382 Mr Murphy on the Inverse Method of 



Again when *>/31 ^ ^M =J dS f>*)^ **&$ =0 , & c . 

 < 7 J aa a/5 «7 



and is then = 2 ; 



and so on ; if we substitute these values in the assumed for- 

 mula for (p, we get successively the Equations, 



<t>»-=fl+f: 



fr=f+f+f 



&c.=&c. 



and therefore f = <p\ 



f = <p»-(p t 



f 3 = <p3~ 0, 



&c. = &c. 



and 



= 0. rfn + (0.-00 — j,£ +(03-0,) - +&C. 



d/8 -*r» -r/ — ^ 



Cor. If the function is also discontinuous below «, as as 

 passes through the decreasing magnitudes a, b, c, &c. <f> assuming 

 then the values <j> , <£_, </>_ 2 , &c. it is easy to see that then 



= (0, - 0,) J a ' ' + (0 2 - 00 jg' ' + &C. 



f , ,_, . dS(a,x) 7j , dS(b,z) „ 



j + (0o-0-.) ,/ a + <0--0-») J// +&c. 



(3) Geometrical Illustrations of the theory of Discontinuity. 



(Vide PI. 24). 



23. It only remains to add a few examples, to illustrate the 

 application of the principles of this Section. 



(Fig. 1.) Let ACa, BCb, be the equal sides, produced, of an 

 isosceles right-angled triangle ; we may thus find the Equation 

 of the black part ACB, as distinguished from that of the dotted 

 part aCb. 



