SOLIDS ON SURFACES. 357 



y/(a?L> -h a L +a>L ') , s/(a*L; -h p;M; -+- y, u A?) • 



Finally, it is easy to verify the following values of these 

 same quantities: 



k = Lx+L'x +LV, /,• = Lfr + My^N?,. 



These last expressions are susceptible of an obvious geome- 

 trical interpretation, and show that k and A- are the projections 



of the two radius vectors of the point of contact upon the com- 

 mon normal at thai point. 



The conditions (20) of a common normal moreover give 

 (30) 



kAx = kUlp x, 4- Bfi y, -f- C f c z , ) . 

 kAx 1 = k(Aa'x, + Bb'y H-C^c'sJ, 

 kJBt'x' = kfcfl'^-i-BJb'y.+ Cs't,)} 



kA,x = k(Aax -h A'a'x' + A"a"x") , 

 k B y = k (A b x -+- A'b'x' -f- A'b'x") , 

 k C z = k (A e x -+- Ac 'x' -+- A"c"x") . 



From each of these triplets may be obtained expressions for 

 the ratio of the two projections which may occasionally be 

 useful. If we add the first three equations together after mul- 

 tiplying the first equation by x, the second by x', and tin 

 third by x , reducing by means of equations ( t) and tin- equa- 

 tion Of the surface of support, and proceed !>v an analogous 



method with the second triplet, employing In tin- reduction 

 iln surface of the moving body, we dial! obtain the two equa- 

 tions 



k = (x -t-£ ) f, -+- (y,-r-n,) ]h -*- (*, -K,) jr. , 



