Mathematics. — ''On hyperelliptic integrals of deficiency p=^1, 

 reducible by a transformation of order r = 4." By Prof. J. C. 

 Kluyver. 



(Communicated in the meeting of Sept. 29, 1917). 



The conditions that an hyperelliptic integral of deficiency p = 2 

 is reducible to an elliptic integral by a transformation of order 

 r = 4, have been assigned by Bolza '), who used direct algebraic 

 methods and also by Igel '), who based his deductions on the trans- 

 formation of the double thetas. 1 will show that the geometry of a 

 linear system of conies affords the means to solve the problem, and 

 that geometric considerations enable us lo add some results to those 

 previously obtained '). 



Let the integral be of the form 



XdiV 



ƒ; 



where X is a linear quantic and tf?i,tfj,,i|?, are binary quadrics of 

 the variable x = x\ : .%\, then the integral is reducible under the 

 following three conditions: 



1. There are three quadrics Si,S,,§„ each of which is a perfect 

 square, such that the quartics ^itpi,§,V',>^8V'8 are linearly connected. 

 Otherwise slated, these quartics are elements of an involution J 

 of order 4. 



2. The mvolution J contains an element T\ a quartic being a 

 perfect square. 



3. The numerator X of the integral has a determinate form 

 depending on ipi,T|'2,if'3. 



In fact, supposing the first and the second of these conditions to 

 be fulfilled, we can take in J any two elements whatever R^ and 

 /?,, and k,,k„k^,h being certain constants, we have 

 ^^x^,^=:R—k,R, , §,xp,^R-k,R, , i,^, = R,-k,R, , T'=zR,--hR,, 



then putting 



R' 



1) Math. Ann., Bd. 28, p. 447. 



2) Monatshefte fur Math. u. Phys., 11, p. 157. 



3) For a summary of the researches on reducible Abelian integrals see : W. C. 

 Post, Dissertation, Leyden, 1917. 



