KUNGL. SV. VKT. AKADEMIENS HANDLINGAR. BAND 58. N:0 3. 31 



Evidently we have to put 



' - ; """ / ( „ ) + ^3^+Ä^<°i + --- 



Using formulae V (21), (22), (23), (24), (25) we find for case I of approximation 



+ ^R 5 (r } ))+(J Od (3R 3 (r}) + GrSR 5 (r}) + r*R 1 {r j )) + 24:(i t0 +ti 3l 2*rR 2 (y) + 



(48) 



+ p 22 12(RAy)+r*RAv)) + Pi3(l2rRAy) + ±r*RAr ) )) + p 0i (3R i (ri) + (ir*R (r}) + 

 + r*R a (r i )J:il+{l 09 R a (r } ) + {! 9i R t (r})l 



If the denominator be inverted according to (33) and the terms arranged accord- 

 ing to powers of rj we obtain 



»'„(£)„ = 3 + 24/? 40 - 12 ft, — 6r 2 + 3r* + 36r/9 13 — 24r/? 81 + 36r 2 /? 22 — 72r s /? 04 — 60r»/J ia + 96r*/S 0J - 



- ij (12/S 2I + 24r/?, — 36r/? 12 — 36r 2 /? 21 + 72r 2 /? 03 + 60r 3 /? 12 — 96r*/? 03 ) + 



+ /; 2 (6r l — 6> 4 + 12ft 2 + 24r/?„ — 72rft 3 — 72r 2 & 2 + 216r 2 /S 04 + 180rVti + 384r 4 .;„J 

 (49) - r/»(12rft 2 + 12r*&, - 36r 2 /? 03 - 40r s /S I2 + 84r 4 /? 03 ) + 



+ if(r* + I2r/? IS + 12r 8 /? 22 — 48r 2 /? 04 — 60r 3 /? 12 + 168rVoJ~ 



— «j B (4r s /? 12 — I2rV M ) + 

 + i ? 6 (4r 3 /? !3 -16rVoJ- 



Here the powers of 9 higher than the sixth order are multiplied by factors t ha t 

 are of the order neglected in case I. 



Here also we may check by multiplying (49) b^v F 2 (r)dr and » ( F, (/)rf/ and inte- 

 grating. Tt is only necessary to remember that 



oo oo co 



j v'A£)r,F 2 ( n )dr t - J j §* F(§, 9)^1^-24^, + 3; 



— 00 00 — OO 



00 00 00 



j\v,(i)„F 2 (,^,= j f?tiF(§,ri)d§dii= -24r/? 30 -12/? 21 -[24/? 4l ]. 



(50) 



The result vvill be found satisfactory. 



Using formulae (34), (34* I), (40 I), (46) we find with the same approximation 

 as before 



vÅ$)vlv = — V (6r&o— 6r«/? 21 + 6r 3 /? I2 — 6r*/?„) + 



-, i 8 (6r/y 31 --12r 3 /J 22 + 18r s /S l3 -24r*/?<u); 



