316 Mr MURPHY'S THIRD MEMOIR ON THE 



an arbitrary constant must be also represented by the same equation, 

 this function, which is here found, is altogether different in its form and 

 properties from Laplace's coefficients. 



The great class of reciprocal functions above alluded to possess the 

 remarkable property, that their integrals vanish between any of their 

 own maxima or minima values. 



In this Section I have noticed some curious trigonometrical func- 

 tions of which the properties are very elegant, particularly as affording 

 simple means of representing by Definite Integrals the general differ- 

 ential coefficients of rational and integral functions ; another applica- 

 tion of trigonometrical functions is made, in representing the sum of 

 the divisors of any given number, by means of a Definite Integral. 



The seventh Section is on Transient Functions. The way of forming 

 reciprocal functions by means of arbitrary coefficients, when the form of 

 the general term was given, has been shewn in the Second Memoir on 

 this subject. To this I have here added the method of finding the 

 functions which shall be reciprocal to any proposed one, and applied 

 the method to the cases where the given function is r, (log. t)", and 

 cos" {t) ; the reciprocal functions which thence resulted are transient, that 

 is, they have but a momentary existence between the limits of inte- 

 gration ; that existence is however sufficient to make their integrals 

 finite, and to endow them with remarkable properties. They are capa- 

 ble of representing the electrical state of a body when an electrical 

 spark is infinitely near, and about to form a part of the system ; they 

 are also capable of representing, under continuous forms, the state of a 

 body considered as composed of absolute mathematical centres of forces, 

 separated mutually by infinitesimal intervals. 



The eighth and last Section is on the Resolution of Equations which 

 contain Definite Integrals; the first method for this purpose is to de- 

 compose the integrals into elements, and then determine the unknown 

 functions by elimination. This tedious process is useful in verifying 

 results otherwise obtained, and in giving numerical approximations in 

 the most difficult cases. Afterwards I have considered separately, 



