t 439 ) 



l'everse. In coiiseqnciU'C ol this (n/r/ (lilfereiilicul quolieiits I - , 



Kdrjic \dryi, 



etc. in Mac-Laurin's expansion into series, henee also the coefficients 

 a, e, etc. for d and d' , will get O[)posite signs. 



If we begin by assnming tlie coefficients in d' to be different from 

 those in (/, tlie calculation gets onlv much more elaborate, but 

 finally we get also a = a' , b =z h' , r = r', etc., of which we have 

 convinced ourselves for the greater security. 

 With 



X z=z ax -f trr' + ''■''* + ••• 

 y = br^ + <lx' + . . . 

 we get : 



d = 1 -f •^'^- + '^^ I '^ —d 7=2 (l-.f— v) I d^ d: = 2 (l + 2.y) 

 d' = 1 — 2.V + 27/ I 3 — (/' == 2 (l + ,v-7/) : d - d: = 4a;. 

 And so (1"), viz. (3 — (/) (3 — r/') ((/ + (/') = «//?, passes into 

 (l-.t--.v)(l-f.7;-^)(lH 2r/) = m, 



oi- also, because (1 — .i- — //) (1 -|-,;,' — //)= (1 — i/f — ,c- = 1 — 2//— (.t-"— ƒ ), 

 into 



Now substituting the values of ,/■ and // iji this, we get: 

 l — 4{lrT'-{-2bdz') — {l +2/>r^ + 2t/r^) [(aVM- 2a(jr^ + r:V''-|-2ar'T«) - 



— (//'r^-f 2/^r7r")] =: I — t'-\ 

 when in the expansion into series we take oidv the terms with r" 

 into consideration. As r::=' i -///, m is = 1 — t-. 



Working this out, we get: 



1 _ 4//'T^ — Shdr' — (1 \-2Uj" -\-2dx') \<rx- f {2ac - Ir) x' + 



4- (c- + 2ar'-2A(/) t'] = 1 — x\ 

 i. e. 



1 - a-x' - (2rr/>-f 3//- + 2r/r) t^ - {4ahc-2h'-^c'-j-2a'd-{-(ibd-{-2<v')x' = I-t'. 

 So it ft)llows from this that 



a' = 1 2a^/H-3/>M 2m- = Aabc — 2b' -^- c' + 2ar-d + Öbd + 2ar = 0, 



and we find already immediately from o(piation (1") alone : 



a — \ . 



Hence the two other- above equations pass into: 

 2/> + W + 2r- = : Abe - 2b' + t--^ + 2r/ ^- G/x/ ^ 2c' = . . [u) 



For the determiuaiion of the other coefïicients we must now use 

 the second equation, viz. (2). This becomes after substitution of the 



