333 ' 



and T, and the numerator A' of the integral, which we iia\ e found 

 to be 



is identical with the linear quantic ^ê^. 



Obviously, we may now conclude that as soon as tiie given 

 integral is reducible, thej-e are three other integrals of deticiency 

 p z= 2 connected with the involution J, i.e. the integrals 



I , dx , I dx , I dx , 



which can be reduced to elliptic integrals. Moreover, it will be evident 

 that the transformation of the four integrals will be effected by one 

 and the same transformation formula, and we may notice that 

 likewise the integral 



T 



J. 



dx 



of deficiency /? = 3 becomes elliptic by that transformation. 



In order to find how the involution »/ can be constructed from the 

 given quadrics i|.',,tf?2,if'3, 1 will proceed with the analj^tical investiga- 

 tion of the system S. It is always possible by adjoining suitable 

 constant factors to the quadrics c,^, .sj, £, to ensure the identical 

 relation 



i/l + I/f, -f 1/I3 - 0, 

 and hence the relation 



1.^ + §/ + §,^ - 2§,^, - 2^35, - 25,§, = 0, 

 an identity in the variable x that denotes at the same time the 

 conic K in the trilinear coordinates si,^»>Si- 



The point A^ on K, the coordinates of which in the system 

 ^,,^^,, ^, are (0,1,1), has its conjugate with respect to the system .S 

 at the point A\ where the tangent §1 of K meets the double line 

 T of the system. 

 Supposing T to have the equation 



T=U,^ M^,+'Ni,:=0, (1) 



this point A\ has the coordinates (0, —N, M), hence the coefficients 

 of the equation 



.If,* + ^Ir + c%,' f 2i^£,s% + 2(?^,§, + 2h:,^, = 0, 

 representing any conic of S, underly the condition 

 BN- FM -f FiN- M) — 0, 



or 



