434< The Rev. J. A. Coombe on an Inextensible String. 



Hence we have 



T (d s y d* x — d s x d s 2 y) + X d s y — Y d s x~\ . . 



+ RV(ia d s y — d y u d s x) = 0. J ' 



So T (d s xd s 2 z — d s zd?x) + Z d s x — X d, z~\ . . 



+ RV(d z ud s x — d x ud s z) = 0, J ' ' * *' 



and T{d s zd?y — d s yd s *z) + Yd s z — Zd a y\ . 



4* RV(<^«<&» — d z iid s y) *&0, J ' ' ^ ' 



Hence (5.) rf,« + (6.)^w + (7.) d x u — gives, on sub- 

 stituting for T its value derived from (4.), 



vd z u(d s yd 9 x — d s xd 9 y) d z u(Yd s x — Xd s y)^ 

 + vd y u{d' s xd s *z — d s zd / ?x) = + d y u(Xd s z — Zd s x) I (A.) 

 + vd x u(d s zd s *y — d s yd?z) -\-d x u(Zd s y — Y d s z)J 



This equation, together with u = 0, are the equations to the 

 curve of double curvature into which the string is arranged. 

 II. To find the pressure on the surface at any point, 

 = (1 .) cos « + (2.) cos /3 + (3.) cos y, 

 gives — R = X cos a + Y cos /3 + Z cos 7 



+ T {d s * x cos a + d*y cos /3 + d, 9 z cos y } . 

 Now if p be the radius of absolute curvature at the point 

 (xy z) and A, //,, v the angles it makes with the axes, we have 

 cos \ = pd/x cos fi = p d s 2 y cos v ■=. pd 9 2, 



.♦. — RSs = X8s.cosa + Y8s. cos/3 + ZSs.cosy 

 ■ T -K 



H {COSaCOSX -j- COS/3cOS/4 + COSyCOSv}. 



Now let 9 be the angle between the radius of absolute cur- 

 vature, in which direction the resultant of the tensions on the 

 extremities of Is acts, and the normal to the surface, then 



cos = cosacosA + cos/3cos/a + cosycosv. 

 Then pressure on the portion 8 s of the surface 

 = resolved force in the normal, 



+ resolved tension 



III. There is one particular case which deserves especial 

 notice, from the peculiar character of the result. It is when 

 the resultant K of the forces X, Y, Z acts in the normal to the 

 surface at the point (xy z), so that 



X = RVrf,KY = KV^«Z = KVrf,B, 

 in which case the equation (A.) becomes 



d z u{d s yd^x—dsxd^y) d z u(d p ud s x—d x ud s y)KV "j 

 + d y ii(d s xd s ' 2 z—d s zdJ i x)==+d 2/ u(d x ud s z—d z ud s x)KV I (8.) 

 + d x u(d s z d?y — d s yd s 2 z) + d x u {d z iid s y — d y ud s z) K V = J 

 or substituting A, B, C for the coefficients of d tV u, d y ii, d z u, 

 the equation is 



