( 661 ) 



hence we Ihid 



and as before 



c = J). 



Had we inteuralcil llio product //„,(. /v/) //„(////), where in -\- ii is 

 even, iirstead of y„, {.rti) ƒ/,„ {//tt) the result wouhl have been siniihir, 

 oidy synimetrv wonhl have beeji h)st. 



We may reniai-k that llie z in e^piatioji (V) is still an al,i>ebraical 

 function. Fur reniembcrini»' that 



ƒ/^• ('0 = !ik-i (")' 



we deduce by repeated partial integration 

 oi' fiiiallv 



7^^- 



:^— 2 



iiA'! v = 



+ (-iy^— .'/2/L-+l(//)-+- 

 // 



+ ( __ 1 )k -1 .,2/.-i j ■" ~:s''],,i^ f ('A j^ 1 ,/,^. (0) + ;,,,^ (,f) r {.,-) j . ( VI) 



( ;/ = I V ■'■ / - ' 



From this e([uation we infer tliat the product --'^ty/ is a rational 

 ijitegral function of ./• and // of degree 4 /: -j- 2, and ,ü;enerally speaking 

 the equatioii re[)resents an algebraical surface .S' of that degree. But 

 it should be noticed that this surface ;S' in reality is composed of an 

 iidinite number of [jartial surfaces, having contact more or less close 

 along a system of plane curves C. And in fact the hxrger tlic integer 

 /• be chosen the closer will l)e the contact of the partial surfaces. 

 Equation (VI) contains the equations of all the partial surfaces, but 

 each of them has a distinct ecpiation the coeHicients of which are 

 made up tVmn the integers 



{x\, [//| and 



(ft= 1,2,3, 



|..J). 



Hence we pass froju one partial surface to an adjacent one in 

 all places, where one at least of these integers increases by unity. 



