INT 



INTERPOLATIONS.!' Woodhouse, Vines.) 



If a, a', a", &c. are successive values of a quantity a, differing by a ooa- 

 stant interval 1, and if the 1st, 2d, 3d, &c. differences be d' t d", d"' t &c. i 

 then any intermediate value (#), distant from a by the interval x, is equal 



. .. 



A r ofe In taking- the differences, the preceding quantity must always 

 be subtracted from the succeeding} they will .". be positive or negative 

 according as the series of quantities is increasing or decreasing. 



If the law of the quantities be such that their last differences always 

 become o, we shall get at any intermediate time the accurate value of 

 that quantity ; but if the differences do not at last become accurately ~ 

 o t we shall then get only an approximate value. 



In general the quantities d', d", &c. diminish very fast, and it will not 

 often be necessary to proceed farther than d'". 

 Ex, 1. Given the squares of 2, 3, 4, and 5, to find the square of 2|. 

 4, 9, 16, 25 ...... quantities 



5, 7, 9 ......... 1st order of differences. 



2, 2 ......... *. 2ddo. 



.............. 3d do. 



Here a = 4, d 1 5, d" =. 2, d'" = 0, x the required interval ; .". 



y - 4 + i X 5 1 X 2 = 6, 25. 



Ex. 2. Given the log. of 110 2.04139, of 111 = 2.04532, of 112 ~ 

 2.04922, and of 113 = 2.05308] required the log. of 110.5. 

 2.01139, 2.04532, 2,01922, 2,05308 

 .00393, .00390, .00386 

 . 00003, . 00004. 

 Here a - 2.04139. d' = .00393, d" - .00003, and x = $, /. 



?/ - 2.01139 + .1 X .00393 J- X -00003 = 2.043359. 

 Ex. 3. Given five places of a comet as follows ; on Nov. 5th at 8h. 

 llm. in Cancer 2. 30' = 150' ; on the Gth at 87*. \~*m. in 4. 1' = 247' ; on 

 the 7th at Q/t. Urn. in 60. 20' = 380' ; on the 8th at 87*. 17w. in 9. 10' 

 550' ; on the 9th at 87*. 170*. in 12. 40' = 760'. To find its place on the 

 7th at 14ft. 17z. 



First subtract 5d. 87*. Mm. from Id. 14ft. I7w., and there remains 5d. 6ft, 



^ 2,25 for the interval of time between the first observation and the 



given time at which the place is required ; this .'. is the value of A; t^ 



which we want to find the corresponding value of y ; hence 



150, 247, 380,. 550, 760 



97, 133, 170, 210 



3G,' 37, 40 



1, 3 

 153 



