45(> 



or, after development into series ^^•ith respect to the powers of v' 



'3 3 



8'^2' ;■ ' 8 





9 63 , 



^ 64^ ^ '^ 160 ^ 



9 , 3 



(2y'-3x') + 2y'«' 



^'' + 



+ 



(/->«') + y'^' 



9 



'-'"-ö(3v'-2»«')^^''+--. 



(36) 



(37) 



x,=i(i+0[(i-rr+^1+^-(2'<'-^')-^7(^7"-'iy'+--^^'+ö'^'-3^^^ 



2 bo 



for which hist expression we wiile : 



X, = ^o4-^:«' + ^, '•' + (39) 



Determination of the phitpoiut and rJass/'jirat/'on of the 

 clijl'ereni possih/c cose.s. 



17. For cahMihitint;- tlie coordinates ?-'^, and ./'^^ of the plaitpoint 

 we ha\e the foUow in.ii- rcUitions : ') 



o./.-" ^ or' ö.f 



ö>' av 



01' Ö.V. ov 



m 





a'lf' ö>' 





(40) 

 (41) 

 (42) 



wiiere m represents') the taii.ücnt of the aiiuic loniicd liv the (r', ,?■)- 

 projection of the comnion tantient of s|)iiio(hil and coinio(hil cnrxe in 

 the phiit|)oint with the /"'-axis. 



If l)v means of (32), (36) and (37) we intro(hice in tiiese eqnations 

 everywhere the vahies of the ditfei-ential (|iiotients at tii'st approxi- 

 mation, in which, as ap})ears, ///, ./'^ and r'^, may l»e treated as 

 small cpiantities of llie same order, we (ind : 



ni 3 

 — --(2y'-3x') = 



(43) 



1) D. J. KoRTEWEG. Ufbcr pallfiipmiktc Wiriicr Sitzungshoiielitt', Bd. 98, 

 Abt. II, (1889), p. 1171. 

 -j See 1. c. p. 11 G3. 



