72 
ANALYTICAL RESEARCHES CONNECTED WITH 
[114 
or finally, if 
S 2 = £ 2 + 7 2 - 4s£ ^7 + = 13 2 + 7 2 ~ 4s (7 + 0£) £, 
which is in fact the case. 
Writing the equations for 
A 1 A 1 
A + A’ A A’ 
in the form 
and substituting in 
a 1 28 . 1 2 . «/O \ 
A + -r- = r-s- A - — = (7 - 2£s), 
A (fi — v) /3s A (ya — 1/) £s 
F = TF j( A “ i) (af + fa + 7f) - ( A + x) S ’ (f + ?>} • 
we have 
u= 1(7 - 2/3s) (af+0V+7f} _ (f+?)1 
= -gj ((— /3 + 2S7 + 2<y>7) f + (7 - 2s/3) 1) + (- £ + 2«7 + 4s£/3) f J; 
and consequently, multiplying by 
Is = 2 ^(7'+ <£/3') (7" + </>£") £73") 
we have 
VA' 2 + BY 1 - C' 2 VA" 2 + B" 2 - G"°- 
= | V(7' + ^/3')(7" + '#>/3")W"l {(- /3 + 2s 7 + 207) ? + (7 - 2s/S) I, + (- /3 + 2«7 + 4s^) f|, 
or collecting the different terms which enter into the equation 
A'A" + B'B" - C'G" = \f A' 2 + B ,2 - C'- VA" 2 + 5" 2 - (f 2 
the result is 
(A" + £73") f + (7 73" + 7 "£' + 20 / e / /3") v + (£73" + 77") £+ ^ («£ + £*7 + 7?) 
- | V(7 + 2¿WW + 20£")£'£"1 {(- £ + 2s 7 + 207) £+(7 - 2s£) t? + (- £ + 2s 7 + 4s0£)£} = 0, 
o 
which, combined with the first equation written under the form 
(«£ + £77 + 7O 2 - S 2 [(£ - £) 2 + *7 2 ] = 0, 
determines the ratios of £, 77, £ that is, the values of F + Z and YZ.