372 Mr. J. C. Moore on Glacial Submergence. 



And the first two terms of the middle expression squared, coeffi- 

 cients only being retained, are 



a^A'+i^p-lJAA' (2) 



m -I - 1 t) 

 Multiply (1) by £— , and we get (2); that is, mul- 

 tiply the former expression by (m+l)jo, and the latter by m(p—l), 

 and the results are the same. And these multipliers are New- 

 ton's factors, as is easily seen by making m equal to 1, 2, 3, &c, 

 in succession, and putting n, n—1, n—2, &c, in succession, 

 for jo. 



An interesting deduction from this general property is, that 

 if Newton's condition hold or fail for the first three terms of any 

 equation, it will in like manner hold or fail for the first three 

 terms of every transformed equation which can result from in- 

 creasing or diminishing the roots by any quantity whatever. 

 For, the proposed equation being 



A^ + A n _ 1 ^- 1 + A n _ 2 ^- 2 + . . . 4-A =0, 



we know that the first three coefficients of the transformed 

 equation will be the last three of the derived functions (I) when 

 r (the transforming factor) is substituted for x, and therefore that 

 they will be of the forms 



A n , ar + A n _!, Z>r 2 + cr + A n _ 2 . 



But, by the above principle, if the condition hold or fail for the 

 three original coefficients, A n , A w _ 1 , A w _ 2 , it must equally hold 

 or fail for these, inasmuch as 2n times the product of the first 

 and third differs from n—1 times the square of the second 

 by exactly the same amount that 2n A n A n _ 2 differs from 

 (n — 1)AJ__ 1 . Hence for every transformation these functions of 

 the coefficients always have the same constant difference. 



LIV. On Glacial Submergence. 

 By John Carrtck Moore, F.G.S* 



THE mathematicians have responded to the geologists, and 

 we are now in a condition to test the value of the theory 

 which attributes the marine shells and drift, at heights 2300 feet 

 above the present sea-level, to a submergence due to the attraction 

 of an ice-cap. I take the formula of Archdeacon Pratt, because 

 his figures are most favourable to the theory, and because his 

 hypothesis of ice thinning out from the pole seems the more 

 reasonable. 



* Communicated by the Author. 



