[ 347 ] 



2. The numeral coefficient of any produ£t A^ B^ C — 



1.2 /'+$'+r-X/'-{-iy + 3rX/)+-2^ + 3r— I -- 



1.2 p+2q-\-'^ry.p.p—i>i l-x- qy.q—ixq—2 irxr— I - i 



I. 2 p + a + r . , 



= — i- 1 — ^=: = the number 



p.p — r - - r I + ^ X q — I - - I x^X r — i - - I 

 of permutations of AAA (p thingsj ^ B (q) CC (r). Thefc 

 laws of continuation are the fame as ftated by De Moivre*, 

 and deduced by him from the application of the multinomial 

 theorem. 



Example VII. From the mean anomaly of a planet to deduce Fig. 

 the eccentric anomaly in a feries afcending by the powers of 

 the excentricity. 



Solution. Let APB be the femi-elliptic orbit defcribed 

 about the focus S and centre C, and P the planet ; then drawing 

 RED perp. to A B meeting the circle defcribed on the diameter A B, 

 the "^ A C R will be the eccentric anomaly. Let the mean ano- 

 maly — m (rad. == i) the eccentric anomaly = c, ACi = i, and 

 CS the eccentricity = e- Then m: circumference :: area ASP: 

 area of the ellipfe : : area A S R : area of the circle •.• becaufe 

 CR= I, w= 2 area ASR = 2ACR + 2CSR=:BRXCR + 

 CS^DR = f"Pfj-, corffz^c+fj, f. 



X X a Let 



* Philofophical Tranfadlions, Vol. XX. p. ipo. 



