division 

 in the case of n=3. I have here investigated the general case. 

 The formulae will be best understood by reference to the memoir 

 itself. 



XIV. " On the Action of Hydriodic Acid upon Mannite." By 

 J. A. WANKLYN, Esq., and Dr. ERLENMEYER. Communi- 

 cated by Dr. FRANKLAND. Received October 24, 1861. 



Mannite may be regarded as a six-atomic alcohol, or better as 

 hydride of hexyl, in which six atoms of hydrogen are replaced by 

 six atoms of peroxide of hydrogen. 



Mannite= 6 H 8 (HO) 6 . 



2K2 



