326 



METHODS OF INTERPOLATION. 



Logarithms of powers of integers, 4'c. — Contiuued. 



'X 



loga;2 



loga;3 



loga;« 



log x^ 



log x^ 



■ log x^ 



log x^ 



31.0 



2. 9827234 



4. 4740851 



5. 9654468 



7. 45680^5 



8. 9481702 



10. 4395319 



11.9308930 



31.5 



2. 9966211 



4. 4949317 



5. 9932422 



7. 4915528 



8. 9898033 



10. 4881739 



11. 9804844 



32.0 



3.0103000 



4.5154499 



6. 0205999 



7. 5257499 



9. 0308999 



10. 5360498 



12.0411998 



32.5 



3. 0237067 



4. 5356501 



6. 0475334 



7. .5594168 



9.0713002 



10. 5831835 



12. 0950669 



33.0 



3. 0370279 



4. 5555418 



6. 0740538 



7. 5925697 



9.1110836 



10. 6295976 



12.1481115 



33.5 



3. 0500896 



4. 5751344 



6. 1001792 



7. 6252240 



9. 1502688 



10. 6753136 



12. 2003585 



34.0 



3. 0629578 



4. 5944368 



6. 12591.57 



7. 6573946 



9. 1888735 



10. 7203524 



12. 2518311 



34.5 



3. 0756382 



4 6134573 



6. 1512764 



7. 6890955 



9. 2269146 



10. 7G47337 



12. 30255M3 



35.0 



3. -0881361 



4. 6322041 



6. 1762722 



7. 7203402 



9. 2644083 



10. 8084763 



12. 3525444 



35.5 



3.»1004567 



4. 6506851 



6.2009134 



7.7511418 



9. 3013701 



10. 8515985 



12. 4018-^68 



36.0 



3. 1126050 



4. 6689075 



6. 2252100 



7.7815125 



9. 3378150 



10. 8941175 



12. 4504200 



36.5 



3. 1245857 



4. 6868786 



6.2491715 



7.8114643 



9. 3737572 



10.9360501 



12. 4983429 



.37.0 



3. 1364034 



4 7046052 



0. 2728069 



7.8410086 



9.4092103 



10.9774121 



12.5456138 



37.5 



3. 1480625 



4. 7220938 



6.2901251 



7. 8701563 



9.4441876 



11.0182189 



12. 5922591 



38.0 



3. 1595672 



4, 7393508 



6.3191344 



7. 8989180 



9. 4787016 



11. 05848.52 



12. 6382688 



38.5 



3.1709215 



4. 7563822 



6.3418129 



7. 9273036 



9. 5127644 



11. 0982251 



12. 6836858 



39.0 



3.1821292 



4. 7731938 



6. 3042584 



7. 9553230 



9. 546CSi6 



11. 1374522 



12. 7285169 



39.5 



3. 1931942 



4. 7897913 



6. 3883884 



7. 9829855 



9. 5795826 



11. 1761797 



12. 7727768 



40.0 



3. 2041200 



4. 8061800 



6. 4082400 



8.0103000 



9. 6123599 



11.2144199 



12. 8104799 



40.5 



3.2149100 



4. 8223651 



6.4298201 



8. 0372751 



9. 6447301 



11.2521852 



12. 8596402 



41.0 



3. C255677 



4. 8383516 



6. 45U354 



8. 0639193 



9.6767031 



11.2894870 



12. 9022709 



41.5 



3. 2360962 



4. 8541443 



6. 4721924 



8. 0902405 



9. 7082886 



1 1. 3263367 



12. 9443848 



42.0 



3. 2464986 



4. 8697479 



6. 4929972 



8. 1162465 



9. 7394957 



11.3627450 



12. 9859943 



42.5. 



3. 2567779 



4. 8851668 



6. 5135557 



8. 1419447 



9. 7703330 



11. 3987225 



13.0271114 



43.0 



3. 2669369 



4. 9004054 



G. 5.338738 



8. 1673423 



9. 8008107 



11. 4342792 



13. 0677476 



43.5 



3. 2769785 



4. 9154678 



6. 5539570 



8. 1924463 



9. 8309355 



11.4694248 



13. 1079141 



44.0 



'S. 2869054 



4. 930.3580 



6. 5738107 



8. 2172634 



9. 8607161 



11.5041687 



13. 1476214 



44.5 



3. 2967200 



4. 9450800 



6.5934400 



8.2418001 



9. 8901601 



11. .5385201 



13. 1868801 



45.0 



3. 3064250 



4. 959S375 



6. 6128501 



8. 2660626 



9.9192751 



11. .5724876 



13. 2257001 



45.5 



3. 3160228 



4. 9740342 



6. 0320456 



8. 2900570 



9. 9480084 



11. 6060798 



13.2640912 



46.0 



3.3255157 



4. 9882735 



6.6510313 



8. 3137892 



9. 9765470 



11.6393048 



13. 3020627 



46.5 



3. 3349059 



5. 0023589 



6.6698118 



8. 3372648 



10. 0047177 



11. 6721707 



13. 3396236 



47.0 



3. 3441957 



5. 0162936 



6. 6883914 



8. 3604893 



10. 0325871 



1 1. 7040850 



13. 3767829 



47.5 



3. 3533872 



5. 0300808 



6. 7067744 



8. 383468U 



10. 0601617 



11. 7308553 



13.4135489 



48.0 



3. 3624825 



5. 0437237 



6. 7249649 



8. 4002062 



10.0874474 



11.7686887 



13. 4499299 



48.5 



3. 3714835 



5. 0572252 



6. 7429670 



8. 4287087 



10.1144504 



11. 8001922 



13. 4859339 



49.0 



3. 3803922 



5. 0705882 



6. 7607S43 



8. 4509804 



10. 1411765 



11.8313726 



13. 5215686 



49.5 



3. 3892104 



5. 0838156 



6. 7784208 



8. 4730200 



10. 1676312 



11. 8622304 



13. 55084 16 



50.0 



3. 3979400 



5. 0969100 



6. 7958800 



8. 4948500 



10. 1938200 



11. 8927900 



13. 5917000 



Before leaving tlie subject of iuterpolation by the "first method," it 

 may be well to notice that the formulas (A), (B), &c., which require that 

 the assumed groups should be of equal exteut, can be used for the pur- 

 pose of ordinary interpolation from single terms which are equidistant, 

 and thus may take the place of the ordinary formula for interpolation 

 by finite differences. We have only to regard the single terms as being 

 represented by Si, S2, &c., respectively, and to take % and 7^ both equal 

 to unity. Suppose, for instance, th:it from the three equidistant terms— 



13 21 35 



we wish to interpolate the value midway between the two last. The 

 ordinary formula is — 



and we have — 



a = 13, Ai= 8, A2= 6 



so that the equation of the series becomes — 



■it = 13 + 8a? -f ox {x — 1)' 

 and for .r = f we get tt = 27^, the value sought. 



