284- Mr. S. Roberts on the Development of 



affect the temperature of the fluid? And does not the ma- 

 thematical investigation contained in this communication suf- 

 ficiently account for the observed velocity of sound on purely 

 hydrodynamical principles ? 



Cambridge Observatory, 

 March 17, 1848. 



XLI. On the Development of Functions of the form Y{z-\-w). 

 By Samuel Roberts, B.A. Manchester New College*. 



IT is the object of the present paper to derive Laplace's 

 development of ^u, where u=F{z + ai^if), by a simple 

 combination of differential and integral processes; and like- 

 wise Lagrange's and Taylor's theorems, which are particular 

 cases of that form, but may be immediately arrived at by a 

 similar method. 



Let F(^ + .x') be any function o( (z + x), z and x being vari- 

 able ; without inquiring whether s and x are dependent or 

 independent, we have 



F{z + x)-Fz=J^^j^. F{z + t)dt, 



regarding t as the variable concerned in the differentiation 

 and integration. 



From the manner in which z, x enter into the function, 



Hence 



Similarly, 



dY{z + t) _ dF{z+t) 

 dt ~ dz 



F{z + x)-F^=fJ^Jiz + t)dL 

 Fiz + t)-Fz=^'^J{z + t')dt' 



is true for any value of ^, and so on. 

 By successive substitution we obtain 



Jq dzJ q dzj q dz 

 + &c., 

 * Communicated by the Author. 



