( 455 ) 



The simplest ease of tlie i'etliicil)le hyperelliptlc iiiteciTal, p = 2, 

 r =z 2, was already known to Legendre. Again the eurve ƒ is of 

 the fourth order, but it has now a node from which six tangents 

 can be drawn to the curve. If there is a reducible integral each of 

 the first three functions R is made up by a pair of these tangents, 

 as fourth function R4 we must take one of the two double elements, 

 counted twice, of the involution which is now necessarily formed 

 by the three pairs of tangents. 



The equation of the curve being 



f= xf - (.r - 1) (.. - /•/) (. - /.:) {X - /) = 

 we get in this way 



R^ — X, R,— (:x — 1) (.(• — />•/), /.':; = {x — k) (r — I), 7?,, = {x ± \//cl)^ 

 and the rcilueildi' integral is 



' (./• qr [/ Id) dx 



W 



-ƒ' 



xy 



In accordance with what resulted from the theorem of Weierstrass 

 for the case p = 2 two independent reducible integrals are obtained. 



Also the case p = 2, )■ =: 3 has been considered from various sides. 

 As before the integral is relative to a nodal (]uai'tic ƒ the equation 

 of which we take in the form 



(■'■ — «!)(■'' — '"l) '/ = (•'■ — "2)0'' — «3)(-i' — ^2)0'' — fy) • 



BuRKHARDT^) has pointed out the invariant relation existing between 

 the binary cubics (.?— aj(.r— «^)(.r— 03) and (.;■— ci)(.c— f2)('^— «3) when 

 one of the integrals is reducible. Previously Goursat ") had treated 

 a more or less particular case of the reducibility and finally Burn- 

 side^) indicated in connection with his more general researches a 

 remarkable form which the reducible integral can always assume. 



After the deduction of some of their results a few lemarks will 

 be added. 



The curves fi, each of which breaks up into thr(^c right lines, must 



») Matli. Annal, Vol. 36, 1890, p. 410 



2) Uoiiipt. Rendus, 100, 1885, p. C22. 



') Pioc. Loud. Math. Soc, 23, 1892, p. 173. 



