( 109 ) 



2 



LP.ZP, r, z, + ^% r. 



T, 



Tlie development is symmetrical with respect to z^ and z^ and 

 tlieir derivatives, and resolves itself into two parts, which have the 

 same form, and which depend besides on r^, only on the valne of 

 z and those of the derivatives of z at one point. 



If the following series 



^ - ^. ^ + {^^ + 2.,) ^ - (8.A + 11^.^ + 8i c,iv + ,^3^ ^ . . 



where only the odd powers of the variable t occur, be denoted by U^r) 

 and the corresponding series for z^ and its derivatives by Uj{x), we get : 



and the ratios of the triangles may be expressed in the following 

 way in these functions Ü : 



t^P,ZP, Lr,(r.) + t/.(T.) 



AP;^ = "'=f/.(r,)+£/.(r.) • • • • ('^'^''> 

 and 



AP3ZP, "^^ U,{x,) + f/3(r,) • • • • ^ ^ 



In the series Ü{x) only such differential quotients occur as can be 



rationally expressed in j) and . By means of the known differential 



a 



equations of the i«^ and of the 2"^^ order for v 



.^ _ 2 1 i> . , .. _ P 1 



we obtain by differentiating z = 



r r 

 1 



~ = 32M9 — 4- — 5- 

 a r ^ 



hi 40 ^' 



while from the differential equation c"' =: 5 tt ^ ~ •^^^ '^J diffe- 



z 9 2^ 



rentiation with lespect to t and by elimination of -p^ the following 



expression is found for c'^' . 



8* 



