ON ELLIPTIC AND HYPEKELLIPTIC FUNCTIONS. 317 



and also if M^, M/, M/', M/" are what M, M', M", M'" become when v, v' 

 and w, w' are substituted for v, v' and iv, iv', then 



M +M' =M, +M;, (1) 



M"-M"'=m;'-m;", (2) 



M -M' =M/' + M;", (3) 



M"+M"'=M^ -M; (4) 



It is a good way to prove form (1) by writing down the fully expanded 

 forms of <p^^ 3, 03 , and then applying the principles of Section 1. Then 

 Eosenhain has shown how to deduce (2), (3), (4) by merely changing the 

 periods. 



Section 7. — By increasing the arguments by semiperiods Eosenhain has 

 deduced an immense number of formulaj, which he has placed in a table at 

 the end of his memoir. We shall endeavour, first, to explain how this table 

 is formed, and, secondly, how to iise it. We remark especially that if 



V, v, v", v" are each augmented by -^, then i\ is augmented by jV, and 

 Vj', ?'/', Vj'" remain unchanged ; but, on the other hand, if v, v', v" are 

 augmented by -^, and v'" diminished by p-, then t\, i\', v^' are also in- 

 creased each by p- and v"' diminished by -^. Again, if while v, v' re- 

 main the same v" is increased and v" diminished by -^, then v^, v^ also 



remain the same, and v^' is increased, v^" diminished by -^. So that the 



four equations of section 6 remain true when the variables are thus changed 

 and the functions M transformed. Now, then, we will consider the Table. 

 Formula la consists of the values of M, M', M", M'" written down as 

 given in section 6. Formula Id is obtained by augmenting w, iv' , iv" by 



-^ and diminishing w'" by -„ , formula 2a from la by augmenting v" by -^ and 

 diminishing v'" by -^ in la, formula 2d from \d by augmenting v" by -^ 



and diminishing v'" by -^ • We need make no special remarks respecting 



3a, 3rZ, 4«, 4f?, which are proved in a similar manner. But when 

 we come to 5a we meet with a change. The formulte of page 410 

 (numbered 80), are then called in, and the arguments augmented by the 

 quantities which render the ratios of the functions doubly periodic, and 

 which we have discussed at full in the third section in reference to ^3 3 {v, w). 

 We thus obtain 5a, and from this, by changing the arguments as before by 



adding and subtracting -^, we arrive at 5cZ, 6a, Qd. 

 Now consider Qd particularly. It gives us 



M-M'=M," + M,"', 



