ISO 
tHEORY OF TIDES. 
mathematics, being always positive, and this qase requiring an 
alternation of positive and negative values, the common forms 
employed by Euler, Arbogast, and others, completely fail ; and 
the difficulty seems to be rather in the nature of the problem, 
than in the mode of investigation, for the formula which an- 
swers the conditions most completely for one part of the 
circle, seems to be incorrect at others. Thus we may put 
S*x= aSx-{-bS3x4-cS5x . . . omitting the even multiples, since 
they would afford different values for the corresponding parts 
of the first two quadrants, and take the successive orders of 
fluxions, whence we have 
S*x—aSx-{-bS3x + cS5T+. . . 
2 Sxgx=z Ggr+3£g3x+5cc5,r+ . . . 
2—4 SPx == — aqSx— gbS3x+25cS5x . . . 
— 8 Sxc y x= — a ;x — 2Jjb\'3x — 1 25c%5x . . . 
1 6S'x — 8=a&r + 8l/ , S3a;+625ciS , 5.r . .. 
If in these five equations we make x alternately = 9O 0 , and 
=0, we may find five coefficients, a = 7861 b = — '2 598, 
c = — O3709, d = '01612, and e = '00/32, which represent 
the ordinates of a curve agreeing with the curve proposed at 
the vertex and at the origin in situation, in curvature, and in the 
first and second fluxions of the curvature : and yet the curves 
differ surprisingly from each other in the intermediate parts; 
the ordinate at 45° becoming less than 4 instead of '5. 
Approxima- On the whole, the best mode of determining the coefficients 
tion * viz. ('4) ('5) appears to be, to divide the quadrant into as many 
parts as we wish to have coefficients, and to substitute the corres- 
ponding values of the arc inthegeneval equation; we thus obtain 
a = '8484, b = —'1696, c — — *0244, d = — '0681, e = 
— OO29, and f == — '0013. Then if we make, as before, the 
square of the time in the entire forced vibration of the point of 
suspension to the square of the time of the spontaneous vibra- 
tion of the pendulum as n to 1, the distance of the pendulous 
body will be expressed by-JL- when that of the pdint of sus- 
pension is unity ; and accordingly as n — 1 is positive or negative, 
the body will be on the same side of the vertical line with the 
point of suspension, or on the opposite side : and the same 
will be true with respect to the displacement corresponding to 
the first term of the series expressing the resistance, substituting 
th< 
