358 Astronomical and Nautical Collections. 



from these investigations, as far as you can judge from the 

 specimen which I send you, containing what the author con- 

 siders as the fundamental proposition cf his method ; a method 

 of which the object can only be, as he observes, *' to determine 

 the law or the form common to all the series of which a given 

 function can be the limit of expression ; and which is therefore 

 reducible to the solution of this single 



PROBLEM. 



Supposing X to be any function of any number of variable 

 quantities, and representing by Fa; any function of x, and con- 

 sequently of the same variable quantities that enter into the ex- 

 pression of X ; to determine the form, or the general law, com- 

 mon to all the series of which Fx can be the limit of expression. 



SOLUTION. 



Taking any state whatever of the magnitude of x to serve as 

 a term or limit, to which all the others may be referred, we 

 shall designate it by the name of the Primitive State, and all 

 the others by the denomination of varied or derivative states. 

 Then representing by x the primitive state of the quantity or 

 function indicated by x, and any of its derivative states by 

 X + u, we shall have Fx for the primitive state of the function 

 of X indicated by the characteristic F, or the magnitude of Fx 

 corresponding to the primitive state of the function represented 

 by X, and F (x + w) will represent the magnitude of Fx cor- 

 responding to X + u. Now, as the increment m is absolutely 

 arbitrary, we may consider it as capable of admitting states of 

 magnitude less than any other that may be assigned : and, 

 therefore m is a variable without any limit to its diminution ; 

 whence it follows that x rs: lim (x -|- ?t); and Fx = lim 

 F (x+u). It may, consequently, be inferred that F(x+u) must 

 be equivalent to Fx more or less a function of x and u, or of « 

 only, without limit to its diminution ; so that, considering the 

 most general form of F (x -{- u) after its separation into two 

 parts, we shall have 



F (x -I- u) - Fx -t- VF' (x, m) 

 V being a function of u without limit to its diminution, an 



