
[M‘poxazp] NUMBER OF REPRESENTATIONS OF A NUMBER 83 
Since 2 w, isa period of p u, p (u—w)) = p (u + WA) and 
p' (2u) +p'(u + w))= 2 
p (2u)+2p (u+twy) =F or 
But we have Schwarz, loc. cit. : 
‘ad 
p Qu) +2pu=p2wt+w))+2pu= +=. log p'(u + w,)) 

5 p' Qu) + pu + w;)-; 
or / CLR À 
+ (5 log p (w+w,)) = À | p Qu) —pu + (4) | 
or p' (2u) + p (w+ wy) p' (u + wy) 
2 — EE 0 Se 
p Cu) — p (u + wy) Dake =f wy) 
It is not necessary to determine which sign is to be taken (which 
could be done from the expansions in series of p, p', p’’), as that is inde- 
pendent of wy and does not affect the cross-ratio value. We have, there- 
fore, to find the cross-ratio of the four quantities 
p' (w+) 
put WA) 

Denote w,, w,, ws indifferently by w,, Do we have 
CEE eg) Cy ey) 
EU a PT 
which is found by differentiating the relation 
NC eg) (ee ED) 
p (u — CAE 

p(utw,)=e,+ 
Differentiating again : 
= (yep Sy ey) Loue) p'u—2 (ue) pl 

p" (u ore = (pu TF eins 
and finally 
Dp (u+ Ww Li p'u 
0 arg DRE Stee 
pws) pt pu—e,, 
Denoting the four quantities by 0, 1, 2, 3, we form the cross-ratio [0312] 
and find it to be 
€, — 
€, — 

. i.e., independent of wu. q.e.d. 
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