INVERSE METHOD OF DEFINITE INTEGRALS 377 



(2) Let AB be a right line perpendicular to the bounding planes, 

 which terminate a solid composed of parallel strata of indefinite extent, 

 but uniformly dense throughout that extent ; and let the law of den- 

 sity of the different strata be such that there is no action on any 

 point Q„ within. 



Let the solid be decomposed into n + 1 equal portions in which the 

 densities are as before represented by po, pi, p% /o„. 



In this case the quantities ao, flj, 02 ci„ are all equal, and putting 



them equal to unity, we have 



u = sm9 + sin39 + sm56 + SiC. 



So = po, Si=poCOS9, \ = 1» 



S2 = 2cos9 . S^ — XiSo=poCos29, X8 = l, 



Ss=-2cos9.Sz — \2Si = poCOs39, X3=l, 



and generally, S„=pocosn9, and\„ = l. 



Hence the solution is pi = 0, p2 = p»_i = 0, pn = po' 



And if E be the whole mass and A the area of the bounding planes, 

 which is supposed very great, we have 



E = 2iA.po. 



This result is analogous to the well-known fact, that electricity can 

 reside only on the surfaces of bodies, and affords another instance of 

 a transient function. 



The method of decomposition may always be applied to obtain 

 numerical approximations in cases which involve Definite Integrals; 

 for instance, in the distribution of electricity on bodies, and in esti- 

 mating the forces between bodies which are electrised. 



(2) By means of Reciprocal Functions. 



43. Equations which contain only one definite integral. 



Let f(f, a) be a function involving a variable f, and an arbitrary 

 parameter a; F{a) a function containing a only, and (p (t) a function 



