28 DR THOMAS MUIR ON THE 



and therefore by addition obtain 



±07'8'9' - ±0'456 = -±88'9'10 - ±344'5 , 

 = -±77'8'12 - ±155'6. 



Again, from the identities, 



06' = 512 - 89, 07 - -29' + 4'5', 

 we derive 



-±04W = -±4'5'512 + ±4'5'S9, 

 -±0789 = ±2899' - ±4'5'89 , 



and thence by addition 



-±04'5'6' - ±0789 = -±44'6'lT + ±1788'- 



Now the two alternative forms thus obtained, viz. 



-±77'8'12 - ±155'6 , - ±44'6'11 + ±1788', 



though no simpler than the original, are readily seen to be equal, because 



±1788' + ±77'8'12 = ±4678' = ±4'589' 

 and 



- ±44'6'll - ±155'6 - ±5'67'9 = ±4'589'- 



(31) The simplified form of the eliminant to which we are thus led contains twenty- 

 one of the twenty-two terms given in § 28 as being common to the two original expan- 

 sions, and nine others which take the place of the fifteen remaining ; and if, further, we 

 substitute for - ±0456' -±4589 its equivalent -±44611, we have finally 



0000 





-±126'8' 



+ ±00111 





+ ±1556 



-±0048 - 



-±0048 



-±1556' 



+ ±0123 





- ±16711 



-±0158 - 



-±0158 



-±167'IT 



+ 0456 





+ ±16711 



+ ±04911+ ±04911 



+ ±167'iT 



+ 0789 





+ ±1788 



+ ±0789' 





+ ±17'88 . 



+ 07'8'9' 





-±44611 



+ 0123 





+ ±44611 



- 0'456 





+ ±4488 



+ ±1268' 





- ±471142 



-±126'8 





-±77812. 



