334. REPORT— 1873. 



will become, by the substitution of these values, 



d{u^ + V^ + W^ Wp + ^+"'p. ''i.i '■p,p)«(«,-^, ■^ip-^p, '■,,,'-p,p) 



= 2(a)e(r(j Up, Ti,, rp,p)JX«, + tf, ^(p + Wp, r,,, rp,p)a, 



a 



■where the coefficients (a) are to be determined. 



To determine these coefficients Konigsberger adopts a method from Weier- 

 strass as follows. 



Taking the ratio A i 2- • ■ • p/^c ^ ^^^ remembering its value as given in 



«(('jV, f'p) ' 



Weierstrass's paper (Crelle, xlvii.), or in the first section of the paper we are 

 now considering, we see that it will be infinite when one of the quantities 

 x^, x^. . . ..Vp is infinite, and zero when they become equal to ««. 



From this Konigsberger deduces the two equations corresponding to these 

 conditions : — 



0(VjVj, .... Vp) = to the first, 

 and 



diy^v^. . . .Vp)a=0 to the second, 



which last may be written 



ln\ +.... +^,4"-' +!<> + . . . . l<r . . .) = o. 



Konigsberger then states that, if the symbol (1,3.5. ...2p — l, e^e^ep) is 

 called e-, and S being supposed to be any whole number, y equal to every 

 symbol of the form Se, and therefore taking 2*" forms, then 6(v^v^„Vp)^yyi = 0, 

 when v.v,^. . . .Vp vanish, y' and y" being different. To show this we remark 

 that the increments of the arguments v^v^ . . . .Vp are partly numerical, partly 

 consist of definite integrals. When y and y" are different, the numerical 

 part becomes entire ; and therefore when v,v., . . . .Vp vanish, d vanishes by a 

 proposition of Weierstrass for the expansion of 0, when the arguments are in- 

 creased by semiperiods of definite integrals. (See Crelle, xlvii. p. 30.) When 

 y and y" are the same, they counteract each other and produce no effect. 

 From these considerations Konigsberger deduces the values of the coeffi- 

 cients (a)*. 



I shall illustrate the Table, p. 28, by deducing from the last equation 

 of p. 27 :— 



Put Wj = — Vj in the equation mentioned, e'=4,a=l,/3 = 5, 6(v^ + w^, • • • . )e-, a, ^ 

 becomes d(v^v. . . •)i,4,5 = 0,,i (see remark at the end of our remarks on sec- 

 tion 2). Since we are dealing with hyperelliptic functions of the first order, 

 Cj and e^ wiU become and 2 ; hence y becomes in succession in the four 

 terms of the formula, 5, 0, 2, 02, ya/3 (omitting /3=5 and y = 5), 1, 01, 12, 

 012, or 1, 01, 12, 34, as we shall see, ae'y becomes 145, 140, 142, 1402, or 

 14, 23, 03, 3 ; fte'y becomes 545, 540, 542, 5402, or 4, 04, 24, 13, which 

 give the indices required. 



* Konigsberger has been very brief in this paragraph from Weierstrass. I am not sure 

 of his meaning. I hope to add sometliing in the Supplement. 



