



/. H. Alexander on a New Formula for Interpolations. 27 



ing series and the curve corresponding to it, pass beyond the sec- 

 ond term and meet the ascending ones through the third, in 

 some stationary and tangential point whose abscissa is that of the 

 solstice. The tangent at this point is, of course (i. e. its abscissa 

 is) less than that of the third term : how much below, the nega- 

 tive differences of declination afford probable data for determining. 



Having four abscissae and three intervals, we have also for the 

 variation in length of these abscissas during the first interval, 13 . 

 and during the third, 8 ; as the actual motion is elliptical, the 

 variation during the second, will be the square root of the differ- 

 ence of the other two (i. e. • v /13-8 = v'5;)^2-23 ; stopping in 

 an equation of the second degree, at the second decimal place. 

 This is the whole change in declination between the second and 

 third abscissa; and the difference of this (2-23), and tfie apparent 

 difference of the two abscissas (2), is the distance in declination 

 below the third : corresponding to an actual abscissa of the sol- 

 stitial point of (31751-0-23) « 3175077 parts. With this de- 

 termination, the case is removed from its anomaly ; and there is 

 no obstacle to the employment of the formula, involving all four 

 terms, to find the corresponding ordinate. With the ordinary 

 methods, there would still be required the resolution of ft cubic 

 equation ; under the present formula, all terms containing a power 

 of n higher than the second, are eliminated. 



It is manifest that as we have now the absolute numerical value 

 of the abscissa at the point required, the sum of the terms con- 

 taining n (the ordinate of the same point), which for differentia- 

 ting was made equal to zero, becomes now equal to the said ab- 

 scissa. So that, recurring to the last solution, we have thence 

 these two equations : 



j „3 _ 6 . n i +31 £ n _ 31791 = 31750 77 ; 

 |w 5 + 4 w 2 -21^4-31776=31750-77. 



Arranging all the terms involving n, on one side, we have 



i« 3 -6iw 2 +31in = 40-23; 



£»*4>4 « 2 -21in- -25-23. 

 Subtracting the latter from the former, and then changing the 

 signs, we have left 



101 w 3 +52iw=- 65-46 

 and dividing n 2 4- on = -6-23; an equation with no 



power of ■ higher than the second, and which, although its sec- 

 ond member has an apparently negative sign, leads yet to a posi- 

 tive result, viz. w = 2-64; corresponding when reduced to time 

 to June 201, \5\ 21 n '-6, a result sufficiently approximate to be 

 employed. 



Such are the examples and illustrations which seemed to me 

 °f sufficient interest to be connected with the formula and with 

 th-s account of it. 



