ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 107 



From this theorem Betti deduces the equations ; 



Oi. i(2+M') e^, ,(z-tv)=e\ ,z d\o(tu)-d\„(z) e\ ,(w), 

 0o,i(~+"') eoM-u')=e\oz 0\i(«o-0\i - (i\oV', 



and a multitude of other formulae which it will bo unnecessary to consider 

 further here, as they occur in Part II. of this Eeport in a slightly different 

 form. 



Section 7. — Two of these equations are as foUows : — 



Gi,i{z-\-w) di,o(z-w) = ei_-,Z 01,0= do,iW eo,o''(' + Oo,iZ do,o^ 6>i,i7y 0i,o^y, 



Oi,o(^ + iv) 6i_i{z-iv)=di,iZ 01, oS Oo,itu Oo,oW-do,iZ Oo,o~ 0,_iW di,o^v; 



putting z+iv for z in these equations, adding them together, then dividing 

 by 2w and putting iu = o, we have 



" ~~d^ ' ~~d — ' ' 



Betti deduces a large number of equations in a similar way, and especiallv 

 these (A. D. M. 3. 128) :— J i J 



criog^ei.i(r) 6\oZ _ cP\og,d, ,oZ 6\,z 



dz' '^e\,Z~ dz' '^ 0\oZ 



_ dUog^e„,-^z ^ e\oZ_ d-\og,Go,oZ , j.6\iz 



~ dz' ^o\iz- dp +"e^—^=^' • • • («) 



0, 0~ 



where C is an arbitrary constant. To determine its value put 



Cz2 



2x' 2 



The value of C is found without much difficulty to be = i^^, and the 



(j^ ill 



Xo^ 



preceding equations become ; 



cZ=log,Y z 6^ z dnog.x - 6' z 



&^ ~ d\,z dz" ~ e\oz 



° -^0, 1 0.0 "^■^O.O 7 2 0.1 



dz-" e\,is' dz^ d-. 



0, 0^ 



It is then shown (A. D. M. 3. 130) that these equations lead to the fol- 

 lowing:— s,:ariW6:. ■ 



