[SULLIVAN] CONCERNING THE INTEGRALS OF LELIEUVRE 175 



Vmv ' Vmv 



r' = - log VdJ^w^^ , A' = - log Vd^^^+dz' , 



dq dp 



^n_^' dq dqj ^,,_ V 'dp dp'/ 



Vmv ' Vmv 



We have seen that the vanishing of F" is the condition that the 

 family /? = const, be straight lines. It therefore follows that 







\ — — I must vanish identically. 

 dq dq^ y 



We shall first investigate equation (A) of range 1. In this case 

 the invariant X (p, q) vanishes; the integrals of (A) are then of the form 



(10) ^r = Mr(^) + ^r(ç) ^=1. 2, 3. 



The vanishing of V" implies that Hr and v^ satisfy the condition 



(11) / , dv, rf2,3X 

 V dq dq' / 



From (11) we conclude that 

 (12) 



(d V2 d? J/3\ , /(/ Vz d^ Vi\ , /d Vi d^ P2\ , / dv2 d^ V3\ ^ 



The point (/xi, n^, Ms) must therefore either describe a plane curve 

 whose position in space is determined by the functions (Vi, V2, V3), 

 and is naturally independent of the value of q, or else the coefficients 

 of (12) must vanish separately. In the latter case it is evident that 

 the locus of (fi, V2, V3) is a straight line. In the former case let us 

 assume the plane to be 



Ix-^-my -\- nz-\- p = 0, 

 where /, m, n, p are constants. 



Then {y2'v3") = l<t>, 



Wvi") = m.(}>, 



(Pl'Vi") = "0. 

 (j'll'2'l'3") = p(f>, 



and, therefore, 



lvi-\-mP2-\-nvs — p. 

 Combining this result with 



/^l + WM2 + »M3 + /> = 0, 



we have 



ldi-{-md2-\-nd,, = 0. 



