a x 2 = 



a 2 = 



2 1 



a 3 - 3 



<^l 2 + «2 2 + a 3 2 



= !)• 



u 2 = v 2 



= w 2 



-gfcl, 



486 DR G. PLARR ON THE DETERMINATION OF 



3l, = , although Mi is expressed independently of P lt P 2 . The equations Pi = , P., = 

 give 



ux 1 =vy 1 = wz 1 . 



Squaring we get by the two first members 



(b 2 + na x 2 + )n 2 a 2 2 a 3 2 — (b 2 + na^)n 2 a 3 2 a^ = . 

 This gives, with the third member wz x treated in the same way, 



(owing to 



Consequently 



and u — v = w, as all three must be positive, and u 2 + v 2 + w 2 = l x = (a 2 + 2b 2 ). 



Let us substitute these elements into M : and verify the signs which can be attributed 

 to the first powers of a u a 2 , a 3 ? 



We have now owing to u 2 = v 2 = w 2 



m m 



Also 



u _ u _ u2 _ i 



V w vw 



= na-> (% 4 — a 2 b 2 ) 



a x x ' 



— {na 2 + 7ia 3 )[# + b 2 u 2 — a 2 b 2 — 6 4 ] 

 + n 2 u 2 a x a 2 a 3 . 



In the second term grouping u^ — a 2 b 2 together, and replacing b\u 2 - h 2 ) by nb 2 a 2 , we get 



— — - — n\a x — a 2 — a 3 ](« 4 — a 2 b 2 ) 



+ n 2 [u 2 a 1 a. 2 a 3 — b' 2 a 1 2 (a 2 +a 3 )] . 

 We have 



M 4 _ a 2J2 = ( a 2 ai 4 _ ^2 a '4) X n ( 



where a' 2 = 1 —a 2 . (This transformation can be shown by considering 



u 2 = b 2 + na 2 , and u 2 = a 2 — n 2 a' 2 . ) 



We divide by n 2 , and replace u 2 by u 2 = a 2 a 2 + b 2 a' 2 ; then we get 



w 4 M 



_i = («j _ a 2 - a 3 )(aV - 6V 4 ) 



+ [a 1 a 2 a 3 (a 2 a 2 1 + & 2 a' z ) — b 2 a x \a 2 + a 2 )] 

 = a 2 [a 1 4 (a 1 — a 2 — a 3 ) + a! 3 a 2 a 3 ] 



+ b 2 r~a' 4 (a 2 + a 3 — aj) — aj 2 (a 2 + a 3 )" 



' pa' 4 (a 2 + a 3 — a x ) — aj 2 (a 2 + a 3 )~j 



L + ft' 2 (ai«2 a 3 J 



= c^a/Kc^ — a 2 )(ai — a 3 ) 



