or (siDce 



and on Go-resolvents. 191 



[Df u, . cc^ - j^— ^^ Ij 1^ \^^ ^ + 7i-aj fc-. u,,„ . x^-^ = 



(xxxiv) 



Next, in (xxxiv), give to r all successive values from zero 

 to infinity and add the results : we find, since 



Uq + Ui X + 11.2 x^ + &c. ad. inf. = u, 



[Z)]n,,_f.!!l^_l 



^:; — ^— + n - a x"" u = (xxx v) 



Hence if in (xx^di) we replace a by i' and, consequently, 

 make the same change in (xxxv) that last equation becomes 



[Dfy^-y —^ Ij [ \, ^ - n-T^^ x-u = - (vi) 



and thus the theorem announced at the commencement of 

 Article 23 is established. The constant r which replaces 

 the constant a will not of course be confounded with the 

 variable integer 'r used in the investigations immediately 

 preceding. 



Brisbane, Queensland, Australia, 

 March 16, 1866. 



Abstract of the Addendum to Chief Justice Cockle's 

 Paper "On Differential Equations," &c., communi- 

 cated to Sir Redmomd Baery. 



In an Addendum to the above paper, dated April 9th- 

 10th, 1866, Chief Justice Cockle gives a co-resolvent of a 

 third form of trinomial algebraic ec|uation undiscussed by 

 Mi\ Harley and the lamented Boole. After obtaining this 

 last co-resolvent by means of Laplace's theorem, the Chief 

 Justice gives a general theorem for the development of 

 functions of two mdependent quantities. Lagrange's the- 

 orem and Laplace's theorem are particular cases of this 

 general theorem. 



