OF DEFLECTIVE FOUCES. ]S5 



If we innltiply the first equation by — y, and the second 



by jr, and add them together, we have — —7, A ~ + 



xddy xy ^ a:dd?/— ?/ddjr 



-d/ + Af =0= \^i — : but d (rdy) = :rd^y + 



dxdy, and d(ydx) = yd-x + dxdy, consequently d (xdy— 



/J 3rdi/ lydjzr 



ydo;) =jrd2y— yd^jT, andO + — =— ^^— , dt being con- 

 stant. Now the equation of the surface gives us xdx + 

 ydy+zdzziO: we have therefore the three equations xdx 



^ A A A A ^A. , da;2 + d?/ + dz^ 

 ■\-yayzz — zdz, xdy — ydx:=.cdt, and j- ■=. c + 



2gz. Adding together the squares of the two first, we 

 obtain x^dx^ + y^dy^ + x^dy^ + y^dx^=zz^dz^ + c'^dt^ziz (^2 + y2>^ 



(da:2 + dy2) -. (^2__22) ^^2 + dy^ =(r2— ^s) |(c + 2^z) df« 



— dz^l : consequently(r2— 22)(c + 2yzX— c'2d^2-. (y.2_22) 



-4- yd^; 

 dz^ -f z^dzs zz r2dz2, and d* =■ " 



V{(r2_z2)(c + 2^z)— c'2^ 

 But it is most convenient to substitute for the denominator 

 s/ j {a—z) (z — 6) (2gz +/) > ; for which we find, by actual 



multiplication, cr^-}-2gr'^z — cz^ — 2^z3— c'2=i:(az— a6— z2 + 



bz) {2gz-^f)-'Zagz^^ 2abgz^2gz^ + 2hgz^ + afz - abf— 



fz"-{-bfz ; then, by equating the coefficients of z (277), 2gr^ 



r~ + ab 

 zz-^2abg + af-{^bf, consequently/ iz 2g. j- : we have 



next, from z^, — c =: 2ag +25^—/, c:zzf—2g (a + b) = 2g 



(r^^ab ,, ^ r^—a'^—ab—¥ , , , 



I z (a + 6) 1= 2q, ; ; and lastly cr' 



^ a-^b ^ ^ a-\-b •' 



— c'* •=. ---abf, whence c'* = «r* + abx2g. -7- = 



