458 



(4) 



These equations are the same as the equations for transform- 

 ing a;2, z/2, z"-, tjz, xz, xy. 



" Introducing into the equations derived from the function 

 V (which contains thirty-six coefficients as above specified), 

 the conditions that the vibrations shall be normal and trans- 

 versal, I obtain the following relations among the constants : 



(«iM = 0, (aici) = 0, (Z/2C2) = 0, ^ 



{aj)^) = 0, (ascs) = 0, (63C3) = ; 

 (iici) + (2«iC2) = (61C1) + {2aih) = 0, 

 (cao-j) + i^haz) = (Caa2) + {'^hcx) = 0, 

 {mh) + (2c3^'i) = {azh) + (2c3a.2) = 0; 

 (a,«2) = {bib.2) = (C1C2) - (a^bs), 

 (M3) = (^2^3) = (a2«3) - {bici), 

 (C3C1) = («30i) = (Ml) - (a-iCi) ; 

 (ai'O = (61^) + (2a,b.,) = (ci^) + (2a,C3), 

 (62') = (^2^) + (262C3) = («2') + (262«l), 

 (C3') = («3-) + (2c3«i) = (bf) + (2C362). 

 The notation will be understood by reference to my former 

 memoir. 



" These equations, twenty-four in number, reduce the 

 function Fto the following (in which A', F, ^ denote jSs - 72^ 

 71 - 03, a.2 - (5i) '• 



2r^JX + Bfi+Cv + 2F<i> + 2Gx + 2^^ 



+ P (X2 - Ml) + Q(Y^-v.,) + R{Z-'- w,) 

 + L {2YZ - (V3 + w;) } + M{2XZ - {w^ + xiz)] 

 + A^(2Xy-(M2 + t'i)J 



K5) 



" This function, containing twelve coefficients, may be 

 reduced to a function of nine coefficients in two different ways, 

 by the aid of the two theorems above given. In fact, let the 

 two ellipsoids represented by the following equations be con- 

 structed : 



Ax^ + By^ + Cz" + 2Fyz + Gxz + 2Hxy = 1 ; 

 Px^ + %' + Hz^ + 2Lyz + 2Mxz + 2Nxy = 1 



>(6) 



