140 Astronomical and Nautical Collections. 

 *1 = Z 



^ = * + Yv 

 dr 2 



-!l = z + * + Xw« 



dr 3 



i!i =s * + Y'v + * + Vv* 



¥l = Z" + * + X'v* + * + Uv l 

 dr 5 



£l = * + F% + * + fv + . . . 



dr 6 



SB Z'" + * + XV + 



dr 



rl1.i 



52. = * + r"t; + . . . 



dr 8 



dr 9 

 It will be convenient to denote the successive results of the 

 differentiation of any quantity, Z, Y, X, with respect to y or z, 

 which introduces a new power of v, by Z,u, Z.,v', Y t v, Yjf, and 

 so forth ; we shall then have 

 Z-Z Z'-YZ 



Yr=Z, Y'=z2XZ+Z\=:{2Y,Z) + (Y 1 Z+Y*)=z3Y l Z+ Y* 

 X=F, X'z=3VZ+ F' l =(3F 2 Z) + (3F 2 Z+3Y 1 F+2F l F) 

 VssXtSsT, — 6F.Z+5F.F 



tf=F 3 r=4f/Z+r 5 =(4r 3 Z)+(6F 3 Z+6i;F)+(5F 2 P 



+ 5 Y*) - 1 F,Z+ 1 1 Y,Y+ 5 F," 

 Z"=F'Z=3F'Z 2 + F a Z 



F"=2X'Z+Z" l =:12F,Z i + 10F.FZ 



+ (3F 2 Z*+6F,FZ) 



+ 2 F, YZ + F 3 ) = 1 5 F 2 Z*+ 1 8 F,F« + F» 



X"=3PZ+ F",= (30 KZ 4 * 33 y a FZ+ 15F, 3 Z) 



+ (15 Y 3 Z*+ 30 F 2 FZ)+ (18 F,«Z 



+ 18Y 2 YZ + 18 y,y») 



+ (3y,y») 

 = 45 y 3 z«+ 8i y 2 yz+ 33 y, 3 z +2 iy,y« 



