of an Abelian Integral. 415 



determined. We may for shortness speak of the surface 



k{& + ip + & + u^ + aa} i + by* + c& + du?=Q 

 as the surface (&). It is clear that we shall then have to speak of 



tf 2 4-2/ 2 + * 2 + w 2 =0 

 as the surface (co ). 



I consider now a chord of the surface (go ) touching the two 

 surfaces (k) and (k') ; and I take 6, <f> as the parameters of the 

 one extremity of this chord; (p, q) as the parameters of the 

 point of contact with the surface (k) ; p 1 \ q 1 as the parameters of 

 the point of contact with the surface (k 1 ) ; and & 9 </>' as the parame- 

 ters of the other extremity of the chord; and the points in 

 question may therefore be distinguished as the points (oo ; 0, <j>), 

 (k;p,q), {k';p',q'), and (cc , 6' , ft) . The coordinates of the 

 point (oo ; 0, <f>) are given by 



x:y: z :w= V (a + 6) [a + </>)-*- V {a— b){a— c)(a— d) 

 i/(b + d){b + <l>)+ \/(b-c){b-d){b-a) 

 V(c+.0)(c + <£)-r- >/{c-d)(c-a)(c-b) 

 */(</+ 0)(</+</>)-r- V(d-a)(d-b)(d-c); 

 those of the point (k ; p, q) by 



x\y \z\ w— ^(a+p){a + q)-r- V ' (a—b)(a—c)(a~-d) Va + k 

 */(b+p){b + q)+ \/(b-c){b-d){b-a) VT+k 

 \/{c+p){c + q)--r */(c—d)(c — a){c — b) Vc + k 

 x/(d+p)(d+q)+ V '(d-a){d-b){d-c) s/d+k', 



and similarly for the other two points. 



Consider, in the first place, the chord in question as a tangent 

 to the two surfaces (k) and (k') . It is clear that the tangent 

 plane to the surface (k) at the point (k ; p, q) must contain the 

 point (k' ;p', q') } and vice versa. Take for a moment f, v\ s £ a> 

 as the coordinates of the point (k ; p } q), the equation of the tan- 

 gent plane to (k) at this point is 



2(a + *)£*=0; 

 or substituting for f, . . their values 



X(x */(a+p){a + q) Va + k-±- V '(a—b){a—c)(a—d)')=0 ; 



or taking for x, . . the coordinates of the point (k', p\ q'), we have 

 for the conditions that this point may lie in the tangent plane in 



question, " 



X( V{a+p)(a + q) </(a+p'){a + q') V(a + k) 



+ \/(a~+F){a--b){a--c){a-d)} = ; 



