( 312 



1111 1 



- 1 



1 + 4 COS 72 



^^•'= ^ + l^-n + 21+2G+-^2.^'^"^''^ + ^"^''°) 



-= + 1.128, 



Slorj2sm 18' 



11111 1 ' 



7,0= —— - — - — H- — — — = — (sin 72^ — 3 sin 36°) + 



2 7 17 22 37 2jr ^ ^^ 



1 + 4 cos 72° 

 + .,\, . ,^o :=- 0,710 



iO// 2 .?iW lb 



11111 1 



7\3= 1 1 = — (— sin 72° + 3 sin 36^) + 



"'^ 3 13 23 33 '38 2rr ^ ^ ^^ 



1 -J- 4 co.s 72° 

 ^ 8 loff 2 sin 18° 



11111 1 



7'.5 4 = f 1 ^ irz — (— 3 67';/ 72° — sin 3G°) — 



14 10 20 ' 34 ' 30 2.T 



1 + 4 cos 72° 



— _j- 0.034. 



8 lop 2 sm 18' 



100 



As a iiiinicrieal test I liavc direct l_v cak-ulated 7\ji . Tlie results 

 were respectively: +1.123, -0.685,'— 0.449, +0.03G. 



From these few particular cases il will he evident that the e(|nations 

 (F) and {G) always permit to evaluate 7\ii and the fact that such 

 a series has in all cases a finite sum may with more or less justice be 

 interpreted thus: Among the integers without quadi-atic factors less 

 than a given large number </, that are solutions of a given congruence 



.V ^ Il . . . . {mod. h) 



the integers made up by an odd number of prime factors are 

 sensibly equal in number to the integers made up by an even number 

 of prime factors. 



Botany. — "The Ascus-form of Aspcnjillus fumlgatu.s Fresenius'\ 

 By Dr. G. Grijns. (Communicated by Prof. F. A. F. C. Went.) 



While during the last course I was occupied with determining 

 fungi in the botanical laboratory under the superintendence of Prof. Went, 

 I noticed that iu a pure culture of Aspergillus fumigatus, a couple 

 of months old, sporefruits had formed ; on inoculation from this 

 culture these same bodies were always produced in the new cultures. 



As nutrient substance I used Koning's malt-canesugar-agar-agar. 



The ascus-form of Aspergillus fumigatus has not yet been described, 

 for the statements of Behrens and Siebenmann are justly doubted by 

 Wehmer, nor do they agree ^vith my result. 



