260 THIRD REPORT — 1S33. 



Thus, if y — a .r — /3 = and y — «' .r — /3' = be the equa- 



A B p p- 



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

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

 form will be expressed by the product 



(y_«^_^) (2,_«/^-_^')^0. (3.) 



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

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

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

 by'^i and y,^, the equation 



y = 



y\ ya 

 yi + ys 



+ 



1 . 1 



2.4 



• (H^f 



1 



2 J V 2 



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

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

 P M for one side and 2^ "* for the other. 



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

 sumes the successive forms ipj, ip,,, f^, &c., for different values 

 of a variable which it involves between the limits « and j3, /3 

 and y, y and 8, &c., Mr. Murphy assumes S («i 2;), S (/Sj z), 



. . 1 . 



S (yi ss), &c., to denote the coefficient of — in the several series 



for 



and supposes 



log 



„ </ S («, «) , /> (/ S (/3, z) 



+ /3 



d S (y, ^) 



+ &c. 



doL '-''' d^ '-''■'' dy 

 If a. be less than z or 2 greater than «, then S (a, ^) = «, 



, , t. d S (u, z) 1 ■r■a^ 1 i.u 4-1 '■^ S (^, s) 



and therefore r"^— ^ = 1 = up be less than z, then ^-^ — - 



d « 



rf^ 



1 : if Y be less than z, then injll = 1 and so on; con- 



' d y 



