494 DR G. PLARR ON THE DETERMINATION OF 



This gives, leaving y a in the place of #>, 



/3=(cos|C+asin£C)y a(cos£C-asin£0), 



~ , . r = ( cos a C + « si n|C)y d (cosAC-asiniC). 



Developing 



/3 = cos 2 iC.y «-sin 2 |C(a 7o a 2 ) 

 + sin 1C cos JC(ayoo - 7 « 2 ) , 



7 = cos 2 ^C.7 -sin 2 ^C(a7 a) 

 + siniCcosiC(a7 -y a). 



Having Sy a = we deduce 



a7o« 2 = — a7 = 7 a 



ay a = 2 a S7 a— 7o« 2 = 7o • 



Hence 



Replacing 



we have 



or putting also 



/3 = 7 a(cos 2 |C -sin 2 |-C) + 2 7o sin f C cos JC . 

 7 = 7 (cos 2 i0^sin 2 iC) + 2a7 sin|CcosJC. 



— A> for a7o 

 /3 = /3 cos C + 7 sin 

 7 = 7 eosC-/3 siuC; 



cosC = c , siu C = c', 

 and replacing for /3 , y their expressions, we get 



/3 = ( - m' + A«o + M'a )c o 



+ (ia'-ja b -ka b')c'. 

 We have thus 



/3 = - m'c +j(a b c - b'c') + Jc(a b'c + b c') 



7 = «»'c' +i( - « V - 6'c ) + K - a b'c' + b c ) . 



This gives us the table of values, including those derived from a 



rtj = a , a 2 = a\ , a 3 = a'b' 



h i=- a'c , b t = a 6 c — b'c' , b 3 = a u b'c + b c' 



c x = a'c , c 2 =— a 6 c' - b'c , c 3 =- a b'c + 6 c . 



§xv. 



We will now express the quantities u 2 , v 2 , w 2 , W/, W 2 ', W 3 ', Y', Z', Y u Z, 

 in function of the three angles A, B, C, and their dependents a , a', b Q , V, e ' } c'. 

 If we examine the values of a 2 , b 2 , c 2 , and compare them to those of a B , b 3 , e s , 



