85] THE PRINCIPAL SCREWS OF INERTIA. 75 



which is of course 



2M4&amp;gt;i = ; 



whence 6 and &amp;lt;/&amp;gt; are reciprocal, and the same being true for each pair of 

 principal screws of inertia we thus learn that they form a co-reciprocal 

 system. 



We can also show that each pair of the Principal Screws of Inertia are 

 Conjugate Screws of Inertia. 



It is easy to see that 



x ^ Pi^-if^i x v Pi^iA 1 ! \ 



~~i // ** i + 77 j2t - ~ = i , ^ 



& 1 ~- x PI x x x PI -~ x y p^ x \ 



As each of the terms on the left-hand side of this equation is zero, the 

 expression on the right-hand is also zero, but this is equivalent to 



whence we show that 6 and &amp;lt;/&amp;gt; are conjugate screws of inertia and the 

 required theorem has been proved. 



85. An algebraical Lemma. 



Let U and V be two homogeneous functions of the second degree in n 

 variables. If either U or V be of such a character that it could be expressed 

 by linear transformation as the sum of n squares, then the discriminant 

 of U + \V when equated to zero gives an equation of the nth degree in \ 

 of which all the roots are real *. 



Suppose that V can by linear transformation assume the form 



x 2 _j_ x 2 -\- x 

 and adopt x ly x.,, ... x n as new variables, so that 



TT ,- /vi 2 i ft /-.i 2 i 9/f ** 



U Ct jjwj |~ Lv nJi *2 i~ ^i(rj2 l * / i t * 2 



The discriminant of U+\V will, when equated to zero, give the equation 

 for \, 



a n + X, a 12 , . . . a m j = 0, 



; ft 2 i , $2-2 &quot;I&quot; * ^2i 



and the discriminant being an invariant the roots of this equation will be 



* A discussion is found in Zanchevsky, Theory of Screws and its application to Mechanics, 

 Odessa, 1889. Mr G. Chawner has most kindly translated the Russian for me. 



