114 Mb MURPHY'S SECOND MEMOIR ON THE 



in a finite extent; such functions correspond to an extended and 

 curious class of pheenomena in nature, when any principles of action 

 which have been observed, under peculiar circumstances cease to produce 

 the observed effects, as when equal charges of opposite electricities 

 are communicated to a body, or when a body electrised by influence 

 is removed from the vicinity of the influencing system ; or lastly, as 

 when heat in its thermometric effects disappears in the chemical 

 changes which bodies undergo. 



The properties of this class of functions are of great use and 

 importance in analysis, as they conduct directly to the theory of 

 reciprocal functions. This term I have here employed to denote such 

 functions, two of which being multiplied together the integral of the 

 product vanishes, except in one particular case. That function which 

 is in this sense reciprocal to another, is also in general different in its 

 nature. There are however many functions which are reciprocal to 

 functions of their own nature, and to this class belong the only two 

 species of reciprocal functions hitherto introduced into analysis ; namely, 

 the sines or cosines of the multiples of an angle, the integral of the 

 product of which always vanishes (when taken between proper limits) 

 except in the particular case of equimultiples; and secondly, such 

 functions as satisfy the well-known partial differential equation in the 

 third book of the Mecanique Celeste; where the integral of the product 

 also vanishes except in the particular case where the functions are of 

 the same order. It is this exception which renders reciprocal func- 

 tions particularly useful, as is evident from the application of the 

 trigonometrical functions in the theory of heat, and of Laplace's functions 

 in investigations relative to the distribution of electricity. In the same 

 Section I have shewn generally the means of discovering all species 

 of reciprocal functions, and given several examples : as an instance of 

 one of the most simple species possessing properties very analogous to 

 those of Laplace's functions, but giving a simpler integral in the case 

 where that integral does not vanish, it is proved in the succeeding 



h 



Section that if T„ be the coefficient of h" in - — j , then when n and in 

 are vmequal ftT„T„ = 0, but when n = vu ftT„T„ = l. 



