Depression in Capillary Titles. 423 



For the coefficients of this equation, 5 terms of the first series 

 were used ; 7 terms of the second ; and 5 of the third ; but as 

 all these last terms were diverging, I reckon the coefficient about 

 half the true value. Means are afforded of forming a nearer 

 estimate ; but, as the cost would outgo the profit, I shall decline 

 the labour of the calculation. We have next an equation to 

 solve. Truly, the Series ought to hav° for its peculiar motto, 

 iMbor omnia vincit improbus. 



The solution of the equation is, y = 0*1997j 

 and the proof . . . . . , , . . , '7083 



247 

 21_ 



•7351 



Now, a =-^ = '00815, the same number found by both 



Ty </ my 



rules. 



The number -1997 is no doubt a little too great, on account 

 of the omissions: it may be -1996, or -1995, or even less; but 

 this is not material. 



Next, let 2a; = -6 ; then qx^ = 17*64 ; and we get this equa- 

 tion, -735 = 5-574 y + 12-07 y^. 



No fewer than 7 terms of the first series and 9 of the second 

 were used to find the two coefficients of this equation, of which 

 the last is only true in the integers. In the next coefficient-series, 

 the last term set down, or the tenth, is a whole number of two 

 figures ; so that we could only have a rough estimate of the co- 

 efficient of ?/5. I shall decline the calculation, by which the un- 

 certainty of the approximation would not be removed, although 

 it might be a little diminished. The solution is, 



y = -12/4; and the proof .. .. '7101 



249 



•735 

 The number 0-1274 is too great ; but I make a large allowance 

 when I say that the true value of y is greater than 0-1264: and 



if both the numbers be substituted in the formula, a — ^' ., we 



get, for the depression -00433 and -00429, between which the 

 truth certainly lies. Now my number -0043 1 is between these 

 limits; and the series itself, with fair computation, has vindicated 

 my rules in this instance also, against the charges of its inventor. 

 1 observe that my antagonist neglects the just grounds of calcu- 

 lation, and amuses himself with making conjectural additions to 

 another series different from the first. He ought to recollect, 

 that the accuracy of the conclusion will depend entirely upon the 

 exactness of the first series. 



My 



