VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 2)5 



thesis must be, because there is but one equation, in which 4 indeterminate 

 quantities/, g-,jb, q, are involved. And indeed there might be many (nore than 

 4 indeterminate quantities, if we should assume more than two terms in the 

 general equation of the densities d = frf + gri + hr', &c. 



21. In order to apply the foregoing theory to the earth, it might seem at first 

 sight, that by the assistance of observations made for measuring the length of 

 the pendulum, we might have other equations, which with the foregoing 

 equation a, would determine the coefficients and exponents now mentioned ; 

 but we shall soon see the impossibility of this, on two accounts : first, there 

 need be only two observations, as to what concerns the length of the" pendu- 

 lum. For because, by art. 17, the augmentation of the gravity from the 

 equator to the pole, is proportional to the square of the sine of the latitude, 

 two observations as much determine the problem as an infinite number can do: 

 so that we could have but one other equation besides the foregoing. This 



5—pf* , 5-qg» 



■n 1 /■ \ P—P 3+PX5+P 3 + 9x5 + ? 

 equation will be (b) = -^_ ^ — — = • 



P p-'^f' , _/_ 1 _£_ I 1-^S 1_ 



3 + px5+p~^ 3+p "*" 3 + 9 3 + qx5 + q 



The first member of this equation expresses the gravity at the equator sub- 

 tracted from the gravity at the pole, and divided by the gravity at the equator ; 

 a quantity which may be known in numbers, by determining the length of the 

 pendulum at two difl^erent latitudes. The other member of the equation is an 

 expression of the same quantity, as it is deduced by the preceding calculus. 



Secondly, This new equation b cannot be of any service in determining the 

 coefficients and exponents J", g, p, q, &c. For we shall now show, that the 



foregoing ratio ^ — - has such an immediate connection with a, that one of them 

 being determined, the other will necessarily be so too, independently of the 

 values o{ f, g, p, q, &c. This may deserve our attention, and the proof is 

 thus : 



The Figure of the Spheroid being known, the Augmentation of Gravity from the 

 Equator to the Pole will be known also ; and so vice versa, 

 22. Because the ratio of the gravity to the centrifugal force is very great, and 

 is expressed by m, in the equation a we may reject the third and fourth 

 terms ; by which means the equation will be reduced to this, 

 f g imf» imgit 



— — + ^-, — = :r-; — ■ -r-r= + Tt^^'t^t^- ^"° " "■°'" this equation 



3+p 3+9 3+PX5+P 3+9X5+9 ^ 



we deduce the value either off or g, and substitute it in the equation b ; having 

 first rejected the first and fourth terms of the denominator, as in this case may 

 be done ; we shall have, after the calculation is made, whatever is the number 



