

of an Abelian Integral 417 



instead of m, 



X(x + a)(x + b){x + c)(x + fy + (px—a)\x—k)(x—l<!) 

 =yfr(x-p){x-q){x-p t ){x-q l ). 

 So that, putting 



dx 





^{x + a){x + b)(x + c){x + d)(x— k)(x— W) 

 xdx 



v'(x + a)(x + b)(x + c)(x + d)(x—k){x-k 1 )' 



we see that the algebraical equations between p, q; p', cf are 

 equivalent to the transcendental equations 



Up±Uq± Up 1 ± Uq' = const. 



U { p ± Hfl ± U t p ! ± U,q' = const. 



The algebraical equations which connect 6, </> with p, q ; p\ q' } 

 may be exhibited under several different forms ; thus, for in- 

 stance, considering the point (go ; } cj>) as a point in the line 

 joining (k ; p, q) and (k' ; p' } q'), we must have 



S(a+p)(a + q)-*-Sa + k, S(b+p)(b + q) + Sb + k, . , . =0, 



*/(a+p'){a + q') -*- »/a~+l<!, ^{b+p'^b + q 1 ) + Vb + W 



V(a + 0)(a + <l>), */(b + d)(b + $) 



i. e. the determinants formed by selecting any three of the four 

 columns must vanish ; the equations so obtained are equivalent 

 (as they should be) to two independent equations. 



Or, again, by considering (co ; 0, <f>) first as a point in the 

 tangent plane at (k ; p } q) to the surface (k), and then as a point 

 in the tangent plane at (k' ; p 1 , q 1 ) to the surface (k 1 ), we obtain 



S((i-c)(c-£O(rf-ft)^(a+i»)(fl + j)V^^^(a + ^(fl + 0))==O 



2((6-c)(c-d)(rf-i) S(aTpW+7) ^(^)V^+0)(^+^)=O. 



Or, again, we may consider the line joining (co ; 0, <p) and 

 (k',p,q)ov (k ! ; p' } q'), as touching the surfaces (k) and (k 1 ) ; the 

 formulae for this purpose are readily obtained by means of the 

 lemma, — 



"The condition in order that the line joining the points 

 (£) V> %i w ) an d {£> V*> %> »0 ma y touch the surface 

 a^ 2 + b?/ 2 + c^ + dw; 2 =0 



2ab(fV-^) 2 =0, 



the summation extending to the binary combinations of a, b, c, d." 



But none of all these formulae appear readily to conduct to 



the transcendental equations connecting 0, <f) with p, q; p' } q l . 



