SOLIDS ON SIIUACES. 175 



If r is not small, it is constant as we have seen, and we hav< 



tffc = d>z, — °r<h — ri +$(rp -+- dq) . 

 ,1 17, = (l' r „ -+- 2rd£ — r\, -+- l(rq — <lp) . 



Substituting for p and q their values (32) and employing tin 

 abridgments & — £</ = u. r„ — £b" = », we shall find 



tffc == d'u—2r<lv — f u. 

 ffff, = d' v -+- 2rdfo — n'. 



By means of these values, and the values of £ and ^ obtained 

 from the abridgments last employed, the two equations of pro- 

 gressive motion are converted into linear equations of tin 

 second order involving a", b", u, v and /. At the same time 

 the two equations of rotatory motion are transformed, by the 

 substitution of the values of p and q, into two other linear 

 equations of the same order involving the same variables. In 

 this way we shall obtain 



(Vu _ dp . , _ » 



-fir— 2r ^ H-a/tt + A ga = o, 



dF"+- 2r IF ^-^-hfi"gb" = o: 



(^l)^H-(^2) d ^+(^3)^-h(^4)a +(d5)V+(d6)v + IY = o, 



(Bl)*£+(B2)^+(B3)% + (B4)b + (B5)o> + (B6)u + Gr = o; 



four linear equations with constant coefficients whose values 

 are 



vol. in. — o c 



