136 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



IVol. XIV. 



If in Z eq. (155) we add and subtracu -g-e — [n'os']) 

 1 



C = 



Substituting this value of f in eq. (156), 



(^c-[n'd2']) + '^c-c'+(j.-[n'dz']) 



,>-|e + K.V]-^c + c' = — f 



(|.-[.'.V]) 



+ Series 



1+2(^^ + 5,^) 



Substituting the last equation in eq. (145), we obtain Z 98, eqs. (159), (160), and (161). In 

 eq. (160) the factor (c — c) is an approximation for - (^ — Zo) '> in our work we have used the latter. 



Since [ndz\ is the series in eq. (156) multiplied by the factor 



-; > 



1 — W 



= ^[ndzl+{: 



Table XXXV. — With the exception of the two coefficients under the heading w', all the 

 bracketed quantities are functions of other coefficients in parentheses or brackets, or they are 

 functions of additional terms. The two coefficients excepted seem to be in disagreement through 

 some numerical error by v. Zeipel. 



Table XXXVI. — Since the mass factors have not been kept explicit, it may be well to remark 

 that only the zero degree term of third order has been included under the heading w~^. 



The bracketed quantities are numerous. Aside from the accumulation of discrepancies 

 already discussed, the disagreements are to be attributed, in general, to the relative extent of 

 the computations. It is found from computation that as the number of terms included in a 

 product is increased the resulting coefficient for a given argument is numerically larger. For 

 the most part our values are larger than v. Zeipel's. Hence the discrepancies are explained by 

 assuming that our computation is more extensive. On the other hand, the function is com- 

 puted much more accurately than is necessary, and many of our disagreements are less important 

 than they appear to be. 



Table XXXVII.— The comparison of Tables XXXVII is similar to that for Tables XXX"\T[ 

 with the exception that our values are not, in general, numerically larger. Some are larger 

 and some are smaller. Below are brief tables showing to what extent we used the necessary 

 series. The 0, 1, 2 signify the degrees of the terms included. 



l^{,nlz-[nSz]] 



