210 Mr. Herapath on the Failure of Lagrange's 



errors, which must be applied to the observed pendulums to 

 make them equal to the calculated lengths : 



About 50° of latitude all the formulas very nearly agree with 

 one another, and with observation. In receding more from 

 the equator, the pendulums are shorter in the formula we are 

 considering, than in the others; but the effect of this is not very 

 perceptible except at the highest latitudes. At Port Bowen 

 and Spitzbergen, the errors are — -00239 and — -00302. 



It appears then that, by lengthening the equatorial pendu- 

 lum, we approach a little nearer to the anomalous experi- 

 ments, and recede from all the rest within 50° of the equator. 

 The foregoing investigation proves that all the 40 observed 

 pendulums can never be represented, without great discre- 

 pancies, by any one ellipticity. In my last formula all the 

 pendulums, except the five anomalous ones, are placed on 

 one and the same surface within the limits of the errors of 

 observation. But no one pendulum within 50° of the equator 

 answers to its proper place in the surface of which the ellip- 

 ticity is =gj This surface is merely an imaginary mean, the 



offspring of calculation. It can have nothing to do with what 

 is called local attraction : for it makes an excess of gravity at 

 five particular stations, and, as far as experience enables us 

 to pronounce, a defect of gravity every where else within the 

 limits mentioned. 

 Feb. 14. J. Ivory. 



XXXIV. Failure of Lagrange's Method of integrating Linear 

 with Constant Coefficients*. By John Herapath, Esq.f 



TF we suppose the roots r, r tf r 2 of the equation 

 r 3 + Ar 2 + Br + C = 



* It should have been mentioned in my last paper,that the constants in 

 (18) are not precisely the same as in (17), except the first two. In the other 

 terms several factors in the denominators were suppressed and understood 

 in the constants, for brevity and ease of printing. 



f Communicated by the Author. 



to 



