( 464 ) 

 so that the roots of the ^",-11111011011 uro detcriiiiiiLMl by 



>--2 — f.i =^ P"i — P"J, ' '■3 — ^1 =^ P"ó^P"'l I *i — ^2 ^ P"i — P'^i • 



By interchanging the «'s and the o's we can obtain the second 

 reducible integral. 



Proceeding to consider reducible hyporelliptic integrals of deficiency 

 2)=^^, it is clear that by the same methods also these can be con- 

 structed without gviint difficulty. If the curve f is given by the 

 equation 



(.(.■—«1) {.r—il,) (./•—«;.) //^ = {.v~a.i) (.*•— «J (■/•— «s) > 



hat is: if the curve ƒ is a quintic with a triple point, from which 

 eight tangents can be drawn, each of the curves R^, R^, R^, R^ in 

 the case r^2 is to be made up by a couple of these tangents. 

 The twofold condition for the reducibility expresses that these four 

 pairs of tangents ai'c in involution and it is easily verified that 

 when A.e^ + Bx -\- C defines the double elements of the involution, 

 the reducible intearal will have the form 



ƒ- 



Ax~ + Bx + C 



dx 



{.,: — Oi){,;- — a2){x — a-, )>j 



The investigation of the next case p = 3, /■ = 3, is closely allied 

 to that of the case /> = 2, r = '0 treated before. 

 Suppose the equation of the curve to be 



(,(.— «,)(•''— «2)0'— «b),r = {■'■— u\)ix — u'o)ix — a' ■^){.i: — Ci){x — C2). 



We regard the product of three tangents (.c— ai)(,(-— a2)(*'— «3) 

 as the curve Ri, in the same way we join the second three 

 (;j;_a',)(.r — a'.i)(x—a'^) to a curve R-i ; the third curve R-^ will be the 

 product of the next tangent {-c — c^) and of a line through the node 

 {w—l^), counted twice, similarly the curve R^ is furnished by the 

 tano'ent (-? — «2) and the double line (x — in). 



The twofold condition for the reducibility is that the four binary 

 cubics 



R^ = (.c -ai)(>(.-— a3)(j.'— («a) , R2 = {x—a\){x—u'c,){x—a\) , 



R,, = {x-Ci){x-bif , Ri = (.*—(•;:)(.'— ^2)- 



