170 Mr. Ivory on the Ellipticitij of the Earth 



to f. The mean of the nine stations must be an approxima- 

 tion sufficient for our present purpose. Let -01330 be sub- 

 stituted for A in the equation (2), then/ = '2057, which will 

 serve for a first value of f. Now put 



a m -01330, A = a + s, 



b = -2057 / = b + t, 



s and t being the corrections of the approximate quantities : 

 then since A and f must satisfy the equation (2), and a and 

 b likewise satisfy the same equation, we get, 



s + -4686 t = 0. 



The formula (2) is one of the equations of the method of 



the least squares applied to all the 34- experiments ; and in 



order to find the other equation of the same method, let all 



the terms of equation (1) be multiplied by the coefficient of 



f: then, 



A sin 2 A +/sin 4 A = 8 sin 2 A + <? sin 2 A; 



by substituting the values of A and^J 



5 sin 2 A -f- t sin 4 A = sin 2 A (8 — a— b sin 2 a) + e sin 2 A : 



and the sum of the like equations for all the 34 stations will 

 be the equation sought, viz. 



5 2 (sin 2 A) + t 2 (sin 4 A) = 2 . (sin 2 A (h-a — b sin 2 A)), 



the term containing e being neglected. Now, 



2 (sin 2 A) = 15-9316, 



2 (sin* A) = 10-4381; 

 wherefore, 



15-93165 + 10-4381 t = 2 . (sin 2 A (£-«-£ sin 2 A)) : 



and, by a former equation, 



15-9316 5+ 7-4654 r = 0; 

 consequently, 



2-973 t = 2 . (sin 2 A (%-a-b sin 2 A)). 



The sum on the right-hand side consists of thirty-four terms, 

 which must be separately calculated by substituting for % and 

 sin 2 A their values at the several stations. The computation 

 being made, and all the results combined according to their 

 respective signs, I have found 



2-973 t= - -00061. 

 Hence, t = — -00020 ; s = — -4686 t = + -00009 ; 

 / = -2055, A = -01339. 



If 



