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DR THOMAS MUIK ON TJ1E 



or 



S-ABC ABC 



A(l-B 2 )(l-C 2 ) S-ABC C(l-B 2 ) B(l-C 2 ) 



B(l-C 2 )(l-A 2 ) C(l-A 2 ) S-ABC A(l-C 2 ) 



C(l-A 2 )(l-B 2 ) B(l-A 2 ) A(l-B 2 ) S-ABC 



which latter can be simplified, as Professor Nanson shows, into 



S+2ABC 



A 



B 



C 



A + 2BCS 



S 



C 



B 



B + 2ACS 



c 



S 



A 



C + 2ABS 



B 



A 



S 



This, how T ever, can be obtained much more directly from the use of another expres- 

 sion for cos (a + ft + y), viz. : — 



cos(a + /3 + y) = cosaCos(/3 + y) + cos/3cos(y + a) + cosy cos( a + /3) — 2 cos a cos ft cos y , 



where nothing but cosines appears, the angles being 



a, ft, y; /3 + y, y + a, a + ft; « + /3 + y. 



Making in this equation the substitutions 



( a = a + ft + y, ( a = — y, 



f a = -ft, 



\ I 8 = -y. -i ft = a + ft + y, J, ft = -a, 



I 7 = —ft' I 7= —a, L y = a + ft + y , 



we obtain three other perfectly similar identities '* connecting the same seven cosines 

 the complete set of four identities being in the notation above employed 



S + 2ABC - Acos(/3 + y) - Bcos(y + a) - Ccos( a + /3) = ^ 

 A + 2SCB - Scos(/3 + y) - Ccos(y + a) - Bcos(a + /3) = 

 B + 2CSA - C cos (/3 + y) - S cos (y + a) - Acos(a + /3) = 

 C + 2BAS - Bcos(/3 + y) - Acos(y + a) - Scos(a + /3) = J . 



From these cos (/3 + y), cos (y + a), cos(a + /3) can be eliminated, and the desired result 

 at once obtained. 



14. It may be noticed in passing that the substitution of 90°— a, 90° -ft, 90°- y 

 for a, ft, y gives the similar relation between sin (a + /3 + y), sin a, sin/3, siny. 



It should also be noted that the corresponding expression for cos («+ ft) in terms 

 of cosct and cos/3 is obtained from an identity of a different type, viz., sin(a + /^) 

 = sin a cos ft + cos a sin ft, the set of equations being 



sin ft + cos (a + /3). siu a — cos asin(a + /3) = ] 

 sin a — cos/3sin(a + /3) = V 



cos(a + /3).sin/3 + 

 cos a . sin ft + 



cos ft . sin a — 



sin(« + /3) 



-o 



* In effect the substitutions are the same as the circular substitution ( . „ ~ ') if we consider cos(0 + 7 

 cos (7 + 0), cos(a+/3)as invariant. 



