Mr, J. Cockle on Transcendental and Algebraic Solution. 135 



The formula expressing the results of experiment was found 

 to be 



v=-075h; 



so that for h = l, 2, 3, 4, 5, 6, 7, the values of v are *075, *15, 

 •225, '3, *375, *45, -525 respectively; but by experiment these 

 velocities were found to be respectively '078, *15, '22, *3, '36, 

 •43, -52. 



[To be continued.] 



XVIII. On Transcendental and Algebraic Solution. — Supplement- 

 ary Paper. By James Cockle, M.A., F.R.A.S., F.C.P.S. %c* 



IT is not, for the purposes of my paper in the last May Num- 

 ber, necessary to deal with more than one root oifx = 0. 

 Assume 



dx 

 da 

 form the equation 



fx=(p + qx + rx* + . . . + tx n ~ l )¥x, 

 and put it under the formf 



P + Cb + IU 2 + . . . +Ta^" 1 =0 

 by eliminating x 11 , x n+1 , x n+2 , &c. Then the n linear equations 



P = 0, Q=0, R = 0, ...T = 

 will determine the n quantities p, q, r, . . . t. Thus, for the cubic 



st*—3x + 2a=0, (1) 



— -j- =p + qx + rx* + . . . + tx n ~ 1 ; 



we have 



fo=2, F#=3(l-# 2 ), 



dx 

 -- r =p + qx + rx*, Y = 3(p + 2aq)-2, 



da 



Q=6(ar-q), R=-3(p + 2r), 

 whence J, clearing of fractions, &c, 



* Communicated by the Author. 

 t The deduced form 



¥ + Qx+Rx 2 + . . .+Tx»-i 

 of a rational function of x is the remainder after a division hyfx. Hence 

 it is attainable by division and readily (more particularly where the coeffi- 

 cients are all numerical) by Horner's synthetic division. When x n , x n + i , 

 &c. are eliminated by substitution, the higher powers should be eliminated 

 first, and in order of magnitude. 

 X For 



2 = -a -1 



P 3(1 -o 2 )' q 3(1 -a 2 )' 3(1 -a 2 )' 



