362 



Major P. A. MacMahon. A certain Class of [Nov. 23, 



Bathurst, River Gambia, lat. 13 28' N., long. 16 37' W. 



The station was on a large piece of open ground and near the 

 centre of McCarthy Square. All the observations taken were made 

 on April 20, 1893. 



Decimation at 8.16 A.M. L.M.T. = 18 50' W. 



Horizontal force at 8.44 = 0'30514 c.g.s. 



Dip at 8.17 = Needle 1, 28 43'"4. 



at 9.14 = 2, 2842'-l. 



III. " A certain Class of Generating Functions in the Theory 

 of Numbers." By Major P. A. MACMAHON, R.A., F.R.S. 

 Received November 3, 1893. 



(Abstract.) 



The present investigation arose from my ' c Memoir on the Compo- 

 sitions of Numbers, 5 ' recently read before the Royal Society and now 

 in course of publication in the ' Philosophical Transactions.' The 

 main theorem may be stated as follows : 



If X,, X 2 , . . . . , X M be linear functions of quantities &,, a? 2 , . . . . , a? 

 given, by the matricular relation 



(X 1} X 2 , , X) = ( a u 12 .. a ln ) (x lt # 2 , , ), 



a n a 22 



that portion of the algebraic fraction 



1 



