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[ 349 ] 



Example VIII. From the mean anomaly of a planet to deduce 

 the true anomaly in a feries afcending by the powers of the eccen- 

 tricity. 



Solution. Let the femi-axis major AC= i. The eccentri- 

 city C S = ^, the anomaly KST = a, m = the correfponding mean pig, 

 anomaly meafured in the circle the rad. of which = i, and the 

 periphery P. Then as the areas are proportional to the times, 

 and therefore to the mean anomalies : 



Flux, area AST: area of the ellipfe : : »? : P •.' ^ S T ^ flux. ^ 



A c^ /■ \ a ACT- '^^ ^^^^ of the ellipfe 

 AST (a) = flux, area A S T = 5 J— = m 



V 



xi i/i — fz or a x^ SH'^ ~ m s/i — ei. But by the prop, of the 



elhpfe S T = Hence — ,, = m or i— f * x a 



I — ecs,a I — ecs,a\ 



X I + 2 e cs, a + :^ e"- cs";, a + ^e^ es\ a+ 8cc. = m. Let A = fl 



acsa, B= iiacs^, a, C = & acs'^.a, &c. and L= i — e\'' . Then 

 <7 + 2^A+3^^B + 4i'3C+ &c. = hm. Ytom this equation the 

 feries is to be deduced by fucceffively taking its fluxions by Prob. 2. 

 making e flow uniformly &c. e — o 



1. <7+ 2eA = L m = o 



2. « + 2. 2 (? A + 3. 2 tf» B = 3 £■* ffl 



3. a 



