70 Central Forces. 



x=0 in (6), I have Q,„=y~2'3 — ^Tir ior the general form of the 

 quantitiesQ.,Q,...Q. etc.j ( ^-^^g-^-^ = the value of j^^^g-^-^. 

 when x=0;j \etn = l,2, 3,4, and so on successively, then Q,=--t-j 



^2 ~i 2dx"^ Q3 = i_2.3t?a;3' ^"^ ^° °"' ^>" substituting these values 



ofQ,,Q2,&c. m (a) It becomes Q=Q<i-x-^+ j-^-j^+j-^' 



d'Q' 



—J— J -\- etc. (c). Let r=fy=^ any function of y; and y^^^iu-^-xr) 



[d), = any function of u-{-xr; then if [d) is solved with respect to y, 



y will equal a function of u and x, but as there is no given relation 



between v, and x they are independent variables; also Q=F2/=: a 



function of u and a?. Again, (by solving {d) with respect to u-^xr;) 



u-\-xr= a function of y, let 2 denote this function; then [d) becomes 



dz 

 z=^u-\-xr (e). The differential of (e) with respect to u, gives -y- X 



dy dr dy ^ , . . . dz dy 



-r~=l-\-x-j- X ^' and its differential relatively to x, gives -7- X-7- = 



f?r (Zy £??/ % f/Q dQ dy 



JQ dy dQ dy 1 . ^Q dy dQ \ dQ dQ dQ 



j~^T~=^Xi— '<3"= since T~X-j-= J— J r'-y- ; hence -j- =r--j- 

 dy ax dy du \ dy du du I du^ ax du 



(g). Let R= any function of r; then R= a function of ?/ = a func- 

 / dQ\ 

 '^■[^'d^j dR dy dQ (dQ\ dR 



tion of u and x; hence —J^' ^^^X^X^+^^-^^i^j = d^ 



dx 



I dQ,\ 



LtlAt U/U l/'Jy \UjiAj j U/ti 



du 

 dQ\ 



^^= — d^= — ^byW5^"^by(^)^=^'rfV ''-^ 



