of KEPLER'S PROBLEM. 241 



Aflume X = J/ + A X /3 -f B xVs 



A and B being indeterminate quantities, not depending on 7. : 

 Then, neglefling the quantities which the degree of exa(fi:nefs 

 prefcribed permits us to negleft, we fliall find, 



xi =/' + 3 A>X^H--3 (B + A') x' J^ 

 a;- =j5 + 5.AXJ7, : 

 xT ■ziy''.'- ■ ' 



If now we fubftitute thefe values in the equation between x and 

 y, and omit the terms common to both fides, there will refult, 

 o = (A -f Ky^) X x/3 -I- (B + B/ + A^/) X x' ;/-% 



"^ ^ (t "^ ? ^ ' ) X ^ J'^ — 2 (A -f A r ) X x' y\ 



+ 3{j + ly')xy^^}'' 



Hence 



A= -X— ^ + ?X 



3 1 +J' 5 I +/' 

 B = li X -4_ + -I3^^1^^ - A^ X -^ 



But, fince V = tan - : therefore — -; — - — cof * - — i — fin^-. 

 ' ■' 2 ' I -\-y'- 2 2 



and — ^ — , =: fin = - : confequently 



I +J' 2 



A = ^— ^fin^f, 



_ II 254-. 3Z , 16/", z 16 ^ .% ^■-' 



B = -^ fin' ~-\ fin ♦ - — — fin ^ -. 



«5 315 .2 45 2 225 2 



18. Suppose v =z z -\- w, w expreffing the difference of the 



two angles v and z, which, it is obvious^ depends on X, and is 



to be reckoned of the fame order with that .quantity : Then. 



X = tan - — tan — - — : but J = tan - ; therefore, accordmg 



to Taylor's theorem, rejeding the quantities that ought to be 



reje^ed, 



