316 Sir William Rowan Hamilton on Fltictuating Functions. 



[28]. It is also possible to extend the foregoing theorem in other ways ; and 

 especially by applying similar reasonings to functions of several variables. Thus, 

 if 7, 7^'> ... be each a function of several real variables a, a^", . . . ; if p and n be 

 still respectively functions of the kinds supposed in former articles, while p<'\ 

 n'", ... are other functions of the same kinds ; then the theorem (w) may be ex- 

 tended as follows : 



\ c^aV da('>...N.^N'<./i)...F„,„(i),.,. =0, (y) 



the function f being finite for all values of the variables a, a^", ..., within the ex- 

 tent of the integrations; and the theorem (x) may be thus extended : 



Ja Ja('> •■^0 *■ (Z) 



•••/a,a('\..VI7; J 



in which, according to the analogy of the foregoing notation, 



— 00 



and L is the coefficient which enters into the expression, supplied by the princi- 

 ples of the transformation of multiple Integrals, 



while the summation in the first member is to be extended to all those values of 

 or, d?''^, . . . which, being respectively between the respective limits of integration 

 relatively to the variables a, a^", ... are values of those variables satisfying the 

 system of equations 



7., x(», . . . = 0, yllln), ...=0,.... (c*) 



And thus may other remarkable results of Cauchy be presented under a gene- 

 ralized form. But the theory of such extensions appears likely to suggest itself 

 easily enough to any one who may have considered with attention the remarks 

 already made ; and it is time to conclude the present paper by submitting a few 

 general observations on the nature and the history of this Important branch of 

 analysis. 



