354 Mr Murphy on the Inverse Method of 



interpolated differential coefficients of fractional orders, for such 

 functions as may be simply represented by Definite Integrals of 

 a peculiar form. If we extend this principle to all functions of 

 operation of the distributive kind*, (that is, such whose action 

 on the whole, is the sum of the actions on the parts), and 

 if we can represent a given algebraical function by a Definite 

 Integral of the proper form, this view will be complete. 



Again, the phenomena of the physical sciences generally re- 

 sult from an infinite number of the elementary actions of the 

 particles forming the system under consideration. Such an inverse 

 calculus would conduct us from the observed phenomenon, to 

 the laws of the elementary actions. 



The following researches have been conducted, with this object, 

 chiefly, in view. The analysis consists of two parts, corresponding 

 to two distinct classes of phenomena in nature. Namely, such 

 as result from sensible, or developed powers, the action of which 

 is generally insensible, at infinitely great distances; and such as 

 are referred to latent, or neutralized powers, which only become 

 sensible at indefinitely small distances from the component par- 

 ticles of the system, whence the actions emanate. Of the first 

 part, which I now lay before the Society, the following is a brief 

 abstract. 



Adopting and 1 throughout as the limits of integration, 

 there is a character common to all the integrals of the functions 

 commonly received in analysis; if the function be multiplied 

 by f, that is, any positive power of the variable, the integral of 

 the product, which is a function of x converges to 0, for infinitely 

 great values of x. To revert from the function outside, to the 



* Annales de Math. Tom. V. Servois' paper; Vid. also Tom. III. Francais' paper. 



