με where μ= cos 0,-, and represents any positive integer whatever, provided is never greater than (+1) Though we have thus the solution of every equation in the system (41), yet that of the first may be obtained under a simpler form by writing therein for λ, its value (2) deduced from (45). A, We shall then immediately perceive that it is satisfied by In consequence of the formula (45), the equation (42) becomes d'P 8-1-np2 dP (2 (8) (2° (8) +8—2) 0= + dp3 p (1-p2) · dp p* k + P - which is satisfied by making k=− λ, − (¿(®) + 2w) (¿(®)+2w+n−1), and + 2w.2w - 2 × 2¿°(s) + 2w+s− 2. 2¿(®) +2w+s-4 2.4 × 2i(+4w + n + 3. 2i + 4w+ n − 5 where a represents any whole positive number. it now only Having thus determined all the factors of p, remains to deduce the corresponding value of H. But H, the particular value satisfying the differential equation in H, will be had from by simply making therein = since in the present case we have generally a, a'. Hence, it is clear that the proper values of 0,, 02, 03, &c. to be here employed are all constant, and consequently the factor entering into is likewise constant. Neglecting therefore this factor as superfluous, we get for the particular value of H, and P. represents what P becomes when p is changed into a. Substituting this value of H, in the equation (25), No. 10, there results since a2=a2+ h2 Thus the complete value of V for the particular case considered in the present number is and the equation (27), No. 11, will give for the corresponding where P', ',,', &c. are the values which the functions P, ,,,, &c. take when we change the unaccented variables ,,,,..., into the corresponding accented ones ',,..... &', and P = ...... n ·8+20-1 n-s+1.n-s+ 3 or the value of P when p=1; where as well as in what follows i is written in the place of ¿"). The differential equation which serves to determine H when we introduce a instead of h as independent variable, may in the present case be written under the form ď H dH 0 = a3 (a" — a'3) + {i (i + 8 − 2) a* — (i + 2w) (i + 2w + n − 1) ao} H, s and the particular integral here required is that which vanishes when his infinite. Moreover it is easy to prove, by expanding in series, that this particular integral is provided we make the variable r to which A" refers vanish after all the operations have been effected. But the constant k may be determined by comparing the coefficient of the highest power of a in the expansion of the last formula with the like coefficient in that of the expression (46), and thus we have k' = Ka'↔2~ ( − 1 ) ∞ n + 2i+2w −1 . n + 2i+2w+1 ... n + 2i+4w−3 2.4.6... 2w Hence we readily get for the equivalent of (47), n+2i+2w-1.n+2i+2w+1...n+2i+4w-3 2.4.6...20 ... × Ka'1** (—-1)*a*A*a* [„daa1*** (a3 — a”3) In certain cases the value of Vjust obtained will be found more convenient than the foregoing one (47). Suppose for instance we represent the value of V when h=0, or a=a' by V。. Then we shall hence get n+2i+2w-1.n+2i+2w+1...n+2i+4w-3 2.4.6...20 since in the formula (8), r ought to be made equal to zero at the end of the process. By conceiving the auxiliary variable u to vanish, it will become clear from what has been advanced in the preceding number, that the values of the function V within circular planes and spheres are only particular cases of the more general one (49), which answer to s=2 and s=3 respectively. We have thus by combining the expressions (48) and (49), the means of determining V, when the density p' is given, and vice versa; and the present method of resolving these problems seems more simple if possible than that contained in the articles (4) and (5) of my former paper. |