practical Instrument of Physical Research. 487 



notation. Now compare the following proof with the Car- 

 tesian one given in Ibbetson's i Elasticity,' § 237. We have 



v = ID^ + JD, + KD f (24) 



Hence 



0A=2D f . I=2{D { (£I) -<£D*I}, 

 or in full 



0A = D f (PI + NJ + MK) + D„(NI + QJ + LK) 



+ D f (MI + LJ + RK)-^(D f I + D,J + D f Kj . . (25) 

 Since the curve rfc is a principal line of curvature on each of 



the surfaces tj and £ (Dupin's theorem) it is obvious (see 

 figure) that if a point move with unit velocity along it, 

 carrying the system I, J, K of vectors, this system will be 

 rotating with (vector) angular velocity 



Hence 



/. D f I + D„ J + D^K = I(^„ + gar*) + J(,w c + ,w f ) + K(^ + fw,) ; 

 and therefore 

 ^Dfl + D^ + DfK) 



= I{P(^ + ^)+N(^ + ^+M(^ + ^)}+J{} + K{}. (27) 

 Hence 



^A==Ij(D^^- |OTc )PH-(D,-,^-^)N + (D c -^-^)M} 

 + I{-^N-^M + ^Q + ^E}+J{ } + K{ }; 



