80 



PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 



finding the volume of the ellipsoid into which an originally spherical group of 

 points has been distorted in time t. 



For this purpose, we refer again to the equation of deformation 



$T = T-*S(tV)o", 



and form the cubic in <p according to Hamilton's exquisite process. We easily 

 obtain, remembering that f is to be neglected, 4 - 



or 



= <p 3 - (3 - tSVa) <p + (3 - 2*SV<r) <p - (1 - tSVa) , 



= (<p-l) 2 (<p-l + ASVo-). 



The roots of this equation are the ratios of the diameters of the ellipsoid whose 

 directions are unchanged to that of the sphere. Hence the volume is increased 

 by the factor 



1 - ^SVo- , 



from which the truth of the preceding statement is manifest, 



23. As the process in last section depends essentially on the use of a non- 

 conjugate vector function, with which the reader is less likely to be acquainted 

 than with the more usually employed forms, I add another investigation. 

 Let 



JJT = <pr = r — tS (tV) a . 



Then 



t = <p~V = ot + *S(arV)<r . 



Hence since if, before distortion, the group formed a sphere of radius 1, we 

 have 



Tt = 1, 



Thus, in Hamilton's notation, X, /x, v being any three non-coplanar vectors, and m, m„ w, the 

 coefficients of the cubic, 



— mS.'Kfiv = S.<p'X<p'/xp'v 



= S.(\ - tVSko)(ji - tVSfia)(p - tVSva) 

 = S.(\ - tVS\a)(Yfiv - tVfxVSva + tVvVSfio) 

 = S.Xfiv - t[S./ivVS\a + S.vXVS/xa + S.fytVSw] 

 = S.Xfiv - tS.[XS.fxvV + fxS.vXV + j/S.\/iV]o- 

 = S . X/xv — tS . X/xvS Ver . 

 m ft. X/xv = S.X<p'/x<p'v + S./x<p'v<p'X + S.vf'X<p'/x 



= S.X(Y/xv - tY/xVSva + tYvVS/xa) + &c. 

 = S.X/xv — tS.X/xVSva — tS.vXVS/xa + &c. 

 = 3S.\/iv - 2t8Va-S.Xfiv. 

 — m^.X/xv = S.X/xcp'v + S./xv<p'X + S.vXcp'/x 

 = S.X/xv — tS.X/xVSvcr + &c. 

 = 3S. Xpv — tSVaB.X/xv. 



