108 REPORT— 1870. 



fPv Z dy Z dy Z 



^|^V2p/-^ + 2^<l-P)%: + (^-H/.V)x„_„z=0. 



From these equations we may expand ^ (~) &c. in powers of (r), from 

 which 0j '^^^ ™^y ^^ course bo deduced. 



This beautiful process was first given by Dr. Weierstrass, in the fifty- 

 second volume of CreUe's Journal, with a different notation, which I hope to 

 explain to the reader when I treat of the hyperelliptic functions. The pro- 

 cess of Betti, however, does not essentially difter from that of Weierstrass. 

 The actual calculation of the coefficients in the expansion of y z has been 

 given by Weierstrass at considerable length in the memoir here mentioned. 



2x'_ ^2 

 Section 8. — Let >;= , ' " , then one of tlic equations (a) of section 7 

 X w 



will give us °'^-^^ = —-—?c'^ ^^-^ ; whence, remembering the values of 



dz' u> a 1,0^ 



a ^3 



0], i", Oi,o(~)> also that r|—=/i.'^ (ScheUbach, p. 73), and integrating, wo have 



r]Z d loge di, oZ 

 d^ 



Jc'^ \dz sin^ ara z= "L ''"^ + 0* ; and taking the integral from z=o 



Ic'^ I dz sii 



w r^ y'd)/ 



to z= ;^, wo have 7j= — 2A" 1 , - — ) , — 

 2' ' Jo s/l-y' Vl— Z:y 



Again, from equations (a) we may deduce 



<^'log Qo,oZ_ T] ^ e'o. ,;r _ 



dz^ (o Qo,oZ 



we also have 



cPOo^oZ _ i ■^'^ dOo^ 



dz' ^;?" dq ' 



whence Betti deduces (A. D. M. 3. 136) 



dot (r) + Jc^ut)to 



* It will be easily seen that i\o%^ Oj {/) vanishes if j • ^ C^y Tanishes, whicli takes 

 place when c=o, Q-iz=-. 



