370 ON THE MOTION OF 



dyj, = dyi,—t,dP\ 



d% = d"l . 

 d"-^ = d\: 



whence we obtain 



i% = d"&'— gda'd/, dV, = d\-+~^dbdt. 



By means of these expressions and the equation $ = g. the 

 two first equations of motion (34) become 



#-?-dT + ^'— ^ 6 = °- 



, ihe same time the equation W = o (15) becomes 



Cd'R + Fd'a"—Gd'b" = o. 



Substituting, in the expressions for Z7 and F(15). f/6' for/). 

 — da for q, and for dr its value derived from the preceding 

 equation, we shall find 



^qC-G^^^-^H+GFy^ — CMgBr, = o, 



( EC - F') d ^ + (Cff+ G.F) ^ - CMg4„ £ = o ; 



which, together with the two equations above involving the 

 same four variables, constitute four linear equations of the 

 second order, with constant coefficients. It is Avell known 

 thai such (({nations are in all possible cases integrable in finite 

 terms by the method of D'Alembert or other analogous pro- 



