INVERSE METHOD OF DEFINITE INTEGRALS. 885 



where eo, e^, e.,, &c. are also unknown. 

 The proposed equation then becomes 



+ eoOoBo + eiUiBi + eia-iS^ + &c.) 



Now to the function A„ there may be found a function A„ reciprocal relative to a, 

 and to B„ B„ b. 



Let f„AaB„ = U„ a function of b only, 

 ftBoA,, = V„ a only. 



Hence, f^AoE (a, b)-CoaofaAoAo = eoaoUo + eia^Ui + 6.^0^112 + kc. ad inf. 

 ft BoE {a, b) - e^a^ fiBoB^ = c^a^ K + c, a, F; + c^a^ F; + &c. ad inf. 



Let t/„ be the function of b, which is reciprocal to f7„, 

 V„ of «, V^. 



\L k {Ao U,E {a, b) - c,a,A,A^ t7"„) = e„a„ /j t7„ U^ 

 Hence, \ \, 



\fJ,{Bo KEia, b) - e„a,B,B,K) = c„«„/„F„rJ 



by which equations the constants Co, e,, are immediately determined. 



\fa fb (Ao U„E (a, b) - c^a^A^A^ U„) = e„ a„ /j Un UA 



Also, \ >; 



(/„ /, (^0 KE{a, b) - e,a,B,B, K) = c„a„ f„ V„ Vj\ 



and since c„, e^, have been found, the latter equations determine gene- 

 rally the coefficients c„, e„, and therefore the required functions <p{t), 

 ^ {t) are known. 



In like manner by employing reciprocal functions relative to double 

 integration, we may solve equations containing three unknown func- 

 tions, &c. 



The problem of the distribution of electricity on bodies of which 

 the surfaces are not . continuous, introduces equations of this nature. 



