Of KEPLER'S PROBLEM. 241 



Aflume x ~ y -f- A X y* -f- B X Vs 



A and B being indeterminate quantities, not depending on X : 

 Then, neglecting the quantities which the degree of exactnefs 

 prefcribed permits us to neglect, we {hall find > 



^ 5 =r + 5 a x^- 



xi zzy\ 



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- A/) x\p + (B + B/ + A 4 /*) X X*j^, 



' — 2 (^ ■+ -> 2 ) X Xj>3 — 2 (A -f- A/) X X 2 j>s, 



+ 3(j+^ 2 ) Xx ^ 5 ' 



Hence 



A = ? x < + ? x j- 





But, flnce v = tan - : therefore — ; — - ± cof 2 - = 1 — fin 2 - 

 ' y 2 ' 1 -f-j 2 2 2 



v z 



and — 4 — , = fin 2 - : confequently 



1 + y 2 2 



3 J 5 2 



_ 11 254 r a z . 16 /*-- . z 16 ~ , z 



B = -- — -^T fin 2 - H fin*- — . — fin 6 -. 



15 3 r 5 2 45 2 22 5 2 



18. Suppose v = z -f- w ; w exprefiing the difference of the 

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

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



v z4-w z 



x = tan - = tan — — : but y = tan - ; therefore, according: 



2 2 2 J 



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



rejected, 



