'212 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



the interstice ^ will be taken away, and all the other interstices, formed by m 

 circles on hp, with m circles on ag, will remain. If again the circle o be taken 

 away, besides the interstice ^ already taken away, the two c, f will disappear , 

 and every circle more that is taken away, of those remaining on hp, from the 

 extremity of the line, 1 more interstices will disappear. If from the row of circles 

 HP, the extreme circle h be taken away, the 1 interstices a, «, will disappear. 

 And if the circles k, l, m, be taken away successively, every new circle that is 

 taken away, 1 more interstices will disappear, of those formed by the 2 rows ag, 

 HP. Again, if the 2 circles p and h be taken away, the 3 interstices ^, a, a, 

 will disappear ; and every circle more that is taken away, from either extremity, 

 2 more interstices will disappear. Hence, whatever number of circles be taken 

 away, out of m circles on hp, provided they be taken successively, from either 

 or both ends of the row (and when the number of circles on hp is supposed less 

 than that on ag, the deficiency must be at the end, not in the middle of the 

 row, otherwise the circles remaining would not be in the closest arrangement), 

 it is evident that the number of interstices which disappear, of those which 

 would be formed by m circles on hp, with m circles on ag, must be either 

 double the number of circles taken away, or less than the double of that num- 

 ber by I. That is, if m — a be the number of circles left on hp, the number 

 of interstices formed by them, with m circles on ag, is less than the number 

 which would be formed by m circles on hp, with m circles on ag, either by 2a, 

 or by 2a — 1. The number of interstices formed by m circles on each row 

 would be, as has been shown in the preceding case, 1m — 2. Therefore the 

 number formed by m circles on ag, with m — a circles on hp, is either Im — 2 

 — 2fl, or 1m — 2a — 1. That is, ultimately (when the number m — a is infi- 

 nitely increased) 1m — 2a. Now suppose the number of circles on ax to be 

 m — a — b. The number of circles on the 2 rows ag, hp, is 1m — a. On 

 the 3 rows ag, hp, qx, the number is 3m, — la — b. And if the number of 

 circles on rs he. m — a — b — c, the number of circles on the 4 rows ag, hp, 

 ax, rs, will be Am — 3a — lb — c. And, universally, the number of rows 

 being n, and the number of circles on the several rows, m,m — a,m — a — b, 

 m — a — b — c, m — a — b —c — d, &c. successively, the whole number on 

 all the n rows will he nm — a X {n — \) — b X {n — 1) — c X {n — 3), &c. 

 But as it has been shown that m circles on ag, with m — a circles on hp, form 

 1m — la interstices, if the number m — a be infinite ; in the same manner it 

 may be shown, that m — a circles on hp, with m — a — b circles on qx, form 

 1m — la — lb interstices, when the number m — a — Z) is infinite. Therefore 

 the whole number of interstices formed by the 3 rows on ag, hp, ax, is 

 (2ot — 2a) X 2 — lb. And, in like manner, the number of interstices, formed 

 by the circles of 4 rows, will be {1m — la) x 3 — lb X {1 — 2c). And, uni- 



