374 Mr Murphy on the Inverse Method of 



SECTION II. 



PRINCIPLES RELATIVE TO DISCONTINUOUS FUNCTIONS. 



(1) Method of representing Discontinuous Functions of only 



one Break. 



15. To bestow on the principles of this analysis, all the gene- 

 rality, required by physical problems, we must extend it to such 

 cases of definite integrals, as undergo a total change in their 

 values, under given circumstances. The attraction of a spherical 

 shell for instance, on any point is a definite integral. But when 

 the thickness is indefinitely small and uniform, and each particle 

 attracts by a force varying inversely as the square of the distance, 

 this definite integral is when the attracted particle is within, 

 and a finite quantity when without the shell. The law of at- 

 traction is therefore interrupted when the attracted point arrives 

 at the surface of the shell. When the shell possesses a finite 

 thickness there are two breaks in the attraction, or the definite 

 integral by which it is expressed. The attractions of spheroids 

 of every form are similarly discontinuous, when the force follows 

 such a law that it becomes insensible at great distances. So also 

 in heated bodies ; the law of expansion is different for the same 

 body, in the different states of solid, fluid, and gaseous. The 

 electrical phenomena present several similar instances. To ob- 

 viate the difficulty thus offered, we must seek an analytical 

 expression, which without altering its form, may continue to 

 represent the true value of the definite integral, or discontinuous 

 function, under alt circumstances. 



