72 



Let O be the position of a point in the medium when quiescent, 

 and let three rectangular axes passing through it, and fixed in space, 

 be taken for the axes of coordinates. For a small displacement in 

 the direction of x, let the elastic forces, excited in the directions of 

 X, y, z, be cr, b, c ; for an equal displacement in the direction of?/ 

 let the forces be a', l/, d ; and for the same in the direction of z let 

 them be a", h'\ c". Then if a point receive an equal displacement in 

 a du-ection 01 making with OX, OY, OR, the angles a, /3, 7, the 

 forces in the direction oix^ y, z, (denoting then by X, Y, Z, respec- 

 tively,) will be 



X z= a COS. a, -\- a' cos. /3 -f- a" cos. y, 

 F = 6 COS. « -f i' COS. /3 -|- b'' COS. y, 

 Zt ■=. c COS. « 4- c' COS. jS -\- c" COS. y, 



as follows from considering (see Fresnel's Memoir, p. 82.) that the 

 force arising from a displacement in any direction is the resultant of 

 the forces arising fi'om the three displacements in the directions of 

 X, y, z, which are the statical components of that displacement. But 

 since (p. 90,) the force in the direction of one of the axes arising 

 from a displacement in the direction of another, is equal to the force 

 in the direction of the latter arising from an equal displacement in 

 the direction of the former, it follows that a' = b, a" =■ c, b" — d \ 

 and hence 



.Y = a COS. « -f- 6 COS. /3 4- c COS. y, 



Y = b COS. « + 6' COS. /3 -|- C COS. y, 



Z=-C COS. a. -{■ c' COS, /3 + c" COS. y. 



Let a.r* + i'?/* + d'z' + Idyz + Icxz -}- ^bxy =■ 1 be the equa- 

 tion of a surface of the second order. Let 01 intersect it in I, the 

 coordinates of I being x', y', %' ; then the equation of the tangent 

 plane at I will be, by the known formulas 



{ax' 4- hj' + cz') X + [bx' + b'y> -{■ c'z') y + {ex' + c'y' + c"z') 2=1. 



