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# The following theorem of Kummer is known (see [1], p. 11):

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(1)

BY

p

p

p

p

2πi/p

p

p

p

p%

p

p

σ−%

p

n

nσ

n%

n

n

σ−%

n

1

i

n

nσj

naj

j

1

r

σj−aj

nj

σji−ai

σij−aj

n

n

n

nσ

na

σ−a

[187]

(2)

n

wn

n

n

n

n

j

j

j

n

τ

nk

σjτ

τ σj

najk

σj

j−aj

σj−aj

j

σj−aj

nj

σji−ai

σji−ai

j−aj)(σi−ai)

i−ai)(σj−aj)

σij−aj

n

n

n

σ

nσ

1t1

rtr

j

j

j

j

j

σ

r

j=1

### A

a

t1

1 ...atj−1j−1 σtj+1j+1 ...σrtr

σtj j −atj

j σj −aj

j

σ

σ−a

nσ

t11

trr

(3)

t11

tj−1j−1

tj+1j+1

rtr

tjj

tjj

t11

tjj

j+2tj+2

trr

j+1tj+1

tj+1j+1

t11

tj−1j−1

jtj

trr

t11

tj+1j+1

j+2tj+2

trr

r

j=1

t11

tj−1j−1

tj+1j+1

rtr

tjj

tjj

σ−a

r

j=1

### α

j−aj)at11...a

tj−1

j−1 σtj+1j+1 ...σtrr

σtj j −atj

j σj −aj

r

j=1

### A

at11 ...atj−1j−1 σtj+1j+1 ...σtrr

σtj j −atj

j σj −aj

j

n

nσ

σn

σ

σ

a

nσ

a

σ

n

σ

nu

a

σ

n

n

σ τ

τ σ

nσ τ

nτ σ

1u1

urr

i

i

i

i

i

τ −bσ

σ−aτ

u11

urr

r

i=1

u11

ui−1i−1

ui+1i+1

urr

iui

uii

τ

r

i=1

### A

a

u1

1 ...aui−1i−1 σi+1ui+1...σurr σ

ui i −aui

i σi−ai

i

(4)

τ −bσ

r

j=1 r

i=1

### A

i−ai)a

t1

1 ...atj−1j−1 σj+1tj+1...σtrr

σtj j −atj

j

σj −aj au11 ...aui−1i−1 σui+1i+1 ...σurr σ

ui i −aui

i σi−ai

j

r

i=1 r

j=1

### A

j−aj)a

u1

1 ...aui−1i−1 σui+1i+1 ...σrurσ

ui i −aui

i

σi−ai at11 ...atj−1j−1 σj+1tj+1...σtrr

σtj j −atj

j σj −aj

i

σ−aτ

τ

nv

b

τ

nσ

nσ

at11...a

tr

n 1

na

nτ

nb

σ τ

nub+va

ab

aτ

rσ

τ σ

nub+va

ab

bσ

στ

aτ

τσ

bσ

στ

σ τ

τ σ

1−aj

nj

1−aj i

1−ai j

i

j

wn

n

n

1

r

1−aj

nj

wn

n

1−aj

nj

1−aj i

1−ai j

1−aj

nj

1−aj

wn(1−aj)/wn

n(1−aj)/wn

nj

j

wxjn

(1−aj)/wn

1−aj i

(1−aj)(1−ai)/wn

1−ai j

(5)

4

4

8

1

8

8−1

2

8

85

1

2

σ1−a1

2

81

σ2−a2

−4

82

1

4i

4

2

4j

4

−2

1

2

4

2

8

σ12−a2

σ21−a1

m

m

m

n

n

m

n

m

wn

n

n

1wn

1

ab

wn

nwa n

1

ab

n

n

n

n

n

a

n

na

a

n

τ

nj

στ

τ σ

naj

σ

(σ−a)τ

σ−a

a

1−a

na

n

σa∈H

wn

n

### Re¸cu par la R´edaction le 19.12.1993

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