On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College.

George Green

Date:: [1833]

Credit: On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College. Source: Wellcome Collection.

24/36 page 418

$+ 2(^ V3V'0 - V'V30 = (1 - V V4^ V4V 0 — VsV 0 — V V50 — all the finite integrals being taken from r = 1 to ;• = a- + 1, and from r' = 1 to r' = 5 + 1. In order to obtain tbe required value v'v0 - vv'0, it is clear that we shall only have to add the first of the five pi’eceding quantities to the sum of the four following ones multiplied by //*, and to render this more easy, we have appended to each of the terms in the ])receding quantities a number inclosed in a small parentbesis. Now since the accents may be permuted at will, and we have like- wise cr = + h\ it is easy to see that the terms marked (1), (6) and (12) mutually destroy each other. In like manner, (2), (3), (7) and (18) mutually destroy each other; the same may evidently be said of (13) and (16), of (15) and (17), of (9) and (19), and of (8) and (14). Moreover the four quantities (4), (5), (10) and (11) will do so likewi.se, and consequently, we have v'v0 - V v'0 = 0. Hence the truth of the equation (31) is manifest.$