Mr. Lubbock on Bernoulli's Theory of the Tides. 457 



The principles upon which the foregoing precepts depend 

 may be explained in a few words. As the coefficients in Xj 

 are severally obtained by multiplying the corresponding co- 

 efficients in X, each by the exponent in the term to which it 

 belongs, the derived coefficients are regularly decreasing mul- 

 tiples of the original coefficients, the first in X! being n times 

 the first in X; the second in Xj, n— 1 times the second in X, 

 and so on. Now in order to effect the division of X by X 1? 

 we must render the leading coefficients divisible; and for this 

 purpose we multiply all the terms of X by n, and thus make 

 the leading coefficient in X the same as that in the divisor 

 Xj ; so that the first term of the quotient being simply .r, it 

 is obvious that the first remainder will always consist of the 

 second term of X, twice the third term, three times the fourth 

 term, and so on. To render the leading term of this remain- 

 der divisible by that of the same divisor X 15 we multiply the 

 entire remainder by the first coefficient in that divisor, and 

 we thus get a result which, from what has just been said, is 

 the same as would arise from multiplying the second term of 

 X by the first coefficient in X 2 , the third term by twice that 

 coefficient, the fourth by three times the same coefficient, and 

 soon: the second remainder, that is — X 2 , is evidently ob- 

 tained by subtracting from this result the product of Xj by 

 the second coefficient of X; and as this subtraction necessarily 

 destroys the term in the former remainder, neither that term 

 nor the one which annuls it need to have been written down. 

 From these considerations the rule (A.) immediately flows ; 

 the precept (B.) is an obvious deduction from it; and the 

 note is only such a modification of it as is necessary to avoid 

 the introduction of common factors into the terms of X 2 . 



There are means also of abridging the process for deter- 

 mining the other functions X 3 , X 4 , &c. ; and as the method 

 of Sturm is destined to supersede every other hitherto em- 

 ployed for ascertaining the nature and situation of the roots 

 of an equation, such abridgements of labour are well worth 

 attending to : I may possibly advert to them at a future op- 

 portunity. 



October 12, 1835. J. R. YOUNG. 



I 



LVIII. On Bernoulli's Theory of the Tides. By 

 J. W. Lubbock, Esq. V.P. and Treas. R.S* 

 T has been shown that the semimenstrual inequality (which 

 is by far the most considerable) in the time and height 



* Communicated bv the Author. 

 TJiird Series. Vol. 7. No. 42. Dec. 1835. 3 N 



