OF ARTS AND SCIENCES. 397 



The solution of these equations will be of the form (using Q for the 

 Neperiau base and Q for the ratio of circumference to diameter) : 



s = ^,6V + ^,6V + -536V + A0V, i ^ '' 



where z^, z^, z^, z^, are the roots of the equation 



(I _ h)z' + ah' + {yl + g)z' + agz + j'^r = 0, 

 where, for each subscript letter, 



and where four arbitrary constants are determined by the initial con- 

 ditions. 



The roots of the biquadratic equation are all imaginary, and may be 

 written 



«i = — ?i + 'Ji V^^^ 23 = — ?2 + % V^"=-l 



»2 = — Si — 7/i ^^T z^ = — ^,^ — ri, sj^rj 



Expressing the coefficients in terms of the real and imaginary parts of 

 the I'oots, the equation becomes 



z^ -1- 2(|i + ^,)z^ + (4.^,5, 4- I' + 'h' + V.' + V-/>' 



If the terms in 2^ and z were neglected, that is, if a were neglected, 

 the solution of the false equation so obtained would be as follows 

 (where observe the varying sign of r/j) : — 



False z^ = -l (4U; + I' + ^f +ri,^ + %^) ± I (4^2 + 1' + 1.^ 



'/,^ + '/.Vl + 4 



(4sSs^2 + .V + f2--'/i^ + '?2^)-^ 



Now, in the actual case, 7/2 will be at least 100 times rj^, Sj will be 

 quite large, and Sj very small. We may therefore neglect the square 

 of the fraction under the radical ; and we have very closely 



False z,^ = false ./ = - ,,^ + M^^'llzMll'l^vl)^:!^ 

 False Z32 = false ^/ = — yj/ — ^■^ _ J^2 _ ^tt^ _ 



4fi^2 + fl^ + ^2^-V + 'A/ 



