Sum of the Dkifors of Numbers. 
393 
PART V. 
i . S (ot x £)=« x S (£f) 4 - Aim of all the divifors of (3 not 
divilible by u = (2 x S(«} + fum of all the dlvifors of a not di- 
vifible by (3. 
?. S' (a x jG) = « x S' (£)-{- fum of all the dlvifors of (3 divi- 
sible by / but not by u—Slq. 
3 . S (« x (3 x y x 5 x &c.) = *xS(Sx^x Jxf, &c.) + fum 
of all the dlvifors of jS x y x d x e, &c. not divilible bv a zz 
u x (3 x S (y x $ x g, &c.) + lum of all the divifors of /3 x y x S 
x e, &c. not divilible by »+«xfum of all the divifors of 
y x T x e, &c. not divilible by (2 = a x (2 x y x S {l x e, 
fiiin of all the divifors of /2 x- y xT x s, &c. not divilible by 
* + «xfum of all the divifors of y x $ x e, &c. not divilible. 
by (3 + cc x @ x fum of all the divifors of o x g, &c. not divilible 
by y — ctx(2xyx2x S (e, &c,y-f lum of all the divifors of 
(2 x y x $ x e. See. not divilible by-u + x x fum of all the divifors 
of ySe, &c. not divilible by jS+;ax/3xlum of all the divi- 
fors of <5e, &c. not divilible by y-!-«Xj6xyx lum of all the 
divifors of e, &c. not divilible by $ = &c. The law of the 
feries is mnmttfh The letters «, (3, y, S, See. which are net 
contained between the parenthefes, denote prime numbers. 
Cor. If lome of the letters a, (3, y , &c. be luofiftuted 
for others* and others for them, the equations retailing will 
be juft, and coriequently many new equations may be- deduced. 
If in the preceding equations for S be wrote S', and for the 
fum of all ti e divilors of a cerrain quantity not divif.blc by a 
prime number (a, or /3, or y , &c.J be wrote the fum of all 
the 
