348 ON THE MOTION OF 



But if we add equations (9) together after multiplying them 

 respectively by L,, M„ and iV, and reduce by means of (20), 

 we shall find 



Lfo + Mfa + Nfa == Lbx-^L'bx' + L"bx"; 



which the equations of the two surfaces enable us to write in 

 this form, 



L,{bx x — bx / )^.M l (by: — fyj-Kflftdz, — bz) = o. 



This last expression becomes, with the aid of the three equa- 

 tions given above, 



L^l-yPR+zZQ)) 

 M{d^—zpP-^x,m)\ = o. 

 Nlbl-xfiQ + yfiP)) 



which is the same with the result before obtained, and is in 

 fact expressing algebraically that the velocity of the point of 

 contact is nought in the direction of the normal. Care must 

 be taken to distinguish between bx„ by,, bz„ and bx„ by,, bz,. 

 They both denote the variations of the point of contact (P) 

 estimated in the direction of the body-axes ; but the former 

 denote its variation along the surface of the moving body, the 

 latter its variation along the surface of support. The former 

 are of the kind called incomplete variations, the body axes 

 being supposed to remain fixed during any one of these vari- 

 ations, and to vary instantaneously in passing to the next. The 

 latter are the total variations of the actual body coordinates of 

 the point of contact (P). These two kinds of variations ne- 

 ver coincide in value except in the case of rolling motion un- 

 accompanied by sliding. 



If the body is supported upon two, three, four or five given 

 surfaces, there will be as many equations of condition similar 

 to equation (22) as there are surfaces of support : if the body 

 is required to be in contact with six given surfaces, its station 

 and aspect become determinate and motion is no longer possi- 



