SERIERUM SINGULARIUM, 
21 
* 
~ t + etc. 
cum termino m-\- l t0 , qui est = oc' im [m, m). Quum sit 
(m,m) = 1, {m, m— 1) = [m, 1), [m, m—2) =±= (m, 2) etc. 
in. 
progressio ita quoque exhiberi poterit: 
F[x,m) = x im -\- x'^ m ~ l ) [m, 1) -j- ^( m — 2 ) 2) -J- x^ m ~~'*) (m, 3) -f- etc. 
^ J x m—2A+1) 
Hinc fit 
F{x,m) — 1 -j- <#(m, 1) -f- a?(?w, 2) -f- a?'* (m, 3) -j- etc. 
etc. in iniin. 
-f- x*. x rn -f- x. x m ~ x (;m, 1) -f- a?*. a? m—2 (m, 2) -j- etc. 
in. 
Quare quum habeatur (art. 5. II) 
gere palam est. 
ubi fit 
(m, 1) -j- a? m = (m -}-l, 1) 
(m, 2) -f- a? m—1 (m, 1) = (m-f-1, 2) 
(m, 3) -f- a? m ~ 2 (m, 2) = [m-\-1,3) etc., 
’-f- etCi 
provenit 
(l-|-a?*" m+ *) F{x,m) — F[x,m-(-1) [3] 
1 X 8 
■ — etc. 
1 X 
Sed iit -F(a?, 0) = 1 : quamobrem erit 
F{x, 1) = 1 -f-a?* 
F (a?, 2) = (1 -f-x*) (1 -f- a?) 
F(x, 3) = (l —{— x*} (i —(— a?) (1 —j— x*j etc., 
sive generaliter 
quas alia occasione reve- 
F{x,m) = (l-f-a?*)(l-|-a?) (1-j-a?*).. . . [\-\-x* m ) .... [4] 
X m ~ i ) (l — x m ~ 2 ) . 
. , —[— etc. 
XX) (i — x) 1 
10. 
Praemissis hisce disquisitionibus praeliminaribus iam propius ad propositum 
nostrum accedamus. Quum pro valore primo ipsius n quadrata 1,4, 9 — [\[n—l)) 2 
omnia inter se incongrua sint secundum modulum n, patet, illorum residua mi 
nima secundum hunc modulum cum numeris a identica esse debere, adeoque 
4) -j- etc. 
2 cos a k io = cos k io -f- cos 4 k io -f- cos 9 k 10 -j- etc. -j- cos [%[n — 1 )) 2 k 10 
2sinak<Ji = sin&to-j-sin 4 Arto-f-sinQ A'u> + etc. —J— sin— 1 )'fku> 
lisquisitionem ad casum 
iries sernper abrumpatur 
Perinde quum eadem quadrata 1, 4, 9 . . . . (¡|\n — l)) 2 ordine inverso congrua sint 
his (i-(w-f-l)) 2 , 3)) 2 , (f(w-J-5)) 2 . . . . [n— l) 2 , etiam erit