of the hyperelliptic functions. 238 



For the case of p = 3, I have similarly partly considered the 

 consequences o'f supposing X 7 = 0, X 8 = ; and append the results 

 for reference. When we take all possible partial derivatives of 

 the quantities $ 33 3 3 , etc., occurring on the left side of the 

 differential equations there are just thirty equations obtainable 

 by equating the resulting identically equal partial derivatives 

 of the fifth order. These are not all independent, nor are those 

 which are given below ; they will be found to give enough 

 independent equations to express all the functions as algebraic 

 functions of three of them, that is, as rational functions of four 

 connected by a rational algebraic equation ; these four are 

 apparently conveniently taken to be those denoted by f, rj, £, P. 

 It is however to allow choice that all the equations not obviously 

 deducible from one another obtained from the thirty spoken of 

 above have been retained. 



Let £>33 = £, fz2 = V< #>31 = £ g>22 = #, £>21 = V, <@11 = Z, 



P = (2£c + rj (2 V + i\ 5 ) + X 6 £) - (4f + X 6 ), 

 Q = (2&+£{2 V + i\ 5 ) )^(4£ + X 6 ), 



R = {2y v - 2^+4^ )^(4£ + X 6 ), 



then, proceeding just as in the case of two variables above, we 

 find , 



#>333 = /^T, g>332 = M> ffm=P& g>322 = /*-?> $111 = ^Q, @ m = flR, 



^=4£ + X 6j 



and the following sixteen equations : 



2f &■ = - $\£ + (4b + \) v + X 5 £ - (2 V + i\ B )P- \Q, 

 r 



2^ = i\3^-£(4*+X 2 ) + (2tf-4r)P + 47?Q+(4£+X 6 )i2, 



2^ = -fr^- 2z v - 4y£ + 4yP +(8?- 2x) Q-(2 V + |X 5 )P, 



2£&« = 2z£+ 2y v + (2x-^+ \ 4 ) £ + 4£P - (ft, + £X 5 ) Q 

 r 



+ (2£-X 6 )P, 



2^ *-f = - iX^ + iX 3 r + 2yP - |X 5 P, 

 scfcs = -X £- 2*?+ 2yQ + 2^P, 



