(4) 



458 



These equations are the same as the equations for transform- 

 ing a;2, 2/2, z^, yz, 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 : 



{aih) = 0, (aici) = 0, {hc^ = 0, 

 {aj)i) = 0, (osCs) = 0, (63C3) = ; 

 (3ici) + (2aiC2) = {hex) + (2ai53) = 0, 

 (caOa) + (2^3) = (caGa) + (2&2C1) = 0, 

 {a^h) + (2c3^i) = {azh) + (2c3a2) = ; 

 (ajGa) = (61*2) = (ciCa) - {aj)^), 

 ihh) = (C2C3) = («2«3) - (*ici)» 

 (C3C1) = (a^ai) = (6361) - (osCa) ; 

 {a{') = (bi^) + {2aih) = (ci^) + (2aiC3), 

 (62') = (C2') + (262C3) = («2') + (2Mi), 

 (C3') = («3^) + (2c3«i) = (63^) + (2C362). 

 The notation will be understood by reference to my former 

 memoir. 



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

 function V to the following (in which X, Y, Z denote /Bs - 72? 

 71 - a3, 02 - ^i) : 



2r = JX + Bfi + Cv + 'iFj, + 2Gx + ^Hxp 



+ P(X^-ui)+ Q(Y'-v,) + R{Z^-w,) 

 + L{2YZ- (V3 + W.2)} + M{2XZ - (w^ + M3)} 

 + N{2XY-iu.2 + v{)} 



" 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^ + Q2/^ + Rz^ + 2Lyz + 2Mxz + 2Nxy = 1 : J ^ ^ 



(5) 



