Set of Linear Equations connected with Homofocal Surfaces. 265 

 Our identity thus is 



! + &,, + ? x , + ... + » x „ = ( ~ !)"(£ ~ «x)- (£ - o.) (g - gj — (£ - a>- ) > 



I 2 ""> • • • -Wl I 



and equating coefficients of like powers of £ we obtain 



_ (-!)- 



^n — ~ ) 



_( - l )"+ I (2a I + a 2 4-a 3 + ... +a g==L ) 



■^w— i ^ ; 



(Jj jOj rJJj 'y • • • (J/m j 



4. The close connection of these with the cases considered by 

 Murphy will be evident when we quote what he calls his " prin- 

 ciple," which is, "If we make the right-hand member of the £th 

 equation disappear by transposition, the left-hand member is then a 

 function of £ which vanishes when £ is any member of the series 1, 

 2, 3, ..., n : and therefore it must be of the form ~P(x - 1) (x - 2) 

 (x - 3) ... (x-n)." His cases are those where the £th equation is 



(a) ^ + A r + F -% + ... + /^=0, 

 v ' £ 4 + 1 £ + 2 $+« 



(6) l + ^ I + £(£ + l)aJ 2 + ... + £(£ + l)...(£ + w-lK=0, 



(C) l + £E I +i; 2 # 2 + ... +$X = 0. 



18 



