By a similar process to that applied above, we shall find that r is determined by an equation of the nth degree, and that m= (n − 1) (n − 1), and that each of the letters q1› I2In-1 is a rational function of r. Thus, there will be n solutions of the form (e + b2) 3 (e + c2) 3 {en−1 + (n − 1) % ̧e”~2 + There will also be n solutions of a similar form, in which the factors (e+c) (e + a2)1, (e + a2)1 (e + b) are respectively involved. Hence, the total number of solutions of the nth degree will be 4n+ 1. 1 , 2 8. We may now investigate the number of solutions of the degree n + n being any positive integer. These will be of the following forms: three obtained by multiplying a rational integral function of e of the degree n by (e + a2): (e+b2)3, (e+c2)3, respectively, and one by multiplying a rational integral function of e of the degree n−1 by the product {(e + a2) (e + b2) (e + c2)}3. An exactly similar process to that applied above will shew us that there will be n+1 solutions of each of the first three kinds, and n of the fourth. Hence the total number of such solutions will be 3 (n + 1) +n, or 4n+3, that is +1. To sum up these results, we may say that the total number of solutions of the nth degree is 4n+1, n denoting either a positive integer, or a fraction with an odd numerator, and denominator 2. Similar forms being obtained for H, H', we may proceed. to transform the expression EHH' into a function of x, y, z. Consider first the case in which 9. H' = (v' — w ̧) (v' — w2) ..... (v′ — w„). EHH' = (e — w1) (v — w ̧) (v' — w ̧)..... (e — w„) (v — w„) (v' — wn). Now we have shewn (see Art. 4 of the present Chapter) that (e-w1) (v - w1) (v' — w ̧) we see that EHH' is equal to the continued product of all expressions of the form the several values of a being the roots of the equation As this equation has been already shewn to have (n + 1) distinct forms, we obtain (n+1) distinct solutions of the equation V2V=0, each solution being the product of n expressions of the form That is, there will be n+1 independent solutions of the degree 2n in x, y, z, each involving only even powers of the variables. 10. To complete the investigation of the number of solutions of the degree 2n, let us next consider the case in which E The object here will be to transform the product (e + b2) 3 (v + b2)3 (v ́ + b2) 1 (e + c2) 3 (v + c2)3 (v' + c2)3, since the other factors will, as already shewn, give rise to the product of n-1 expressions of the form y2 22 a2 + w b2+w c2 + w 1. Now, by comparison of the value of x2 given in Art. 4, we see that (e + b2) (v + b2) (v' + b2) (e + c2) (v + c2) (v′ + c2) = · (b2 — c2) (b2 — a2) (c2 — a3) (c2 — b2) y2z2. Hence, we obtain a system of solutions of the form of the product of (n-1) expressions of the form. multiplied by yz. Of these there will be n, and an equal number of solutions in which zx, xy, respectively, take the place of yz. Thus, there will be 4n+1 solutions of the degree 2n in the variables of which n+1 are each the product of n expressions of the form n are each the product of (n-1) such expressions, multiplied 11. We may next proceed to consider the solutions of the degree 2n+1 in the variables x, y, z. Consider first the case in which E = (e + a")} {e" + np,e" + " (". = 1) ++; n. (n - 1) 1.2 Here the product (e + a2)1 (v + a2)3 (v + a2) will, as just shewn, give rise to a factor x in the product EHH'. Hence we obtain a system of solutions each of which is the product of n expressions of the form multiplied by x. Of these there will be n+1, and an equal number of solutions in which y, z, respectively take the place of the factor x. Lastly, in the case in which n-1 n-2 E = (e + a2) 3 (e + b2)3 (e + c2) 1 {e"1 + (n − 1) P ̧e"~2 (v+c2) $ (v ́+c2) $ (e+a2)3 (v+a2)1 (v ́+a2)* (e+b2) $ (v+b2)3 (v′ +b2)3 (e+c2)} will give rise to a factor xyz. Hence we obtain a system of solutions each of which is the product of (n − 1) expressions of the form multiplied by xyz. Of these there will be n. Thus there will be 4n+3 solutions of the degree 2n + 1 in the variables, of which (n+1) are each the product of n expressions of the form (n + 1) are each the product of n such expressions, multiplied (n + 1) ... ... ... n are each the product of (n-1) such expressions, multiplied by xyz. 12. Now an expression of the form C. EHH', C being any arbitrary constant, is an admissible value of the potential at any point within the shell x2, y2 z2 = = 1. But it is c2 not admissible for the space without the shell, since it becomes infinite at an infinite distance. The factor which becomes infinite is clearly E, and we have therefore to enquire whether any form, free from this objection, can be found for this factor. We shall find that forms exist, bearing the same relation to E that zonal harmonics of the second kind bear to those of the first. which we suppose to be satisfied by putting U=E, we see that, since it is of the second order, it must admit of another particular integral. To find this, substitute for U, E fude, 1 de + 5 { (e + b2) (e + c2) + (e + c2) (e + a3) + (e+a2) (e + b2)} Ev dv + (e + a3) (e + 8°) (e + o®) (dll. v + Edu). de de |