Formula fo7' the Summation of Infinite Series. 121 

 Oxide of formyl = Cg H^ O = Fo O 



\ C2H4O2 FoO + H,0 



Hydrated oxide 

 Methylene aldehyd 



Formic acid Cg Hg O3 Fo O3 



Chloroform C^U^C\q Fo Clg 



Bromoform Cg Hg Brg Fo Bg 



Formal C4H10O3 3 Mtl. O + Fo O3 



EL'b5ri;r;;;;7i:"i;} ^^^^-^^ sMti.o+Foo 



Chloral and water ... C4 H2 Og CIg + H^ O 

 giving chloroform ... Cg Hg Clg + formic acid Cg Hg O3. 

 By the action of water and the chlorine body from pyroxy- 

 lic spirit, that is, from Cg H4 Og Clg + Hg O, there should re- 

 sult deutochloride of formyl, C3 H3 Clg, and formous acid, 

 C3 H3 O3; but I am not yet satisfied of the Composition of the 

 bodies so formed. 



XXVI. Investigation of Formulce for the Summation of certain 

 - Classes of Infinite Series, By J. R. Young, Esq.^ Professor 

 of Mathematics in Belfast College,^ 



T^HE general formulae established in the present paper are, 

 -■- I believe, new ; and as they are extensively applicable, 

 and involve but a very moderate amount of numerical labour, 

 they may, perhaps, be found useful on many occasions. The 

 investigation, which is very simple, is as follows : 

 Since any series of fractions of the form 



1 .... (A.) 



n [ri + p){n + 2p ... {n + mp) ' ' ' ' ^ '^ 



is equal to — , the difference between a series of the form 

 ^ m p 



n(n+p)... [n + {m-- }) p] ^ '^ 



and another of the form 



(n +p) (71 + 2^) ... {n + mp) • • • • V •>' 

 it follows that a series of the form A^ will be equal to ^ o » 



2 

 the sum of two series of the forms B^ and C^ minus — o — oj a se- 



m^ p^ 



ries of the form B C. Now it is obvious that the two series 

 whose general terms are B' and C% will, together, be equal 



• Communicated by the Author. 

 27iird Series, Vol. 10. No. 59. Feb, 1837. R 



