KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58. NIO 3. 25 



But both in case I and in case II we have already neglected terms whose 

 coefficients are of a smaller order of magnitude than /? 04 . Thus in expanding the 

 denominator we should have to neglect such terms. Hence we have to put: 



In for mula I 



In formula II 



(33) ii _J_ --i-PMri-PoiRM-^PltRM+PliRXx)- 



Now these expressions will not give a good approximation at the extremes of 

 the above ranges. As we may well deem the approximation to be very good if, say, 

 the first term neglected does not amount to more than one fifth of the last term 

 included, we obtain the following ranges within which our formulae (33) may be used 

 with perfect safety. 



Case I. Range for which \^B s (rj) + p ei B t (r))\< 0,2 



|& 3 | = 1:^1 = 0,002 1 1] \ < 3,5 



1^3 1 HA* I = 0,oo.i hl < 3,o 



l/»oil = l/»«|-0,oi6 M<2,ö 



l,'o,| = |^ J = 0,O3O l',l<2,o. 



In this case the greatest error to be feared is that produced by multiplying 

 by 0.8 instead of by 0,84. 



Case II. Range for which \2p 03 p ni B 3 ( v )B t (v) + fiUB,(^[B u (f i ) — B\{fi)-]\<0,t\p M R i (f i ) + 



+ |«.[Ä.(^)-2Ä',(i?)|. 



I .>.»3 1 = 0,006 I ,>' 04 1 = 0,0000 \i\<3,r, 

 I ;' os | = 0,oio | A, | = 0,0001 l',l<3,o 



Mos I = 0,0 17 | /? <| =0,0003 M < 2,5 



|p\> 3 | = 0,058 |^04| = 0.0034 h| < 2,0. 



Here, however, the restriction that the quantities neglected shall amount to 

 only 0.2 of the smallest quantities included is at times unnecessarily punctilious. 

 A positive sign of -S IA will also have the effect of improving the convergence. On 

 the whole the ranges may often be considerably greater than those given above, or, 

 the range being fixed, the magnitude of /S s may be allowed to be greater. 



If now the forms (33) are used for the denominators in (I) and (II) and the 

 factors are multiplied, we shall again have to neglect terms of lower order than those 

 contained in the factors. 



The resulting expressions will be much simplified by arranging the terms according 

 to powers of rj. It ivill be found that in this case the regression will be cubic. The 



K. Sv. Vet. Ak.nl. Harull. Band 58. N:o 3. 4 



