PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 81 



the equation of the ellipsoid is 



T(ttt + *S(raV)<r) = 1, 

 or 



OT 2 -i- 2*SotVSwo- = - 1 . 

 This may be written 



S.OT^ra- = S.^(trr + ^VSoto" + #S(otV)o-) = — 1 , 



where x is now self-conjugate. 



Hamilton has shown that the reciprocal of the product of the squares of 

 the semiaxes is 



whatever rectangular system of unit-vectors is denoted by i, j, k. 

 Substituting the value of x, we have 



-S.(i + tVSia + *S(iV)o-)(j + &c.)(* + &c.) 



= _ S. (t + tWSia- + *S(zV)<r)(« + 2tiSVa - *S(«V)o- - tVSi<r) 



= 1 + 2*SV<r . 



The ratio of volumes of the ellipsoid and sphere is therefore, as before, 



1 



s/l + 2t$Va 



= 1 - *SV<x 



24. Before concluding I may append a generalised form of Green's Theorem, 

 which is obviously fitted to be of use in quaternion investigations. If we put 



T = iP + jY + kV", 



we easily see by the equations at the end of § 5 that 



fffSFV,. V)*fe = -fffV^rd, + J 0rp i S(U*.V)«fe , 



= -fffrVV^ck +ffrS.VV 1 Vv.ds . 

 As a particular case, let 



P a = Sap 

 so that 



VPj = iSia + yS/a + k$ka = — a , 



V^ = , 



we have 



jQfjrSiaVjrds =fffSapV*rds -//Sap${TJvS7)Tds , 



=ffT$.a\Jvds . 



VOL. XXVI. PART I. Y 



