432 Mr. Ivory on the Figure of the Earth, as deduced 



terms to one side after having divided by R ; by which pro- 

 cedure we shall obtain these five equations, 



af l) = -009642 — 3-0662.6 + 0-341 . e 2 

 af v = -026612— 8-1680. 6-0992. e 2 

 .r (3) = -020364-6-2574. e + 2-719. e 2 

 ,r< 4 >= -007875—2-4944 . e— 0*684 . s 2 

 af 5) = -007992—2-4819 . e-2'678 . e 2 

 The symbols af x \ af 2 \ &c. stand for the errors, or rather, for 

 the quantities which the want of perfect exactness in the ex- 

 perimental numbers, makes it necessary to supply in order that 

 the same value of e may produce an equality in all the five ex- 

 pressions. The best mode of solution is to employ the me- 

 thod of the least squares, or to determine s so as to satisfy this 

 equation, viz. 



di dt 



Thus we have, 



= (-009642-3-0662. e + 0-341 e 2 ) ( — 3-0662 + 0-682 s) 

 + (-026612-8-1680 e +0*992 e 2 ) (-8-1680 + 1-984 e) 

 + &c. 

 All the operations being performed, this final equation will be 

 obtained, viz. 



0-413829= 127*666. 6 + 4-795. e 2 ; 

 from which we get I = -00324. And if this value of g be 

 substituted in the foregoing expression of A, we shall find 

 A si 60655-9 x 1-003295 = 60855-7, or 60856. The direct 

 method of calculation here followed, has therefore brought us 

 to the same elements as before, which, as has already been 

 shown, represent all the five measurements with very small 

 errors*. 



It is easy to verify the value that has been found for A ; 

 namely, by combining the original equations so as to eliminate 

 the terms containing e, or so as to render the same terms so 

 small that they may be neglected. Thus, if the expression of 

 the French measurement be multiplied by 2*9, we shall have, 



2179544= A(35-8739-18-2152. 6-30-351. e 2 ); 



and, if the Indian expression be subtracted from this, the re- 

 mainder will be, 



1580914 = A (25-9780-0-0167 . e-36'514 . 6 2 ), 

 from which we get A =60856 as before. We are therefore 

 sure that there is no uncertainty in the values that have been 

 assigned to the two elementary quantities. 



The elements of the elliptical meridian being known, we 



* Phil. Mng. for May, p. 34C. 



may 



