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**Example text**

Of Corollary 3, P a r t In this I, section to certain we c o n s i d e r applications g r o u p s os g e n u s two~ t h r e e t and five. If W1 Z2 x Z 2 let admits a group of automorphisms, G = {~1,~2,~3,~4 } *~ = *~ = *~ = ' l ' thus have the Let following where W1/<*j~ - Wj covers G, i s o m o r p h i c ~1 " i d , and l e t ~2 ~5 = ~4 W0 = WI/S. to and We of surfaces W1 W2 h'3 W4 Wo where each line Pi segment corresponds i s the genus Of (10) Wi~ i = 0 ~ 1 , 2 , 3 , 4 Pl admitting for the quotients.

With these considerations we will indicate hew Lemma 3 of Part I is proven. Lemma 1 : Let (i) describe the map (2) a_~0 = ~I~ a : J(W0) ยง J(WI) where nJ 0 = ~J1 ~. Then I "~2 36 where ~(u) is a~ n t h o r d e r characteristic Proof: If [;] n - 1 transformation first order (2) formally, the theory result theory for is first transformations. , P0 = P l ) ' transformation n th order trans- f o r m a t i o n s 2) . The s l i g h t this degree lemma f o l l o w s go t h r o u g h when w o r k when ~ are from the P0 ~ P l ; and satisfied.

F2 ) and ( 2 f l ) since given in Part III of this 55 E(U) i s t h e c o n s t a n t one by Lemma 1, P a r t IX. Xoj (1/4)~ i s t o be f d e f i n e d by o n e - f o u r t h of the values of I duo where t h e p a t h os x01 integration to points is restricted in t h i s to AO~ f o r i f we s h r i n k 8A 0 and 8A 1 c a s e we o b t a i n ~ i n t h e l i m i t O[n' + ~ a ' ] ( O ; B q ) { O F g ] ( u ; B O' )2 " s (u;Bo')2 The d e p e n d e n c e o f t h e c o n s t a n t s i n g p r o b l e m which we b e l i e v e is open.