ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 325 



p" + s- - ^7-ij"--r p"s" - ^^y^\p-p- + S-S-) 



+ -^^^(P'=s"'» + p-s'^) - \„ZL P'"S"^ 



m '117 K p K p ^ ^ 



lu a similar manner the following equations are obtained : — 

 (^^.-)(P'-S- - FS^) . 



= {P'^ - y,t^/ p"S" + S'^} - {P"^ - ^1±£;^F-S"^ + 8"^},. . (B) 

 (F"S"' + PS)^ 



+ "V •t^'V^^-'V ^'V'"^ jPF'S'S", . . . (C) 



(P"'S"' - ps)^ = ^^^^-^;^\p'2p-= + s'=s'.-2) 



W ■Ztr ^ ' 



Z-"'2 '"S / III 111/1,14 I '/4\ 2 ^'2 I ■> '"2\ 



W^(™'^ + ^"'^") + 2(^^r^f^ + ^^±:^y'F'S'S". (D) 



K p \ 'a 'ST K p ig -nr / 



5fec<ton 12.— Equation (A) gives a relation between P', S', P", S". Gopel 

 proves that no other relation can exist, of a purely algebraical nature, 

 between these quantities (p. 292). He consequently investigates the rela- 

 tions which exist between the differentials of those functions in the following 

 way : — 



Putting 



r(«+2«,K+2J,Lr... 



M ^/(«+(2«+l)K+(2/;4-l)L)» . . . 



we have 



MM =^2r{(,j+i)K+(0: + i)L}^+...^2r{«4-(»,+i)K+(0+i)I4H-.. 



where 



a + «, = )7> 6 + 6j = 0,. 



this is easily seen if we remember that 



(«i-«,)' + (a-«J+| + (rt + fr,)H(« + aJ + i = 2a= + 2o'/ + 2a + i, 

 and also that 

 2(a-rt, + iX6-6, + i) + 2(;« + «, + i)(/. + /), + i) = 4«,6, + 4«ft + 2-Y + k'6+l; 



