[si;i livan] concerning THE INTEGRALS OF LELIEUVRE 181 



If the factor (^ijus') were to vanish, we should find 

 (miV2) + (m/^2) _ , x 

 (aw/) 

 which is impossible. We must therefore have 



(33) T'-lfx/-mti2'-nfxz' = 0. 

 From this and (31) it is necessary that 



(34) Ml'(7-+/^l + W02+«03)=O. 



If Ml' vanishes, the curve (^ti, ^2, Ms) is situated in the plane m = const. 

 If the expression in brackets vanishes, it follows from (33) and (34) 

 that 



(35) /mi + W7M2 + nus = n(q) ; 



and therefore (mi, M2, ms) describes a curve in the plane (35) which must 

 be independent of the particular value assigned to q. Let us consider, 

 as a particular example, the case in which the (v) line is parallel to 

 the (m) plane, and let these be defined as follows: 



(36).. tJ-}=p, M2 = m'. M3 = c, 



v\={), V'z = q, 1^3 = 0. 

 The equations of (5) become 



(37) .r = r(g-M'), 



y = cp, 



s = pij.' — 2n — pq. 

 The equation of the complex (28) becomes 



(38) (x8y-y5x)-\-2c(p5x-}-iJL'ôy)-\-c^z = 0. 

 Hence 



dz y-2cp dz (x + 2cm') 



dx c^ dv \^ 



and therefore 





where / is an arbitrary function. 



The surface (5) defined by (37) is therefore an integral surface of 

 the familiar equation 



(39) 



<dxdyy \dx'^y Vôy/ c^ 

 Among the integrals of this equation are the Cayley Cubic 

 Scrolls (See paper by the Author, Trans. Royal Society of Canada, 

 Series III, Vol. X, p. 130). If we compare the results just referred to 

 with equation (38) of this paper, we shall see that the axes of the com- 

 plexes (38) are parallel and in the xz plane; they are therefore per- 

 pendicular to the (v) line and the plane of the (m) curve. 



Sec. Ill, Sig. 15 



