26 DR THOMAS MUIR ON THE 



-2167'IT [ = ] 



+ 2155'6' 



+ ±1788' 



+ ±16711 



+ ±17'88' 



+ ±1788 



+ ±4488' 



-±44'88 



- ±44611 



+ ±4'589 



-±4589 



-±77812. 



Fortunately the process by which this is accomplished brings to light a simpler expres- 

 sion than either of the two. 



(29) In the first place it can be shown that the aggregate of the first two terms on 

 the left is equal to the aggregate of the first and fifth on the right, the single term 

 ±04911 being an equivalent for either. As a matter of fact, we have, by a well-known 

 elementar) 7 theorem, 



«A C 3 1 • I C AA ! = I Khh I • I c i a ?A i + I a A c s I • I c A/s I . 



i.e., 08' = 56-911, 



and therefore, on multiplying by — 04, 



-0048' = -0456 + 049TT, 

 and consequently 



-±0048' + ±0456 = ±04911; 



and the fact that 



±004'8 + ±0789 = ±04911 



follows in exactly the same way from the identity 



04' = 6l0 - 79 . 



(30) There are four other pairs of identities like this, the full collection being 



-±0048' +±0456 = ±04911 = ±004'8 +±0789, 



±04911 - ±44611 = -±47TTl2 = -±04'9lT - ±77812, 



±1556 +±4488' - ±44611 = ±045'6' + ±4'589 , 



±078'9' -±4589' = -±77812 = ±1788 - ±44'88, 



±1268' + ±17'88' = ±16'7'IT = -±126'8 + ±155'6'. 



Further it can be shown that 



-±16711 + ±1788' = ±16711 - ±155'6; 



