9 *tOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 411 



blem which has called forth the skill of several of the most learned philosophers and 

 astronomers of the last and present age. 



A preparatory step to the solution of this problem is, to find a convenient ex- 

 pression for the reciprocal of the cube, or rather of the n th power, of the distance 

 of any 2 planets. Such an expression was first given by Euler, in series proceeding 

 by the cosines of the multiples, in arithmetic progression, of the angle of commu- 

 tation ; but the calculations of the first 2 co-efficients in it were very laborious, re- 

 quiring the summation of series of the common form, which converged very slowly. 

 Afterwards, other series were discovered by other authors, by which the same 

 co-efficients might be computed with less labour; the best of which, that I have 

 seen, appear to be those that were pointed out to me by Dr. Maskelyne, invented 

 by la Grange, and published in the memoirs of the Royal Academy of Sciences at 

 Berlin, for the year 1781 . Yet the calculation of the first 2 co-efficients, a and b, 

 for the perturbations of Mars, Venus, and the earth, by his method, is not 

 shorter, if it be so short as by my method, to the investigation of which I now 

 proceed. 



Prob. — 2. To determine the values of a, b, c, d, &c. in the equation 



-^ - = z (a + b . cos. z 4- c . cos. 2z + d. cos. 3z, &c.) z being the 



(c — b . cos. z)" v ' ■ ' * ' ° 



arch of a circle of which the radius is 1, and b less than a. 



First, to find the co-efficient a. — 3. The fluent of the right-hand side of this 

 equation is az + b . sin. z + 4c . sin. 2z -f 4d . sin. 3z -f ^e . sin. 4z*, &c. which 

 evidently vanishes when z = O; and when z = 3*14159, & c « tne arcn of 180°, 

 it becomes barely = az, the sines of z, 2z, 3z, &c. being then each = 0. There- 

 fore if the fluent of the first side of the equation be taken, the increase of it, while 

 z increases from O to 3*14159 &c. = w 9 will be = tta; and consequently a will be 

 determined. 



4. Now, to find the fluent of ~ - = — - — =^— — , x being put 



(a— o.cos. z) //(l — xx) (a — bx) n ° r 



= the cosine of z ; in which expressions, while % increases from O to 3*14159, x 

 will decrease from 1 to — 1 . Therefore, to obtain a more convenient expression, 

 put vv = — T7T"J then, while x decreases from 1 to — 1, vv will increase from 



a -^— h to —r-i = 1 J ana< we shall have the following equation : 



-7 A 7-r. = (a + b)~ n X //t , „ : ; the fluent of which may 



y(l — xx) (a — bx) n v ' V(l - vv) V(vv — cc) f 



be found when the value of n is given. 



5. Now, the values of n with which astronomers are most concerned, are 4 and 4. 

 Let therefore 4 be written for n, and the radical quantity ^(1 — vv) be converted 

 into series, then the last expression will be 



2i» 



— % 



vv . 3d 4 , 3 . 5v* . 3.5. 7v * 



* See Euler' s Institutiones Calculi Integrals, vol. 1, p. 150 — Orig. 



3 G2 



