^60 THIRD REPORT — 1833. 



Thus, if y — « .r — j3 = and y ~ «' .r — /3' = be the eqna- 



A B p /» u 



tions of two lines B C and D C, forming a triangle with a por- 

 tion B D of the axis of .r, then the system of lines which they 

 form will be expressed by the product 



^y^^a:-^){y-o}x-^')=^. (3.) 



Now it is obvious that if common ordinates P M, P M' be 

 drawn to the two lines, the least of tliem will belong to the sides 

 of the triangle BCD; if we denote, therefore, P M and P M' 

 by yi and y.^; ^^ equation 



y = 



+ 



2/1 + ya ^-^ ^~^^') --- V 2 



2 . 4' 



'yi±i?) -^2.4. 



■'i'^y 



■will become the equation of the sides of the triangle BCD, 

 when yi and yg are replaced by their values ; for y will denote 

 P M for one side and p m for the other. 



In order to express a discontinuous function <p, wdiich as- 

 sumes the successive forms (p,, ip^j, f-s, &c., for different values 

 of a variable which it involves between the limits « and /3, /3 

 and y, y and 8, &c., Mr. Murphy assumes S (aj z), S (/3i z), 



S (-/i z), Sec, to denote the coefficient of — in the several series 



log ^^±^ -(^^+^1 log (J^+JU^J:^), &e., 



and supposes 



=A' 



d S («, z) 



^^+/. 



d S (/3, z) 



+ /3 



d y 



'^, + &c. 



du '■"''' d^ 



If a be less than z or z greater than «, then S («, x) = «, 



dSia.,z) 1 .p/3 1 1 ^1 4.1 d'S{&,z) 



and therefore / = 1 : if /3 be less than z, then — -j-^ — 



= 1 : if y be less than z, then ''- — p^ = 1, and so on; con- 



i 



